{"diffoscope-json-version": 1, "source1": "/srv/reproducible-results/rbuild-debian/r-b-build.5dAs03Eb/b1/sketch_0.3.7-12_armhf.changes", "source2": "/srv/reproducible-results/rbuild-debian/r-b-build.5dAs03Eb/b2/sketch_0.3.7-12_armhf.changes", "unified_diff": null, "details": [{"source1": "Files", "source2": "Files", "unified_diff": "@@ -1,4 +1,4 @@\n \n- c37f60d7d0ed4169b48d4aa5ca7656f6 194948 debug optional sketch-dbgsym_0.3.7-12_armhf.deb\n- 3ca3ee629f9e72686e4f2dadf0f41c46 1175648 doc optional sketch-doc_0.3.7-12_all.deb\n- 3ef7177fc7c34b931e23a369c3a3b841 51552 graphics optional sketch_0.3.7-12_armhf.deb\n+ df1859d0b2c55b7a3d5658be092fd527 194948 debug optional sketch-dbgsym_0.3.7-12_armhf.deb\n+ c7213f8861b47b2eb550b2dc7c58c863 1175540 doc optional sketch-doc_0.3.7-12_all.deb\n+ 71810ea0618806755e6857be10cd98d7 51528 graphics optional sketch_0.3.7-12_armhf.deb\n"}, {"source1": "sketch-doc_0.3.7-12_all.deb", "source2": "sketch-doc_0.3.7-12_all.deb", "unified_diff": null, "details": [{"source1": "file list", "source2": "file list", "unified_diff": "@@ -1,3 +1,3 @@\n -rw-r--r-- 0 0 0 4 2020-11-07 02:36:18.000000 debian-binary\n -rw-r--r-- 0 0 0 4300 2020-11-07 02:36:18.000000 control.tar.xz\n--rw-r--r-- 0 0 0 1171156 2020-11-07 02:36:18.000000 data.tar.xz\n+-rw-r--r-- 0 0 0 1171048 2020-11-07 02:36:18.000000 data.tar.xz\n"}, {"source1": "control.tar.xz", "source2": "control.tar.xz", "unified_diff": null, "details": [{"source1": "control.tar", "source2": "control.tar", "unified_diff": null, "details": [{"source1": "./md5sums", "source2": "./md5sums", "unified_diff": null, "details": [{"source1": "./md5sums", "source2": "./md5sums", "comments": ["Files differ"], "unified_diff": null}]}]}]}, {"source1": "data.tar.xz", "source2": "data.tar.xz", "unified_diff": null, "details": [{"source1": "data.tar", "source2": "data.tar", "unified_diff": null, "details": [{"source1": "file list", "source2": "file list", "unified_diff": "@@ -128,12 +128,12 @@\n -rw-r--r-- 0 root (0) root (0) 2431 2020-11-07 02:36:18.000000 ./usr/share/doc/sketch-doc/sketch/ex220.png\n -rw-r--r-- 0 root (0) root (0) 2846 2020-11-07 02:36:18.000000 ./usr/share/doc/sketch-doc/sketch/ex230.png\n -rw-r--r-- 0 root (0) root (0) 3579 2020-11-07 02:36:18.000000 ./usr/share/doc/sketch-doc/sketch/ex240.png\n -rw-r--r-- 0 root (0) root (0) 2116 2020-11-07 02:36:18.000000 ./usr/share/doc/sketch-doc/sketch/ex250.png\n -rw-r--r-- 0 root (0) root (0) 2032 2020-11-07 02:36:18.000000 ./usr/share/doc/sketch-doc/sketch/ex260.png\n -rw-r--r-- 0 root (0) root (0) 66246 2012-02-25 02:28:13.000000 ./usr/share/doc/sketch-doc/sketch/index.html\n -rw-r--r-- 0 root (0) root (0) 9205 2020-11-07 02:36:18.000000 ./usr/share/doc/sketch-doc/sketch/transparent.png\n--rw-r--r-- 0 root (0) root (0) 718029 2020-11-07 02:36:18.000000 ./usr/share/doc/sketch-doc/sketch.pdf.gz\n+-rw-r--r-- 0 root (0) root (0) 717962 2020-11-07 02:36:18.000000 ./usr/share/doc/sketch-doc/sketch.pdf.gz\n drwxr-xr-x 0 root (0) root (0) 0 2020-11-07 02:36:18.000000 ./usr/share/doc-base/\n -rw-r--r-- 0 root (0) root (0) 398 2020-11-07 02:36:18.000000 ./usr/share/doc-base/sketch-doc.sketch\n drwxr-xr-x 0 root (0) root (0) 0 2020-11-07 02:36:18.000000 ./usr/share/info/\n--rw-r--r-- 0 root (0) root (0) 40129 2020-11-07 02:36:18.000000 ./usr/share/info/sketch.info.gz\n+-rw-r--r-- 0 root (0) root (0) 40137 2020-11-07 02:36:18.000000 ./usr/share/info/sketch.info.gz\n"}, {"source1": "./usr/share/doc/sketch-doc/sketch.pdf.gz", "source2": "./usr/share/doc/sketch-doc/sketch.pdf.gz", "unified_diff": null, "details": [{"source1": "sketch.pdf", "source2": "sketch.pdf", "unified_diff": null, "details": [{"source1": "pdftotext {} -", "source2": "pdftotext {} -", "comments": ["error from `pdftotext {} -`:", "Syntax Error: ExtGState 'R7' is unknown", "Syntax Error: ExtGState 'R7' is unknown", "Syntax Error: ExtGState 'R7' is unknown", "Syntax Error: ExtGState 'R7' is unknown", "Syntax Error: ExtGState 'R7' is unknown", "Syntax Error: ExtGState 'R7' is unknown", "Syntax Error: ExtGState 'R7' is unknown", "Syntax Error: ExtGState 'R7' is unknown", "Syntax Error: ExtGState 'R7' is unknown", "Syntax Error: ExtGState 'R7' is unknown", "Syntax Error: ExtGState 'R7' is unknown", "Syntax Error: ExtGState 'R7' is unknown", "Syntax Error: ExtGState 'R7' is unknown", "Syntax Error: ExtGState 'R7' is unknown", "Syntax Error: ExtGState 'R7' is unknown", "Syntax Error: ExtGState 'R7' is unknown", "Syntax Error: ExtGState 'R7' is unknown", "Syntax Error: ExtGState 'R7' is unknown", "Syntax Error: ExtGState 'R7' is unknown", "Syntax Error: ExtGState 'R7' is unknown", "Syntax Error: ExtGState 'R7' is unknown", "Syntax Error: ExtGState 'R7' is unknown", "Syntax Error: ExtGState 'R7' is unknown", "Syntax Error: ExtGState 'R7' is unknown", "Syntax Error: ExtGState 'R7' is unknown", "[ truncated after 25 lines; 3 ignored ]"], "unified_diff": "@@ -1,15 +1,15 @@\n Sketch\n Simple 3D sketching\n-Version 0.3 (build 7), Tuesday, November 05, 2024\n+Version 0.3 (build 7), Wednesday, November 06, 2024\n \n Gene Ressler\n \n \fCopyright c 2005 to 2012 Eugene K. Ressler.\n-This manual is for sketch, version 0.3 (build 7), Tuesday, November 05, 2024, a program that converts descriptions of simple three-dimensional scenes into line drawings. This\n+This manual is for sketch, version 0.3 (build 7), Wednesday, November 06, 2024, a program that converts descriptions of simple three-dimensional scenes into line drawings. This\n version generates PSTricks or PGF/TikZ code suitable for use with the TEX document\n processing system.\n Sketch is free software. You can redistribute it and/or modify it under the terms of the\n GNU General Public License as published by the Free Software Foundation; either version\n 3, or (at your option) any later version.\n Sketch is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY;\n without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.\n"}]}]}, {"source1": "./usr/share/info/sketch.info.gz", "source2": "./usr/share/info/sketch.info.gz", "unified_diff": null, "details": [{"source1": "sketch.info", "source2": "sketch.info", "has_internal_linenos": true, "unified_diff": "@@ -5,9794 +5,9794 @@\n 00000040: 7463 682e 7465 7869 2e0a 0a43 6f70 7972 tch.texi...Copyr\n 00000050: 6967 6874 20c2 a920 3230 3035 2074 6f20 ight .. 2005 to \n 00000060: 3230 3132 2045 7567 656e 6520 4b2e 2052 2012 Eugene K. R\n 00000070: 6573 736c 6572 2e0a 0a20 2020 5468 6973 essler... 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If no\n-000003f0: 742c 2073 6565 0a68 7474 703a 2f2f 7777 t, see.http://ww\n-00000400: 772e 676e 752e 6f72 672f 636f 7079 6c65 w.gnu.org/copyle\n-00000410: 6674 2e0a 0a49 4e46 4f2d 4449 522d 5345 ft...INFO-DIR-SE\n-00000420: 4354 494f 4e20 5465 580a 5354 4152 542d CTION TeX.START-\n-00000430: 494e 464f 2d44 4952 2d45 4e54 5259 0a2a INFO-DIR-ENTRY.*\n-00000440: 2053 6b65 7463 683a 2028 736b 6574 6368 Sketch: (sketch\n-00000450: 292e 2020 2020 2020 2020 2020 2020 2053 ). 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T\n-00000520: 6869 7320 6d61 6e75 616c 2069 7320 666f his manual is fo\n-00000530: 7220 e280 9873 6b65 7463 68e2 8099 2c20 r ...sketch..., \n-00000540: 7665 7273 696f 6e20 302e 3320 2862 7569 version 0.3 (bui\n-00000550: 6c64 2037 292c 2054 7565 7364 6179 2c20 ld 7), Tuesday, \n-00000560: 4e6f 7665 6d62 6572 0a30 352c 2032 3032 November.05, 202\n-00000570: 342c 2061 2070 726f 6772 616d 2074 6861 4, a program tha\n-00000580: 7420 636f 6e76 6572 7473 2064 6573 6372 t converts descr\n-00000590: 6970 7469 6f6e 7320 6f66 2073 696d 706c iptions of simpl\n-000005a0: 650a 7468 7265 652d 6469 6d65 6e73 696f e.three-dimensio\n-000005b0: 6e61 6c20 7363 656e 6573 2069 6e74 6f20 nal scenes into \n-000005c0: 6c69 6e65 2064 7261 7769 6e67 732e 2020 line drawings. \n-000005d0: 5468 6973 2076 6572 7369 6f6e 2067 656e This version gen\n-000005e0: 6572 6174 6573 0ae2 8098 5053 5472 6963 erates....PSTric\n-000005f0: 6b73 e280 9920 6f72 20e2 8098 5047 462f ks... or ...PGF/\n-00000600: 5469 6b5a e280 9920 636f 6465 2073 7569 TikZ... code sui\n-00000610: 7461 626c 6520 666f 7220 7573 6520 7769 table for use wi\n-00000620: 7468 2074 6865 2054 6558 2064 6f63 756d th the TeX docum\n-00000630: 656e 740a 7072 6f63 6573 7369 6e67 2073 ent.processing s\n-00000640: 7973 7465 6d2e 0a0a 2020 20e2 8098 536b ystem... ...Sk\n-00000650: 6574 6368 e280 9920 6973 2066 7265 6520 etch... is free \n-00000660: 736f 6674 7761 7265 2e20 2059 6f75 2063 software. You c\n-00000670: 616e 2072 6564 6973 7472 6962 7574 6520 an redistribute \n-00000680: 6974 2061 6e64 2f6f 7220 6d6f 6469 6679 it and/or modify\n-00000690: 2069 740a 756e 6465 7220 7468 6520 7465 it.under the te\n-000006a0: 726d 7320 6f66 2074 6865 2047 4e55 2047 rms of the GNU G\n-000006b0: 656e 6572 616c 2050 7562 6c69 6320 4c69 eneral Public Li\n-000006c0: 6365 6e73 6520 6173 2070 7562 6c69 7368 cense as publish\n-000006d0: 6564 2062 7920 7468 650a 4672 6565 2053 ed by the.Free S\n-000006e0: 6f66 7477 6172 6520 466f 756e 6461 7469 oftware Foundati\n-000006f0: 6f6e 3b20 6569 7468 6572 2076 6572 7369 on; either versi\n-00000700: 6f6e 2033 2c20 6f72 2028 6174 2079 6f75 on 3, or (at you\n-00000710: 7220 6f70 7469 6f6e 2920 616e 790a 6c61 r option) any.la\n-00000720: 7465 7220 7665 7273 696f 6e2e 0a0a 2020 ter version... \n-00000730: 2053 6b65 7463 6820 6973 2064 6973 7472 Sketch is distr\n-00000740: 6962 7574 6564 2069 6e20 7468 6520 686f ibuted in the ho\n-00000750: 7065 2074 6861 7420 6974 2077 696c 6c20 pe that it will \n-00000760: 6265 2075 7365 6675 6c2c 2062 7574 2057 be useful, but W\n-00000770: 4954 484f 5554 0a41 4e59 2057 4152 5241 ITHOUT.ANY WARRA\n-00000780: 4e54 593b 2077 6974 686f 7574 2065 7665 NTY; without eve\n-00000790: 6e20 7468 6520 696d 706c 6965 6420 7761 n the implied wa\n-000007a0: 7272 616e 7479 206f 6620 4d45 5243 4841 rranty of MERCHA\n-000007b0: 4e54 4142 494c 4954 5920 6f72 0a46 4954 NTABILITY or.FIT\n-000007c0: 4e45 5353 2046 4f52 2041 2050 4152 5449 NESS FOR A PARTI\n-000007d0: 4355 4c41 5220 5055 5250 4f53 452e 2053 CULAR PURPOSE. S\n-000007e0: 6565 2074 6865 2047 4e55 2047 656e 6572 ee the GNU Gener\n-000007f0: 616c 2050 7562 6c69 6320 4c69 6365 6e73 al Public Licens\n-00000800: 6520 666f 720a 6d6f 7265 2064 6574 6169 e for.more detai\n-00000810: 6c73 2e0a 0a20 2020 596f 7520 7368 6f75 ls... You shou\n-00000820: 6c64 2068 6176 6520 7265 6365 6976 6564 ld have received\n-00000830: 2061 2063 6f70 7920 6f66 2074 6865 2047 a copy of the G\n-00000840: 4e55 2047 656e 6572 616c 2050 7562 6c69 NU General Publi\n-00000850: 6320 4c69 6365 6e73 650a 616c 6f6e 6720 c License.along \n-00000860: 7769 7468 20e2 8098 736b 6574 6368 e280 with ...sketch..\n-00000870: 993b 2073 6565 2074 6865 2066 696c 6520 .; see the file \n-00000880: 434f 5059 494e 472e 7478 742e 2020 4966 COPYING.txt. If\n-00000890: 206e 6f74 2c20 7365 650a 6874 7470 3a2f not, see.http:/\n-000008a0: 2f77 7777 2e67 6e75 2e6f 7267 2f63 6f70 /www.gnu.org/cop\n-000008b0: 796c 6566 742e 0a0a 2a20 4d65 6e75 3a0a yleft...* Menu:.\n-000008c0: 0a2a 2041 626f 7574 2073 6b65 7463 683a .* About sketch:\n-000008d0: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n-000008e0: 2057 6879 2073 6b65 7463 6820 6578 6973 Why sketch exis\n-000008f0: 7473 2061 6e64 2077 6861 7420 6974 2064 ts and what it d\n-00000900: 6f65 732e 0a2a 2049 6e74 726f 6475 6374 oes..* Introduct\n-00000910: 696f 6e20 6279 2065 7861 6d70 6c65 3a3a ion by example::\n-00000920: 2020 2020 204d 6f73 7420 6665 6174 7572 Most featur\n-00000930: 6573 2073 686f 776e 2061 7320 776f 726b es shown as work\n-00000940: 696e 6720 636f 6465 2e0a 2a20 496e 7075 ing code..* Inpu\n-00000950: 7420 6c61 6e67 7561 6765 3a3a 2020 2020 t language:: \n-00000960: 2020 2020 2020 2020 2020 5379 6e74 6178 Syntax\n-00000970: 2061 6e64 2073 656d 616e 7469 6373 206f and semantics 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You \n+00000820: 7368 6f75 6c64 2068 6176 6520 7265 6365 should have rece\n+00000830: 6976 6564 2061 2063 6f70 7920 6f66 2074 ived a copy of t\n+00000840: 6865 2047 4e55 2047 656e 6572 616c 2050 he GNU General P\n+00000850: 7562 6c69 6320 4c69 6365 6e73 650a 616c ublic License.al\n+00000860: 6f6e 6720 7769 7468 20e2 8098 736b 6574 ong with ...sket\n+00000870: 6368 e280 993b 2073 6565 2074 6865 2066 ch...; see the f\n+00000880: 696c 6520 434f 5059 494e 472e 7478 742e ile COPYING.txt.\n+00000890: 2020 4966 206e 6f74 2c20 7365 650a 6874 If not, see.ht\n+000008a0: 7470 3a2f 2f77 7777 2e67 6e75 2e6f 7267 tp://www.gnu.org\n+000008b0: 2f63 6f70 796c 6566 742e 0a0a 2a20 4d65 /copyleft...* Me\n+000008c0: 6e75 3a0a 0a2a 2041 626f 7574 2073 6b65 nu:..* About ske\n+000008d0: 7463 683a 3a20 2020 2020 2020 2020 2020 tch:: \n+000008e0: 2020 2020 2057 6879 2073 6b65 7463 6820 Why sketch \n+000008f0: 6578 6973 7473 2061 6e64 2077 6861 7420 exists and what \n+00000900: 6974 2064 6f65 732e 0a2a 2049 6e74 726f it does..* Intro\n+00000910: 6475 6374 696f 6e20 6279 2065 7861 6d70 duction by examp\n+00000920: 6c65 3a3a 2020 2020 204d 6f73 7420 6665 le:: Most fe\n+00000930: 6174 7572 6573 2073 686f 776e 2061 7320 atures shown as \n+00000940: 776f 726b 696e 6720 636f 6465 2e0a 2a20 working code..* \n+00000950: 496e 7075 7420 6c61 6e67 7561 6765 3a3a Input language::\n+00000960: 2020 2020 2020 2020 2020 2020 2020 5379 Sy\n+00000970: 6e74 6178 2061 6e64 2073 656d 616e 7469 ntax and semanti\n+00000980: 6373 206f 6620 e280 9873 6b65 7463 68e2 cs of ...sketch.\n+00000990: 8099 2063 6f6d 6d61 6e64 732e 0a2a 2042 .. commands..* B\n+000009a0: 7569 6c64 696e 6720 6120 6472 6177 696e uilding a drawin\n+000009b0: 673a 3a20 2020 2020 2020 2020 2048 6f77 g:: How\n+000009c0: 2074 6f20 7573 6520 e280 9873 6b65 7463 to use ...sketc\n+000009d0: 68e2 8099 2070 726f 6475 6374 6976 656c h... productivel\n+000009e0: 792e 0a2a 2043 6f6d 6d61 6e64 206c 696e y..* Command lin\n+000009f0: 653a 3a20 2020 2020 2020 2020 2020 2020 e:: \n+00000a00: 2020 204f 7074 696f 6e73 2061 6e64 2074 Options and t\n+00000a10: 6865 6972 2075 7361 6765 2e0a 2a20 496e heir usage..* In\n+00000a20: 7374 616c 6c69 6e67 2073 6b65 7463 683a stalling sketch:\n+00000a30: 3a20 2020 2020 2020 2020 2020 4275 696c : Buil\n+00000a40: 6469 6e67 2061 6e64 2069 6e73 7461 6c6c ding and install\n+00000a50: 696e 6720 6672 6f6d 2073 6f75 7263 6573 ing from sources\n+00000a60: 2e0a 2a20 496e 6465 7820 6f66 2073 796e ..* Index of syn\n+00000a70: 7461 783a 3a0a 2a20 496e 6465 783a 3a0a tax::.* Index::.\n+00000a80: 0a20 2d2d 2054 6865 2044 6574 6169 6c65 . -- The Detaile\n+00000a90: 6420 4e6f 6465 204c 6973 7469 6e67 202d d Node Listing -\n+00000aa0: 2d0a 0a41 626f 7574 2073 6b65 7463 680a -..About sketch.\n+00000ab0: 0a2a 2052 6570 6f72 7469 6e67 2062 7567 .* Reporting bug\n+00000ac0: 733a 3a20 2020 2020 2020 2020 2020 2020 s:: \n+00000ad0: 204c 6574 2075 7365 206b 6e6f 7720 7768 Let use know wh\n+00000ae0: 6174 2773 2077 726f 6e67 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\n+00001460: 2020 2020 2020 2020 2020 2020 4472 6177 Draw\n+00001470: 2064 6f74 732e 0a2a 204c 696e 6573 3a3a dots..* Lines::\n 00001480: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00001490: 2020 2044 7261 7720 706f 6c79 6c69 6e65 Draw polyline\n-000014a0: 732e 0a2a 2043 7572 7665 733a 3a20 2020 s..* Curves:: \n-000014b0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000014c0: 2020 2044 7261 7720 6375 7276 6573 2e0a Draw curves..\n-000014d0: 2a20 506f 6c79 676f 6e73 3a3a 2020 2020 * Polygons:: \n+00001490: 2020 2020 2020 2044 7261 7720 706f 6c79 Draw poly\n+000014a0: 6c69 6e65 732e 0a2a 2043 7572 7665 733a lines..* Curves:\n+000014b0: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+000014c0: 2020 2020 2020 2044 7261 7720 6375 7276 Draw curv\n+000014d0: 6573 2e0a 2a20 506f 6c79 676f 6e73 3a3a es..* Polygons::\n 000014e0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000014f0: 4472 6177 2070 6f6c 7967 6f6e 732e 0a2a Draw polygons..*\n-00001500: 2053 7065 6369 616c 733a 3a20 2020 2020 Specials:: 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2020 2020 .* Puts:: \n+000014f0: 2020 2020 4472 6177 2070 6f6c 7967 6f6e Draw polygon\n+00001500: 732e 0a2a 2053 7065 6369 616c 733a 3a20 s..* Specials:: \n+00001510: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00001520: 2020 2045 6d62 6564 2072 6177 204c 6154 Embed raw LaT\n+00001530: 6558 2061 6e64 20e2 8098 5053 5472 6963 eX and ...PSTric\n+00001540: 6b73 e280 992e 0a2a 2053 7765 6570 733a ks.....* Sweeps:\n+00001550: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+00001560: 2020 2020 2020 2044 7261 7720 7377 6565 Draw swee\n+00001570: 7073 206f 6620 646f 7473 2061 6e64 2070 ps of dots and p\n+00001580: 6f6c 796c 696e 6573 2e0a 2a20 426c 6f63 olylines..* Bloc\n+00001590: 6b73 3a3a 2020 2020 2020 2020 2020 2020 ks:: \n+000015a0: 2020 2020 2020 2020 2020 4772 6f75 7020 Group \n+000015b0: 6f74 6865 7220 6472 6177 6162 6c65 732e other drawables.\n+000015c0: 0a2a 2052 6570 6561 7473 3a3a 2020 2020 .* Repeats:: \n+000015d0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+000015e0: 2044 7261 7720 7472 616e 7366 6f72 6d65 Draw transforme\n+000015f0: 6420 636f 7069 6573 206f 6620 6f62 6a65 d copies of obje\n+00001600: 6374 732e 0a2a 2050 7574 733a 3a20 2020 cts..* Puts:: \n 00001610: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00001620: 2044 7261 7720 6f6e 6520 6f62 6a65 6374 Draw one object\n-00001630: 2074 7261 6e73 666f 726d 6564 2e0a 0a53 transformed...S\n-00001640: 7765 6570 730a 0a2a 2053 7765 7074 2070 weeps..* Swept p\n-00001650: 6f69 6e74 733a 3a20 2020 2020 2020 2020 oints:: \n-00001660: 2020 2020 2020 2053 7765 7074 2070 6f69 Swept poi\n-00001670: 6e74 7320 6d61 6b65 206c 696e 6573 206f nts make lines o\n-00001680: 7220 706f 6c79 676f 6e73 2e0a 2a20 5377 r polygons..* Sw\n-00001690: 6570 7420 6c69 6e65 733a 3a20 2020 2020 ept lines:: \n-000016a0: 2020 2020 2020 2020 2020 2020 5377 6570 Swep\n-000016b0: 7420 6c69 6e65 7320 6d61 6b65 206f 7065 t lines make ope\n-000016c0: 6e20 6f72 2063 6c6f 7365 6420 7375 7266 n or closed surf\n-000016d0: 6163 6573 2e0a 2a20 5377 6570 7420 706f aces..* Swept po\n-000016e0: 6c79 676f 6e73 3a3a 2020 2020 2020 2020 lygons:: \n-000016f0: 2020 2020 2020 5377 6570 7420 706f 6c79 Swept poly\n-00001700: 676f 6e73 206d 616b 6520 636c 6f73 6564 gons make closed\n-00001710: 2073 7572 6661 6365 732e 0a2a 2053 7765 surfaces..* Swe\n-00001720: 7074 2062 6c6f 636b 733a 3a20 2020 2020 pt blocks:: \n-00001730: 2020 2020 2020 2020 2020 2053 7765 7074 Swept\n-00001740: 2062 6c6f 636b 20e2 89a1 2062 6c6f 636b block ... block\n-00001750: 206f 6620 7377 6565 7073 2e0a 2a20 5377 of sweeps..* Sw\n-00001760: 6565 7020 6661 6365 2073 706c 6974 7469 eep face splitti\n-00001770: 6e67 3a3a 2020 2020 2020 2020 4669 7869 ng:: Fixi\n-00001780: 6e67 2077 6172 7065 6420 6661 6365 7320 ng warped faces \n-00001790: 7769 7468 2074 7269 616e 676c 6573 2e0a with triangles..\n-000017a0: 0a44 6566 696e 6974 696f 6e73 0a0a 2a20 .Definitions..* \n-000017b0: 466f 726d 7320 6f66 2064 6566 696e 6974 Forms of definit\n-000017c0: 696f 6e73 3a3a 2020 2020 2020 2020 4469 ions:: Di\n-000017d0: 6666 6572 656e 7420 6465 6673 2066 6f72 fferent defs for\n-000017e0: 2064 6966 6665 7265 6e74 2070 7572 706f different purpo\n-000017f0: 7365 732e 0a2a 2046 6f72 6d73 206f 6620 ses..* Forms of \n-00001800: 7265 6665 7265 6e63 6573 3a3a 2020 2020 references:: \n-00001810: 2020 2020 2048 6f77 2072 6566 6572 656e How referen\n-00001820: 6365 7320 6465 6e6f 7465 2074 7970 6573 ces denote types\n-00001830: 2e0a 0a47 6c6f 6261 6c20 656e 7669 726f ...Global enviro\n-00001840: 6e6d 656e 740a 0a2a 2047 6c6f 6261 6c20 nment..* Global \n-00001850: 6f70 7469 6f6e 733a 3a20 2020 2020 2020 options:: \n-00001860: 2020 2020 2020 2041 7474 7269 6275 7465 Attribute\n-00001870: 7320 6f66 2074 6865 2065 6e74 6972 6520 s of the entire \n-00001880: 6472 6177 696e 672e 0a2a 2043 616d 6572 drawing..* Camer\n-00001890: 613a 3a20 2020 2020 2020 2020 2020 2020 a:: \n-000018a0: 2020 2020 2020 2020 2041 2066 696e 616c A final\n-000018b0: 2063 616d 6572 6120 7472 616e 7366 6f72 camera transfor\n-000018c0: 6d61 7469 6f6e 206f 6620 7468 6520 7363 mation of the sc\n-000018d0: 656e 652e 0a2a 2050 6963 7475 7265 2062 ene..* Picture b\n-000018e0: 6f78 3a3a 2020 2020 2020 2020 2020 2020 ox:: \n-000018f0: 2020 2020 2053 6574 7469 6e67 2074 6865 Setting the\n-00001900: 2062 6f75 6e64 696e 6720 626f 7820 616e bounding box an\n-00001910: 6420 3264 2063 6c69 7070 696e 672e 0a2a d 2d clipping..*\n-00001920: 2046 7261 6d65 3a3a 2020 2020 2020 2020 Frame:: \n-00001930: 2020 2020 2020 2020 2020 2020 2020 2041 A\n-00001940: 6464 696e 6720 6120 626f 7820 6172 6f75 dding a box arou\n-00001950: 6e64 2074 6865 2064 7261 7769 6e67 2e0a nd the drawing..\n-00001960: 2a20 4c61 6e67 7561 6765 3a3a 2020 2020 * Language:: \n+00001620: 2020 2020 2044 7261 7720 6f6e 6520 6f62 Draw one ob\n+00001630: 6a65 6374 2074 7261 6e73 666f 726d 6564 ject transformed\n+00001640: 2e0a 0a53 7765 6570 730a 0a2a 2053 7765 ...Sweeps..* Swe\n+00001650: 7074 2070 6f69 6e74 733a 3a20 2020 2020 pt points:: \n+00001660: 2020 2020 2020 2020 2020 2053 7765 7074 Swept\n+00001670: 2070 6f69 6e74 7320 6d61 6b65 206c 696e points make lin\n+00001680: 6573 206f 7220 706f 6c79 676f 6e73 2e0a es or polygons..\n+00001690: 2a20 5377 6570 7420 6c69 6e65 733a 3a20 * Swept lines:: \n+000016a0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+000016b0: 5377 6570 7420 6c69 6e65 7320 6d61 6b65 Swept lines make\n+000016c0: 206f 7065 6e20 6f72 2063 6c6f 7365 6420 open or closed \n+000016d0: 7375 7266 6163 6573 2e0a 2a20 5377 6570 surfaces..* Swep\n+000016e0: 7420 706f 6c79 676f 6e73 3a3a 2020 2020 t polygons:: \n+000016f0: 2020 2020 2020 2020 2020 5377 6570 7420 Swept \n+00001700: 706f 6c79 676f 6e73 206d 616b 6520 636c polygons make cl\n+00001710: 6f73 6564 2073 7572 6661 6365 732e 0a2a osed surfaces..*\n+00001720: 2053 7765 7074 2062 6c6f 636b 733a 3a20 Swept blocks:: \n+00001730: 2020 2020 2020 2020 2020 2020 2020 2053 S\n+00001740: 7765 7074 2062 6c6f 636b 20e2 89a1 2062 wept block ... b\n+00001750: 6c6f 636b 206f 6620 7377 6565 7073 2e0a lock of sweeps..\n+00001760: 2a20 5377 6565 7020 6661 6365 2073 706c * Sweep face spl\n+00001770: 6974 7469 6e67 3a3a 2020 2020 2020 2020 itting:: \n+00001780: 4669 7869 6e67 2077 6172 7065 6420 6661 Fixing warped fa\n+00001790: 6365 7320 7769 7468 2074 7269 616e 676c ces with triangl\n+000017a0: 6573 2e0a 0a44 6566 696e 6974 696f 6e73 es...Definitions\n+000017b0: 0a0a 2a20 466f 726d 7320 6f66 2064 6566 ..* Forms of def\n+000017c0: 696e 6974 696f 6e73 3a3a 2020 2020 2020 initions:: \n+000017d0: 2020 4469 6666 6572 656e 7420 6465 6673 Different defs\n+000017e0: 2066 6f72 2064 6966 6665 7265 6e74 2070 for different p\n+000017f0: 7572 706f 7365 732e 0a2a 2046 6f72 6d73 urposes..* Forms\n+00001800: 206f 6620 7265 6665 7265 6e63 6573 3a3a of references::\n+00001810: 2020 2020 2020 2020 2048 6f77 2072 6566 How ref\n+00001820: 6572 656e 6365 7320 6465 6e6f 7465 2074 erences denote t\n+00001830: 7970 6573 2e0a 0a47 6c6f 6261 6c20 656e ypes...Global en\n+00001840: 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2020 2020 2020 2020 \n+00001940: 2020 2041 6464 696e 6720 6120 626f 7820 Adding a box \n+00001950: 6172 6f75 6e64 2074 6865 2064 7261 7769 around the drawi\n+00001960: 6e67 2e0a 2a20 4c61 6e67 7561 6765 3a3a ng..* Language::\n 00001970: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00001980: 5365 7474 696e 6720 7468 6520 6f75 7470 Setting the outp\n-00001990: 7574 206c 616e 6775 6167 652e 0a0a 4275 ut language...Bu\n-000019a0: 696c 6469 6e67 2061 2064 7261 7769 6e67 ilding a drawing\n-000019b0: 0a0a 2a20 4f76 6572 7669 6577 3a3a 2020 ..* Overview:: \n-000019c0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000019d0: 2020 4275 696c 6469 6e67 2061 2073 7562 Building a sub\n-000019e0: 7374 616e 7469 616c 2064 7261 7769 6e67 stantial drawing\n-000019f0: 2e0a 2a20 4120 7465 6368 6e69 6361 6c20 ..* A technical \n-00001a00: 6472 6177 696e 673a 3a20 2020 2020 2020 drawing:: \n-00001a10: 2020 416e 2065 7861 6d70 6c65 2077 6974 An example wit\n-00001a20: 6820 6669 6e65 2070 6c61 6365 6d65 6e74 h fine placement\n-00001a30: 2e0a 2a20 4120 6869 6572 6172 6368 6963 ..* A hierarchic\n-00001a40: 616c 206d 6f64 656c 3a3a 2020 2020 2020 al model:: \n-00001a50: 2020 416e 2065 7861 6d70 6c65 2077 6974 An example wit\n-00001a60: 6820 7377 6565 7073 2061 6e64 2070 7574 h sweeps and put\n-00001a70: 732e 0a2a 2043 6176 6561 7473 3a3a 2020 s..* Caveats:: \n-00001a80: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00001a90: 2020 2057 6865 7265 2074 726f 7562 6c65 Where trouble\n-00001aa0: 2063 616e 206f 6363 7572 2e0a 0a43 6176 can occur...Cav\n-00001ab0: 6561 7473 0a0a 2a20 4c69 6d69 7473 206f eats..* Limits o\n-00001ac0: 6e20 6572 726f 7220 6465 7465 6374 696f n error detectio\n-00001ad0: 6e3a 3a20 2020 5768 6174 2073 6b65 7463 n:: What sketc\n-00001ae0: 6820 646f 6573 6e27 7420 646f 2e0a 2a20 h doesn't do..* \n-00001af0: 436c 6970 7069 6e67 3a3a 2020 2020 2020 Clipping:: \n-00001b00: 2020 2020 2020 2020 2020 2020 2020 4e6f No\n-00001b10: 2063 6c69 7070 696e 6720 6174 2070 7265 clipping at pre\n-00001b20: 7365 6e74 2e0a 2a20 4869 6464 656e 2073 sent..* Hidden s\n-00001b30: 7572 6661 6365 2072 656d 6f76 616c 3a3a urface removal::\n-00001b40: 2020 2020 2020 496d 7065 7266 6563 7469 Imperfecti\n-00001b50: 6f6e 7320 746f 2066 6978 2e0a 0a48 6964 ons to fix...Hid\n-00001b60: 6465 6e20 7375 7266 6163 6520 7265 6d6f den surface remo\n-00001b70: 7661 6c20 616e 6420 706f 6c79 676f 6e20 val and polygon \n-00001b80: 7370 6c69 7474 696e 670a 0a2a 2053 7461 splitting..* Sta\n-00001b90: 7469 7374 6963 733a 3a20 2020 2020 2020 tistics:: \n-00001ba0: 2020 2020 2020 2020 2020 2050 6572 666f Perfo\n-00001bb0: 726d 616e 6365 206e 756d 6265 7273 206f rmance numbers o\n-00001bc0: 6e20 6465 7074 6820 736f 7274 2e0a 2a20 n depth sort..* \n-00001bd0: 4275 6773 2061 6e64 2061 6e6f 6d61 6c69 Bugs and anomali\n-00001be0: 6573 3a3a 2020 2020 2020 2020 2020 496d es:: Im\n-00001bf0: 7065 7266 6563 7469 6f6e 7320 696e 2074 perfections in t\n-00001c00: 6869 7320 696d 706c 656d 656e 7461 7469 his 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2079 6f75 How you\n-00001d00: 2063 616e 2068 656c 702e 2e2e 2e0a 0ae2 can help.......\n-00001d10: 8098 536b 6574 6368 e280 9920 6973 2061 ..Sketch... is a\n-00001d20: 2073 6d61 6c6c 2c20 7369 6d70 6c65 2073 small, simple s\n-00001d30: 7973 7465 6d20 666f 7220 7072 6f64 7563 ystem for produc\n-00001d40: 696e 6720 6c69 6e65 2064 7261 7769 6e67 ing line drawing\n-00001d50: 7320 6f66 2074 776f 2d0a 6f72 2074 6872 s of two-.or thr\n-00001d60: 6565 2d64 696d 656e 7369 6f6e 616c 206f ee-dimensional o\n-00001d70: 626a 6563 7473 2061 6e64 2073 6365 6e65 bjects and scene\n-00001d80: 732e 2020 4974 2062 6567 616e 2061 7320 s. 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One can\n-00002020: 2065 6173 696c 790a 696e 636c 7564 6520 easily.include \n-00002030: 6172 6269 7472 6172 7920 e280 9850 5354 arbitrary ...PST\n-00002040: 7269 636b 73e2 8099 206f 7220 e280 9854 ricks... or ...T\n-00002050: 696b 5a2f 5047 46e2 8099 2064 7261 7769 ikZ/PGF... drawi\n-00002060: 6e67 7320 616e 6420 7465 7874 206f 7665 ngs and text ove\n-00002070: 722c 2069 6e2c 0a6f 7220 756e 6465 7220 r, in,.or under \n-00002080: e280 9873 6b65 7463 68e2 8099 2064 7261 ...sketch... dra\n-00002090: 7769 6e67 732c 2070 726f 7669 6469 6e67 wings, providing\n-000020a0: 2061 6363 6573 7320 746f 2074 6865 2066 access to the f\n-000020b0: 756c 6c20 706f 7765 7220 6f66 204c 6154 ull power of LaT\n-000020c0: 6558 0a74 6578 7420 616e 6420 6d61 7468 eX.text and math\n-000020d0: 656d 6174 6963 7320 666f 726d 6174 7469 ematics formatti\n-000020e0: 6e67 2069 6e20 6120 7468 7265 652d 6469 ng in a three-di\n-000020f0: 6d65 6e73 696f 6e61 6c20 7365 7474 696e mensional settin\n-00002100: 672e 0a0a 1f0a 4669 6c65 3a20 736b 6574 g.....File: sket\n-00002110: 6368 2e69 6e66 6f2c 2020 4e6f 6465 3a20 ch.info, Node: \n-00002120: 5265 706f 7274 696e 6720 6275 6773 2c20 Reporting bugs, \n-00002130: 204e 6578 743a 2043 6f6e 7472 6962 7574 Next: Contribut\n-00002140: 696f 6e73 2c20 2050 7265 763a 2041 626f ions, Prev: Abo\n-00002150: 7574 2073 6b65 7463 682c 2020 5570 3a20 ut sketch, Up: \n-00002160: 4162 6f75 7420 736b 6574 6368 0a0a 312e About sketch..1.\n-00002170: 3120 5265 706f 7274 696e 6720 6275 6773 1 Reporting bugs\n-00002180: 2061 6e64 2072 6563 6f6d 6d65 6e64 696e and recommendin\n-00002190: 6720 696d 7072 6f76 656d 656e 7473 2e0a g improvements..\n-000021a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================\n+00001980: 2020 2020 5365 7474 696e 6720 7468 6520 Setting the \n+00001990: 6f75 7470 7574 206c 616e 6775 6167 652e output language.\n+000019a0: 0a0a 4275 696c 6469 6e67 2061 2064 7261 ..Building a dra\n+000019b0: 7769 6e67 0a0a 2a20 4f76 6572 7669 6577 wing..* Overview\n+000019c0: 3a3a 2020 2020 2020 2020 2020 2020 2020 :: \n+000019d0: 2020 2020 2020 4275 696c 6469 6e67 2061 Building a\n+000019e0: 2073 7562 7374 616e 7469 616c 2064 7261 substantial dra\n+000019f0: 7769 6e67 2e0a 2a20 4120 7465 6368 6e69 wing..* A techni\n+00001a00: 6361 6c20 6472 6177 696e 673a 3a20 2020 cal drawing:: \n+00001a10: 2020 2020 2020 416e 2065 7861 6d70 6c65 An example\n+00001a20: 2077 6974 6820 6669 6e65 2070 6c61 6365 with fine place\n+00001a30: 6d65 6e74 2e0a 2a20 4120 6869 6572 6172 ment..* A hierar\n+00001a40: 6368 6963 616c 206d 6f64 656c 3a3a 2020 chical model:: \n+00001a50: 2020 2020 2020 416e 2065 7861 6d70 6c65 An example\n+00001a60: 2077 6974 6820 7377 6565 7073 2061 6e64 with sweeps and\n+00001a70: 2070 7574 732e 0a2a 2043 6176 6561 7473 puts..* Caveats\n+00001a80: 3a3a 2020 2020 2020 2020 2020 2020 2020 :: \n+00001a90: 2020 2020 2020 2057 6865 7265 2074 726f Where tro\n+00001aa0: 7562 6c65 2063 616e 206f 6363 7572 2e0a uble can occur..\n+00001ab0: 0a43 6176 6561 7473 0a0a 2a20 4c69 6d69 .Caveats..* Limi\n+00001ac0: 7473 206f 6e20 6572 726f 7220 6465 7465 ts on error dete\n+00001ad0: 6374 696f 6e3a 3a20 2020 5768 6174 2073 ction:: What s\n+00001ae0: 6b65 7463 6820 646f 6573 6e27 7420 646f ketch doesn't do\n+00001af0: 2e0a 2a20 436c 6970 7069 6e67 3a3a 2020 ..* Clipping:: \n+00001b00: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00001b10: 2020 4e6f 2063 6c69 7070 696e 6720 6174 No clipping at\n+00001b20: 2070 7265 7365 6e74 2e0a 2a20 4869 6464 present..* Hidd\n+00001b30: 656e 2073 7572 6661 6365 2072 656d 6f76 en surface remov\n+00001b40: 616c 3a3a 2020 2020 2020 496d 7065 7266 al:: Imperf\n+00001b50: 6563 7469 6f6e 7320 746f 2066 6978 2e0a ections to fix..\n+00001b60: 0a48 6964 6465 6e20 7375 7266 6163 6520 .Hidden surface \n+00001b70: 7265 6d6f 7661 6c20 616e 6420 706f 6c79 removal and poly\n+00001b80: 676f 6e20 7370 6c69 7474 696e 670a 0a2a gon splitting..*\n+00001b90: 2053 7461 7469 7374 6963 733a 3a20 2020 Statistics:: 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It began\n+00001d90: 2061 7320 6120 7761 7920 746f 206d 616b as a way to mak\n+00001da0: 650a 696c 6c75 7374 7261 7469 6f6e 7320 e.illustrations \n+00001db0: 666f 7220 6120 7465 7874 626f 6f6b 2061 for a textbook a\n+00001dc0: 6674 6572 2077 6520 636f 756c 6420 6669 fter we could fi\n+00001dd0: 6e64 206e 6f20 7375 6974 6162 6c65 2074 nd no suitable t\n+00001de0: 6f6f 6c20 666f 720a 7468 6973 2070 7572 ool for.this pur\n+00001df0: 706f 7365 2e20 2045 7869 7374 696e 6720 pose. Existing \n+00001e00: 7363 656e 6520 7072 6f63 6573 736f 7273 scene processors\n+00001e10: 2065 6d70 6861 7369 7a65 6420 4755 4973 emphasized GUIs\n+00001e20: 2061 6e64 2f6f 720a 7068 6f74 6f2d 7265 and/or.photo-re\n+00001e30: 616c 6973 6d2c 2062 6f74 6820 756e 2d75 alism, both un-u\n+00001e40: 7365 6675 6c20 746f 2075 732e 2020 5765 seful to us. We\n+00001e50: 2077 616e 7465 6420 746f 2070 726f 6475 wanted to produ\n+00001e60: 6365 2066 696e 656c 790a 7772 6f75 6768 ce finely.wrough\n+00001e70: 742c 206d 6174 6865 6d61 7469 6361 6c6c t, mathematicall\n+00001e80: 792d 6261 7365 6420 696c 6c75 7374 7261 y-based illustra\n+00001e90: 7469 6f6e 7320 7769 7468 206e 6f20 6578 tions with no ex\n+00001ea0: 7472 616e 656f 7573 2064 6574 6169 6c2e traneous detail.\n+00001eb0: 0a0a 2020 20e2 8098 536b 6574 6368 e280 .. ...Sketch..\n+00001ec0: 9920 6163 6365 7074 7320 6120 7469 6e79 . accepts a tiny\n+00001ed0: 2073 6365 6e65 2064 6573 6372 6970 7469 scene descripti\n+00001ee0: 6f6e 206c 616e 6775 6167 6520 616e 6420 on language and \n+00001ef0: 6765 6e65 7261 7465 730a e280 9850 5354 generates....PST\n+00001f00: 7269 636b 73e2 8099 206f 7220 e280 9854 ricks... or ...T\n+00001f10: 696b 5a2f 5047 46e2 8099 2063 6f64 6520 ikZ/PGF... code \n+00001f20: 666f 7220 4c61 5465 582e 2020 5468 6520 for LaTeX. 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See http://so\n+00002000: 7572 6365 666f 7267 652e 6e65 742f 7072 urceforge.net/pr\n+00002010: 6f6a 6563 7473 2f70 6766 2e20 204f 6e65 ojects/pgf. One\n+00002020: 2063 616e 2065 6173 696c 790a 696e 636c can easily.incl\n+00002030: 7564 6520 6172 6269 7472 6172 7920 e280 ude arbitrary ..\n+00002040: 9850 5354 7269 636b 73e2 8099 206f 7220 .PSTricks... or \n+00002050: e280 9854 696b 5a2f 5047 46e2 8099 2064 ...TikZ/PGF... d\n+00002060: 7261 7769 6e67 7320 616e 6420 7465 7874 rawings and text\n+00002070: 206f 7665 722c 2069 6e2c 0a6f 7220 756e over, in,.or un\n+00002080: 6465 7220 e280 9873 6b65 7463 68e2 8099 der ...sketch...\n+00002090: 2064 7261 7769 6e67 732c 2070 726f 7669 drawings, provi\n+000020a0: 6469 6e67 2061 6363 6573 7320 746f 2074 ding access to t\n+000020b0: 6865 2066 756c 6c20 706f 7765 7220 6f66 he full power of\n+000020c0: 204c 6154 6558 0a74 6578 7420 616e 6420 LaTeX.text and \n+000020d0: 6d61 7468 656d 6174 6963 7320 666f 726d mathematics form\n+000020e0: 6174 7469 6e67 2069 6e20 6120 7468 7265 atting in a thre\n+000020f0: 652d 6469 6d65 6e73 696f 6e61 6c20 7365 e-dimensional se\n+00002100: 7474 696e 672e 0a0a 1f0a 4669 6c65 3a20 tting.....File: \n+00002110: 736b 6574 6368 2e69 6e66 6f2c 2020 4e6f sketch.info, No\n+00002120: 6465 3a20 5265 706f 7274 696e 6720 6275 de: Reporting bu\n+00002130: 6773 2c20 204e 6578 743a 2043 6f6e 7472 gs, Next: Contr\n+00002140: 6962 7574 696f 6e73 2c20 2050 7265 763a ibutions, Prev:\n+00002150: 2041 626f 7574 2073 6b65 7463 682c 2020 About sketch, \n+00002160: 5570 3a20 4162 6f75 7420 736b 6574 6368 Up: About sketch\n+00002170: 0a0a 312e 3120 5265 706f 7274 696e 6720 ..1.1 Reporting \n+00002180: 6275 6773 2061 6e64 2072 6563 6f6d 6d65 bugs and recomme\n+00002190: 6e64 696e 6720 696d 7072 6f76 656d 656e nding improvemen\n+000021a0: 7473 2e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ts..============\n 000021b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================\n 000021c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================\n-000021d0: 3d0a 0a54 6865 2067 726f 7570 2068 7474 =..The group htt\n-000021e0: 703a 2f2f 6772 6f75 7073 2e67 6f6f 676c p://groups.googl\n-000021f0: 652e 636f 6d2f 6772 6f75 702f 736b 6574 e.com/group/sket\n-00002200: 6368 2d75 7365 7273 2069 7320 7468 6520 ch-users is the \n-00002210: 6265 7374 2070 6c61 6365 0a74 6f20 7265 best place.to re\n-00002220: 706f 7274 2062 7567 7320 616e 6420 6d61 port bugs and ma\n-00002230: 6b65 2069 6d70 726f 7665 6d65 6e74 732e ke improvements.\n-00002240: 2020 4120 7365 636f 6e64 206d 6574 686f A second metho\n-00002250: 6420 7468 6174 2077 696c 6c0a 7072 6f62 d that will.prob\n-00002260: 6162 6c79 2070 726f 6475 6365 2061 2073 ably produce a s\n-00002270: 6c6f 7765 7220 7265 7370 6f6e 7365 2069 lower response i\n-00002280: 7320 656d 6169 6c20 746f 2073 6b65 7463 s email to sketc\n-00002290: 6840 6672 6f6e 7469 6572 6e65 742e 6e65 h@frontiernet.ne\n-000022a0: 742e 0a57 6520 7769 6c6c 2074 7279 2074 t..We will try t\n-000022b0: 6f20 7265 7370 6f6e 642c 2062 7574 2063 o respond, but c\n-000022c0: 616e 2774 2070 726f 6d69 7365 2e20 2049 an't promise. I\n-000022d0: 6e20 616e 7920 6576 656e 742c 2064 6f6e n any event, don\n-000022e0: 2774 2062 650a 6f66 6665 6e64 6564 2069 't be.offended i\n-000022f0: 6620 6120 7265 706c 7920 6973 206e 6f74 f a reply is not\n-00002300: 2066 6f72 7468 636f 6d69 6e67 2e20 2057 forthcoming. W\n-00002310: 6527 7265 206a 7573 7420 6275 7379 2061 e're just busy a\n-00002320: 6e64 2077 696c 6c20 6765 7420 746f 0a79 nd will get to.y\n-00002330: 6f75 7220 7375 6767 6573 7469 6f6e 2065 our suggestion e\n-00002340: 7665 6e74 7561 6c6c 792e 0a0a 2020 2046 ventually... 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A recomm\n-00002400: 656e 6461 7469 6f6e 2066 6f72 2069 6d70 endation for imp\n-00002410: 726f 7665 6d65 6e74 2066 726f 6d20 6f6e rovement from on\n-00002420: 6520 756e 6b6e 6f77 6e20 7065 7273 6f6e e unknown person\n-00002430: 2063 6f75 6e74 7320 6173 0a6f 6e65 2076 counts as.one v\n-00002440: 6f74 652e 2020 5765 2075 7365 206f 7665 ote. We use ove\n-00002450: 7261 6c6c 2076 6f74 6520 7461 6c6c 6965 rall vote tallie\n-00002460: 7320 746f 2064 6563 6964 6520 7768 6174 s to decide what\n-00002470: 2074 6f20 646f 206e 6578 7420 6173 0a72 to do next as.r\n-00002480: 6573 6f75 7263 6573 2070 6572 6d69 742e esources permit.\n-00002490: 2020 5765 2072 6573 6572 7665 2074 6865 We reserve the\n-000024a0: 2072 6967 6874 2074 6f20 6173 7369 676e right to assign\n-000024b0: 2061 6e79 206e 756d 6265 7220 6f66 2076 any number of v\n-000024c0: 6f74 6573 2074 6f0a 7375 6767 6573 7469 otes to.suggesti\n-000024d0: 6f6e 7320 6672 6f6d 2070 656f 706c 6520 ons from people \n-000024e0: 7768 6f20 6861 7665 2062 6565 6e20 6865 who have been he\n-000024f0: 6c70 6675 6c20 616e 6420 7375 7070 6f72 lpful and suppor\n-00002500: 7469 7665 2069 6e20 7468 650a 7061 7374 tive in the.past\n-00002510: 2e0a 0a1f 0a46 696c 653a 2073 6b65 7463 .....File: sketc\n-00002520: 682e 696e 666f 2c20 204e 6f64 653a 2043 h.info, Node: C\n-00002530: 6f6e 7472 6962 7574 696f 6e73 2c20 2050 ontributions, P\n-00002540: 7265 763a 2052 6570 6f72 7469 6e67 2062 rev: Reporting b\n-00002550: 7567 732c 2020 5570 3a20 4162 6f75 7420 ugs, Up: About \n-00002560: 736b 6574 6368 0a0a 312e 3220 436f 6e74 sketch..1.2 Cont\n-00002570: 7269 6275 7469 6f6e 730a 3d3d 3d3d 3d3d ributions.======\n-00002580: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a49 6620 ===========..If \n-00002590: 796f 7520 696e 7465 6e64 2074 6f20 696d you intend to im\n-000025a0: 706c 656d 656e 7420 616e 2065 6e68 616e plement an enhan\n-000025b0: 6365 6d65 6e74 206f 6620 796f 7572 206f cement of your o\n-000025c0: 776e 2c20 7468 6174 2773 2074 6572 7269 wn, that's terri\n-000025d0: 6669 6321 0a43 6f6e 7369 6465 7220 636f fic!.Consider co\n-000025e0: 6c6c 6162 6f72 6174 696e 6720 7769 7468 llaborating with\n-000025f0: 2075 7320 6669 7273 7420 746f 2073 6565 us first to see\n-00002600: 2069 6620 7765 2772 6520 616c 7265 6164 if we're alread\n-00002610: 7920 776f 726b 696e 6720 6f6e 0a79 6f75 y working on.you\n-00002620: 7220 6964 6561 206f 7220 6966 2077 6520 r idea or if we \n-00002630: 6361 6e20 7573 6520 796f 7572 2077 6f72 can use your wor\n-00002640: 6b20 696e 2074 6865 206f 6666 6963 6961 k in the officia\n-00002650: 6c20 7265 6c65 6173 652e 0a0a 0008 5b69 l release.....[i\n-00002660: 6d61 6765 2073 7263 3d22 6578 3030 302e mage src=\"ex000.\n-00002670: 706e 6722 2074 6578 743d 2249 6d61 6765 png\" text=\"Image\n-00002680: 2065 7830 3030 206f 6d69 7474 6564 2069 ex000 omitted i\n-00002690: 6e20 7465 7874 2076 6572 7369 6f6e 206f n text version o\n-000026a0: 6620 7468 6973 2064 6f63 756d 656e 742e f this document.\n-000026b0: 2200 085d 0a0a 1f0a 4669 6c65 3a20 736b \"..]....File: sk\n-000026c0: 6574 6368 2e69 6e66 6f2c 2020 4e6f 6465 etch.info, Node\n-000026d0: 3a20 496e 7472 6f64 7563 7469 6f6e 2062 : Introduction b\n-000026e0: 7920 6578 616d 706c 652c 2020 4e65 7874 y example, Next\n-000026f0: 3a20 496e 7075 7420 6c61 6e67 7561 6765 : Input language\n-00002700: 2c20 2050 7265 763a 2041 626f 7574 2073 , Prev: About s\n-00002710: 6b65 7463 682c 2020 5570 3a20 546f 700a ketch, Up: Top.\n-00002720: 0a32 2049 6e74 726f 6475 6374 696f 6e20 .2 Introduction \n-00002730: 6279 2065 7861 6d70 6c65 0a2a 2a2a 2a2a by example.*****\n+000021d0: 3d3d 3d3d 3d0a 0a54 6865 2067 726f 7570 =====..The group\n+000021e0: 2068 7474 703a 2f2f 6772 6f75 7073 2e67 http://groups.g\n+000021f0: 6f6f 676c 652e 636f 6d2f 6772 6f75 702f oogle.com/group/\n+00002200: 736b 6574 6368 2d75 7365 7273 2069 7320 sketch-users is \n+00002210: 7468 6520 6265 7374 2070 6c61 6365 0a74 the best place.t\n+00002220: 6f20 7265 706f 7274 2062 7567 7320 616e o report bugs an\n+00002230: 6420 6d61 6b65 2069 6d70 726f 7665 6d65 d make improveme\n+00002240: 6e74 732e 2020 4120 7365 636f 6e64 206d nts. A second m\n+00002250: 6574 686f 6420 7468 6174 2077 696c 6c0a ethod that will.\n+00002260: 7072 6f62 6162 6c79 2070 726f 6475 6365 probably produce\n+00002270: 2061 2073 6c6f 7765 7220 7265 7370 6f6e a slower respon\n+00002280: 7365 2069 7320 656d 6169 6c20 746f 2073 se is email to s\n+00002290: 6b65 7463 6840 6672 6f6e 7469 6572 6e65 ketch@frontierne\n+000022a0: 742e 6e65 742e 0a57 6520 7769 6c6c 2074 t.net..We will t\n+000022b0: 7279 2074 6f20 7265 7370 6f6e 642c 2062 ry to respond, b\n+000022c0: 7574 2063 616e 2774 2070 726f 6d69 7365 ut can't promise\n+000022d0: 2e20 2049 6e20 616e 7920 6576 656e 742c . In any event,\n+000022e0: 2064 6f6e 2774 2062 650a 6f66 6665 6e64 don't be.offend\n+000022f0: 6564 2069 6620 6120 7265 706c 7920 6973 ed if a reply is\n+00002300: 206e 6f74 2066 6f72 7468 636f 6d69 6e67 not forthcoming\n+00002310: 2e20 2057 6527 7265 206a 7573 7420 6275 . We're just bu\n+00002320: 7379 2061 6e64 2077 696c 6c20 6765 7420 sy and will get \n+00002330: 746f 0a79 6f75 7220 7375 6767 6573 7469 to.your suggesti\n+00002340: 6f6e 2065 7665 6e74 7561 6c6c 792e 0a0a on eventually...\n+00002350: 2020 2046 6f72 2062 7567 732c 2061 7474 For bugs, att\n+00002360: 6163 6820 6120 e280 9873 6b65 7463 68e2 ach a ...sketch.\n+00002370: 8099 2069 6e70 7574 2066 696c 6520 7468 .. input file th\n+00002380: 6174 2063 6175 7365 7320 7468 6520 6261 at causes the ba\n+00002390: 6420 6265 6861 7669 6f72 2e0a 456d 6265 d behavior..Embe\n+000023a0: 6420 636f 6d6d 656e 7473 2074 6861 7420 d comments that \n+000023b0: 6578 706c 6169 6e20 7768 6174 2074 6f20 explain what to \n+000023c0: 6c6f 6f6b 2066 6f72 2069 6e20 7468 6520 look for in the \n+000023d0: 6265 6861 7669 6f72 206f 6620 e280 9873 behavior of ...s\n+000023e0: 6b65 7463 68e2 8099 0a6f 7220 6974 7320 ketch....or its \n+000023f0: 6f75 7470 7574 2e0a 0a20 2020 4120 7265 output... A re\n+00002400: 636f 6d6d 656e 6461 7469 6f6e 2066 6f72 commendation for\n+00002410: 2069 6d70 726f 7665 6d65 6e74 2066 726f improvement fro\n+00002420: 6d20 6f6e 6520 756e 6b6e 6f77 6e20 7065 m one unknown pe\n+00002430: 7273 6f6e 2063 6f75 6e74 7320 6173 0a6f rson counts as.o\n+00002440: 6e65 2076 6f74 652e 2020 5765 2075 7365 ne vote. We use\n+00002450: 206f 7665 7261 6c6c 2076 6f74 6520 7461 overall vote ta\n+00002460: 6c6c 6965 7320 746f 2064 6563 6964 6520 llies to decide \n+00002470: 7768 6174 2074 6f20 646f 206e 6578 7420 what to do next \n+00002480: 6173 0a72 6573 6f75 7263 6573 2070 6572 as.resources per\n+00002490: 6d69 742e 2020 5765 2072 6573 6572 7665 mit. 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If y\n-00002e70: 6f75 2776 6520 6e6f 7469 6365 6420 7468 ou've noticed th\n-00002e80: 6174 2074 6865 2070 726f 6a65 6374 696f at the projectio\n-00002e90: 6e20 7765 2061 7265 2075 7369 6e67 2073 n we are using s\n-00002ea0: 6565 6d73 2065 7175 6976 616c 656e 740a eems equivalent.\n-00002eb0: 746f 2065 7261 7369 6e67 2074 6865 207a to erasing the z\n-00002ec0: 2d63 6f6f 7264 696e 6174 6520 6f66 2074 -coordinate of t\n-00002ed0: 6865 2074 6872 6565 2d64 696d 656e 7369 he three-dimensi\n-00002ee0: 6f6e 616c 2069 6e70 7574 2070 6f69 6e74 onal input point\n-00002ef0: 732c 2070 6174 0a79 6f75 7273 656c 6620 s, pat.yourself \n-00002f00: 6f6e 2074 6865 2062 6163 6b2e 2020 596f on the back. Yo\n-00002f10: 7520 6172 6520 636f 7272 6563 742e 2020 u are correct. \n-00002f20: 5468 6973 2069 7320 6361 6c6c 6564 2061 This is called a\n-00002f30: 20e2 809c 7061 7261 6c6c 656c 0a70 726f ...parallel.pro\n-00002f40: 6a65 6374 696f 6ee2 809d 2e20 2054 6865 jection.... The\n-00002f50: 207a 2d63 6f6f 7264 696e 6174 6520 6178 z-coordinate ax\n-00002f60: 6973 2069 7320 706f 696e 7469 6e67 2073 is is pointing s\n-00002f70: 7472 6169 6768 7420 6f75 7420 6f66 2074 traight out of t\n-00002f80: 6865 0a70 6170 6572 2061 7420 7573 2c20 he.paper at us, \n-00002f90: 7768 696c 6520 7468 6520 782d 2061 6e64 while the x- and\n-00002fa0: 2079 2d61 7865 7320 706f 696e 7420 746f y-axes point to\n-00002fb0: 2074 6865 2072 6967 6874 2061 6e64 2075 the right and u\n-00002fc0: 7020 6173 2075 7375 616c 2e0a 0a20 2020 p as usual... \n-00002fd0: 5468 6520 7265 7375 6c74 696e 6720 7069 The resulting pi\n-00002fe0: 6374 7572 6520 6669 6c65 2063 616e 2062 cture file can b\n-00002ff0: 6520 696e 636c 7564 6564 2069 6e20 6120 e included in a \n-00003000: 4c61 5465 5820 646f 6375 6d65 6e74 2077 LaTeX document w\n-00003010: 6974 680a 5c69 6e70 7574 7b73 696d 706c ith.\\input{simpl\n-00003020: 657d 2e20 2041 6c74 6572 6e61 7465 6c79 e}. Alternately\n-00003030: 2c20 6164 6469 6e67 2074 6865 2063 6f6d , adding the com\n-00003040: 6d61 6e64 206c 696e 6520 6f70 7469 6f6e mand line option\n-00003050: 20e2 8098 2d54 e280 9928 3129 0a63 6175 ...-T...(1).cau\n-00003060: 7365 7320 7468 6520 e280 9870 7370 6963 ses the ...pspic\n-00003070: 7475 7265 e280 9920 746f 2062 6520 7772 ture... to be wr\n-00003080: 6170 7065 6420 696e 2061 2073 686f 7274 apped in a short\n-00003090: 2062 7574 2063 6f6d 706c 6574 6520 646f but complete do\n-000030a0: 6375 6d65 6e74 2c0a 7265 6164 7920 746f cument,.ready to\n-000030b0: 2072 756e 2074 686f 7567 6820 4c61 5465 run though LaTe\n-000030c0: 582e 2020 496e 2061 2066 696e 6973 6865 X. In a finishe\n-000030d0: 642c 2074 7970 6573 6574 2064 6f63 756d d, typeset docum\n-000030e0: 656e 742c 2074 6865 2070 6963 7475 7265 ent, the picture\n-000030f0: 0a6c 6f6f 6b73 206c 696b 6520 7468 6973 .looks like this\n-00003100: 2e20 2028 5468 6520 6178 6573 2068 6176 . (The axes hav\n-00003110: 6520 6265 656e 2061 6464 6564 2069 6e20 e been added in \n-00003120: 6c69 6768 7420 6772 6179 2e29 0a0a 0008 light gray.)....\n-00003130: 5b69 6d61 6765 2073 7263 3d22 6578 3032 [image src=\"ex02\n-00003140: 302e 706e 6722 2074 6578 743d 2249 6d61 0.png\" text=\"Ima\n-00003150: 6765 2065 7830 3230 206f 6d69 7474 6564 ge ex020 omitted\n-00003160: 2069 6e20 7465 7874 2076 6572 7369 6f6e in text version\n-00003170: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen\n-00003180: 742e 2200 085d 0a0a 4974 2069 7320 696d t.\"..]..It is im\n-00003190: 706f 7274 616e 7420 746f 206b 6e6f 7720 portant to know \n-000031a0: 7468 6174 206f 6e6c 7920 7468 6520 226f that only the \"o\n-000031b0: 7574 7369 6465 2220 6f66 2061 2070 6f6c utside\" of a pol\n-000031c0: 7967 6f6e 2069 7320 6e6f 726d 616c 6c79 ygon is normally\n-000031d0: 0a64 7261 776e 2e20 2054 6865 20e2 809c .drawn. The ...\n-000031e0: 6f75 7473 6964 65e2 809d 2069 7320 7768 outside... is wh\n-000031f0: 6572 6520 7468 6520 7665 7274 6963 6573 ere the vertices\n-00003200: 2067 6976 656e 2069 6e20 7468 6520 e280 given in the ..\n-00003210: 9870 6f6c 7967 6f6e e280 990a 636f 6d6d .polygon....comm\n-00003220: 616e 6420 6170 7065 6172 2069 6e20 5f63 and appear in _c\n-00003230: 6f75 6e74 6572 2d63 6c6f 636b 7769 7365 ounter-clockwise\n-00003240: 5f20 6f72 6465 722e 2020 5468 7573 2c20 _ order. Thus, \n-00003250: 6966 2074 6865 2063 6f6d 6d61 6e64 2061 if the command a\n-00003260: 626f 7665 0a68 6164 2062 6565 6e0a 2020 bove.had been. \n-00003270: 706f 6c79 676f 6e28 302c 312c 3029 2831 polygon(0,1,0)(1\n-00003280: 2c30 2c30 2928 302c 302c 3129 0a74 6865 ,0,0)(0,0,1).the\n-00003290: 2070 6f6c 7967 6f6e 2077 6f75 6c64 206e polygon would n\n-000032a0: 6f74 2061 7070 6561 7220 696e 2074 6865 ot appear in the\n-000032b0: 2070 6963 7475 7265 2061 7420 616c 6c2e picture at all.\n-000032c0: 2020 4974 2077 6f75 6c64 2068 6176 6520 It would have \n-000032d0: 6265 656e 0ae2 809c 6375 6c6c 6564 e280 been....culled..\n-000032e0: 9d20 6672 6f6d 2074 6865 2073 6365 6e65 . from the scene\n-000032f0: 2e20 2054 6869 7320 6375 6c6c 696e 6720 . This culling \n-00003300: 6265 6861 7669 6f72 206d 6179 2073 6565 behavior may see\n-00003310: 6d20 7374 7261 6e67 652c 2062 7574 0a73 m strange, but.s\n-00003320: 7461 7920 7475 6e65 642e 0a0a 2020 202d tay tuned... -\n-00003330: 2d2d 2d2d 2d2d 2d2d 2d20 466f 6f74 6e6f --------- Footno\n-00003340: 7465 7320 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a tes ----------..\n-00003350: 2020 2028 3129 204f 7220 666f 7220 4575 (1) Or for Eu\n-00003360: 726f 7065 616e 2075 7365 7273 206f 6620 ropean users of \n-00003370: 4134 2073 697a 6520 7061 7065 722c 20e2 A4 size paper, .\n-00003380: 8098 2d54 65e2 8099 2e0a 0a1f 0a46 696c ..-Te........Fil\n-00003390: 653a 2073 6b65 7463 682e 696e 666f 2c20 e: sketch.info, \n-000033a0: 204e 6f64 653a 2044 7261 7769 6e67 206f Node: Drawing o\n-000033b0: 7074 696f 6e73 2c20 204e 6578 743a 2044 ptions, Next: D\n-000033c0: 7261 7769 6e67 2061 2073 6f6c 6964 2c20 rawing a solid, \n-000033d0: 2050 7265 763a 2048 656c 6c6f 2077 6f72 Prev: Hello wor\n-000033e0: 6c64 2c20 2055 703a 2049 6e74 726f 6475 ld, Up: Introdu\n-000033f0: 6374 696f 6e20 6279 2065 7861 6d70 6c65 ction by example\n-00003400: 0a0a 322e 3220 4f70 7469 6f6e 730a 3d3d ..2.2 Options.==\n-00003410: 3d3d 3d3d 3d3d 3d3d 3d0a 0a4d 616e 7920 =========..Many \n-00003420: e280 9850 5354 7269 636b 73e2 8099 2061 ...PSTricks... a\n-00003430: 6e64 20e2 8098 5469 6b5a 2f50 4746 e280 nd ...TikZ/PGF..\n-00003440: 9920 6f70 7469 6f6e 7320 776f 726b 206a . options work j\n-00003450: 7573 7420 6669 6e65 2069 6e20 e280 9873 ust fine in ...s\n-00003460: 6b65 7463 68e2 8099 2e20 2049 660a 6765 ketch.... If.ge\n-00003470: 6e65 7261 7469 6e67 20e2 8098 5053 5472 nerating ...PSTr\n-00003480: 6963 6b73 e280 992c 2074 6865 2063 6f64 icks..., the cod\n-00003490: 650a 2020 706f 6c79 676f 6e5b 6669 6c6c e. polygon[fill\n-000034a0: 636f 6c6f 723d 6c69 6768 7467 7261 792c color=lightgray,\n-000034b0: 6c69 6e65 7769 6474 683d 3370 745d 2830 linewidth=3pt](0\n-000034c0: 2c30 2c31 2928 312c 302c 3029 2830 2c31 ,0,1)(1,0,0)(0,1\n-000034d0: 2c30 290a 2020 6c69 6e65 5b6c 696e 6573 ,0). line[lines\n-000034e0: 7479 6c65 3d64 6f74 7465 645d 282d 312c tyle=dotted](-1,\n-000034f0: 2d31 2c2d 3129 2832 2c32 2c32 290a 7072 -1,-1)(2,2,2).pr\n-00003500: 6f64 7563 6573 0a0a 0008 5b69 6d61 6765 oduces....[image\n-00003510: 2073 7263 3d22 6578 3033 302e 706e 6722 src=\"ex030.png\"\n-00003520: 2074 6578 743d 2249 6d61 6765 2065 7830 text=\"Image ex0\n-00003530: 3330 206f 6d69 7474 6564 2069 6e20 7465 30 omitted in te\n-00003540: 7874 2076 6572 7369 6f6e 206f 6620 7468 xt version of th\n-00003550: 6973 2064 6f63 756d 656e 742e 2200 085d is document.\"..]\n-00003560: 0a0a 2020 2054 6f20 7072 6f64 7563 6520 .. To produce \n-00003570: e280 9854 696b 5a2f 5047 46e2 8099 2c20 ...TikZ/PGF..., \n-00003580: 7468 6520 636f 7272 6573 706f 6e64 696e the correspondin\n-00003590: 6720 636f 6465 2069 730a 2020 706f 6c79 g code is. poly\n-000035a0: 676f 6e5b 6669 6c6c 3d6c 6967 6874 6772 gon[fill=lightgr\n-000035b0: 6179 2c6c 696e 6520 7769 6474 683d 3370 ay,line width=3p\n-000035c0: 745d 2830 2c30 2c31 2928 312c 302c 3029 t](0,0,1)(1,0,0)\n-000035d0: 2830 2c31 2c30 290a 2020 6c69 6e65 5b73 (0,1,0). line[s\n-000035e0: 7479 6c65 3d64 6f74 7465 645d 282d 312c tyle=dotted](-1,\n-000035f0: 2d31 2c2d 3129 2832 2c32 2c32 290a 2020 -1,-1)(2,2,2). \n-00003600: 676c 6f62 616c 207b 206c 616e 6775 6167 global { languag\n-00003610: 6520 7469 6b7a 207d 0a54 6865 2066 696e e tikz }.The fin\n-00003620: 616c 20e2 8098 676c 6f62 616c e280 9920 al ...global... \n-00003630: 696e 7374 7275 6374 7320 e280 9873 6b65 instructs ...ske\n-00003640: 7463 68e2 8099 2074 6f20 7072 6f64 7563 tch... to produc\n-00003650: 6520 e280 9854 696b 5a2f 5047 46e2 8099 e ...TikZ/PGF...\n-00003660: 2063 6f64 6520 6173 0a6f 7574 7075 7420 code as.output \n-00003670: 7261 7468 6572 2074 6861 6e20 7468 6520 rather than the \n-00003680: 6465 6661 756c 742c 20e2 8098 5053 5472 default, ...PSTr\n-00003690: 6963 6b73 e280 992e 2020 4e6f 7465 2074 icks.... Note t\n-000036a0: 6861 7420 e280 9870 6f6c 7967 6f6e e280 hat ...polygon..\n-000036b0: 9920 6669 6c6c 0a63 6f6c 6f72 2061 6e64 . fill.color and\n-000036c0: 20e2 8098 6c69 6e65 e280 9920 7374 796c ...line... styl\n-000036d0: 6520 6f70 7469 6f6e 7320 626f 7468 2063 e options both c\n-000036e0: 6f6e 666f 726d 2074 6f20 e280 9854 696b onform to ...Tik\n-000036f0: 5ae2 8099 2073 796e 7461 7820 7275 6c65 Z... syntax rule\n-00003700: 732e 2020 5468 650a 7265 7375 6c74 696e s. The.resultin\n-00003710: 6720 e280 9854 696b 5a2f 5047 46e2 8099 g ...TikZ/PGF...\n-00003720: 206f 7574 7075 7420 6973 0a20 205c 6265 output is. \\be\n-00003730: 6769 6e7b 7469 6b7a 7069 6374 7572 657d gin{tikzpicture}\n-00003740: 5b6a 6f69 6e3d 726f 756e 645d 0a20 205c [join=round]. \\\n-00003750: 6472 6177 5b64 6f74 7465 645d 282d 312c draw[dotted](-1,\n-00003760: 2d31 292d 2d28 2e33 3333 2c2e 3333 3329 -1)--(.333,.333)\n-00003770: 3b0a 2020 5c66 696c 6c64 7261 775b 6669 ;. \\filldraw[fi\n-00003780: 6c6c 3d6c 6967 6874 6772 6179 2c6c 696e ll=lightgray,lin\n-00003790: 6520 7769 6474 683d 3370 745d 2830 2c30 e width=3pt](0,0\n-000037a0: 292d 2d28 312c 3029 2d2d 2830 2c31 292d )--(1,0)--(0,1)-\n-000037b0: 2d63 7963 6c65 3b0a 2020 5c64 7261 775b -cycle;. \\draw[\n-000037c0: 646f 7474 6564 5d28 2e33 3333 2c2e 3333 dotted](.333,.33\n-000037d0: 3329 2d2d 2832 2c32 293b 0a20 205c 656e 3)--(2,2);. \\en\n-000037e0: 647b 7469 6b7a 7069 6374 7572 657d 0a54 d{tikzpicture}.T\n-000037f0: 6865 2072 656d 6169 6e69 6e67 2065 7861 he remaining exa\n-00003800: 6d70 6c65 7320 6f66 2074 6869 7320 6d61 mples of this ma\n-00003810: 6e75 616c 2061 7265 2069 6e20 5053 5472 nual are in PSTr\n-00003820: 6963 6b73 2073 7479 6c65 2e0a 0a1f 0a46 icks style.....F\n-00003830: 696c 653a 2073 6b65 7463 682e 696e 666f ile: sketch.info\n-00003840: 2c20 204e 6f64 653a 2044 7261 7769 6e67 , Node: Drawing\n-00003850: 2061 2073 6f6c 6964 2c20 204e 6578 743a a solid, Next:\n-00003860: 2053 7065 6369 616c 206f 626a 6563 7473 Special objects\n-00003870: 2c20 2050 7265 763a 2044 7261 7769 6e67 , Prev: Drawing\n-00003880: 206f 7074 696f 6e73 2c20 2055 703a 2049 options, Up: I\n-00003890: 6e74 726f 6475 6374 696f 6e20 6279 2065 ntroduction by e\n-000038a0: 7861 6d70 6c65 0a0a 322e 3320 4472 6177 xample..2.3 Draw\n-000038b0: 696e 6720 6120 736f 6c69 640a 3d3d 3d3d ing a solid.====\n-000038c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============.\n-000038d0: 0a4c 6574 2773 2074 7279 2073 6f6d 6574 .Let's try somet\n-000038e0: 6869 6e67 206d 6f72 6520 6578 6369 7469 hing more exciti\n-000038f0: 6e67 2e20 20e2 8098 536b 6574 6368 e280 ng. ...Sketch..\n-00003900: 9920 6861 7320 6e6f 206e 6f74 696f 6e20 . has no notion \n-00003910: 6f66 2061 2073 6f6c 6964 2c0a 6275 7420 of a solid,.but \n-00003920: 706f 6c79 676f 6e61 6c20 e280 9c66 6163 polygonal ...fac\n-00003930: 6573 e280 9d20 6361 6e20 6265 2075 7365 es... can be use\n-00003940: 6420 746f 2072 6570 7265 7365 6e74 2074 d to represent t\n-00003950: 6865 2062 6f75 6e64 6172 7920 6f66 2061 he boundary of a\n-00003960: 2073 6f6c 6964 2e0a 546f 2074 6865 2070 solid..To the p\n-00003970: 7265 7669 6f75 7320 6578 616d 706c 652c revious example,\n-00003980: 206c 6574 2773 2061 6464 2074 6872 6565 let's add three\n-00003990: 206d 6f72 6520 7472 6961 6e67 756c 6172 more triangular\n-000039a0: 2070 6f6c 7967 6f6e 7320 746f 0a6d 616b polygons to.mak\n-000039b0: 6520 7468 6520 6661 6365 7320 6f66 2061 e the faces of a\n-000039c0: 6e20 6972 7265 6775 6c61 7220 7465 7472 n irregular tetr\n-000039d0: 6168 6564 726f 6e2e 0a20 2025 2076 6572 ahedron.. % ver\n-000039e0: 7469 6365 7320 6f66 2074 6865 2074 6574 tices of the tet\n-000039f0: 7261 6865 6472 6f6e 0a20 2064 6566 2070 rahedron. def p\n-00003a00: 3120 2830 2c30 2c31 2920 6465 6620 7032 1 (0,0,1) def p2\n-00003a10: 2028 312c 302c 3029 0a20 2064 6566 2070 (1,0,0). def p\n-00003a20: 3320 2830 2c31 2c30 2920 6465 6620 7034 3 (0,1,0) def p4\n-00003a30: 2028 2d2e 332c 2d2e 352c 2d2e 3829 0a0a (-.3,-.5,-.8)..\n-00003a40: 2020 2520 6661 6365 7320 6f66 2074 6865 % faces of the\n-00003a50: 2074 6574 7261 6865 6472 6f6e 2e0a 2020 tetrahedron.. \n-00003a60: 706f 6c79 676f 6e28 7031 2928 7032 2928 polygon(p1)(p2)(\n-00003a70: 7033 2920 2520 6f72 6967 696e 616c 2066 p3) % original f\n-00003a80: 726f 6e74 2070 6f6c 7967 6f6e 0a20 2070 ront polygon. p\n-00003a90: 6f6c 7967 6f6e 2870 3129 2870 3429 2870 olygon(p1)(p4)(p\n-00003aa0: 3229 2025 2062 6f74 746f 6d0a 2020 706f 2) % bottom. po\n-00003ab0: 6c79 676f 6e28 7031 2928 7033 2928 7034 lygon(p1)(p3)(p4\n-00003ac0: 2920 2520 6c65 6674 0a20 2070 6f6c 7967 ) % left. polyg\n-00003ad0: 6f6e 2870 3329 2870 3229 2870 3429 2025 on(p3)(p2)(p4) %\n-00003ae0: 2072 6561 720a 0a20 2025 206c 696e 6520 rear.. % line \n-00003af0: 746f 2070 6965 7263 6520 7468 6520 7465 to pierce the te\n-00003b00: 7472 6168 6564 726f 6e0a 2020 6c69 6e65 trahedron. line\n-00003b10: 5b6c 696e 6563 6f6c 6f72 3d72 6564 5d28 [linecolor=red](\n-00003b20: 2d31 2c2d 312c 2d31 2928 322c 322c 3229 -1,-1,-1)(2,2,2)\n-00003b30: 0a54 6869 7320 6578 616d 706c 6520 7573 .This example us\n-00003b40: 6573 20e2 809c 6465 6669 6e69 7469 6f6e es ...definition\n-00003b50: 73e2 809d 2c20 7768 6963 6820 6265 6769 s..., which begi\n-00003b60: 6e20 7769 7468 20e2 8098 6465 66e2 8099 n with ...def...\n-00003b70: 2e20 2054 6865 7365 20e2 809c 6465 6669 . These ...defi\n-00003b80: 6e65 e280 9d0a 6f72 2067 6976 6520 6e61 ne....or give na\n-00003b90: 6d65 7320 746f 2070 6f69 6e74 732c 2077 mes to points, w\n-00003ba0: 6869 6368 2061 7265 2074 6865 6e20 6176 hich are then av\n-00003bb0: 6169 6c61 626c 6520 6173 20e2 809c 7265 ailable as ...re\n-00003bc0: 6665 7265 6e63 6573 e280 9d20 6279 0a65 ferences... by.e\n-00003bd0: 6e63 6c6f 7369 6e67 2074 6865 206e 616d nclosing the nam\n-00003be0: 6573 2069 6e20 7061 7265 6e74 6865 7365 es in parenthese\n-00003bf0: 732c 2065 2e67 2e20 2866 6f6f 292e 2020 s, e.g. (foo). \n-00003c00: 5468 6520 7061 7265 6e74 6865 7365 7320 The parentheses \n-00003c10: 6465 6e6f 7465 0a74 6861 7420 7468 6520 denote.that the \n-00003c20: 6e61 6d65 7320 7265 6665 7220 746f 2070 names refer to p\n-00003c30: 6f69 6e74 733b 2074 6865 7920 6172 6520 oints; they are \n-00003c40: 7265 7175 6972 6564 2e20 2054 6865 7265 required. There\n-00003c50: 2063 616e 2062 6520 6e6f 0a77 6869 7465 can be no.white\n-00003c60: 2073 7061 6365 2062 6574 7765 656e 2074 space between t\n-00003c70: 6865 6d20 616e 6420 7468 6520 6e61 6d65 hem and the name\n-00003c80: 2e0a 0a20 2020 4173 2079 6f75 2063 616e ... As you can\n-00003c90: 2073 6565 2c20 636f 6d6d 656e 7473 2073 see, comments s\n-00003ca0: 7461 7274 2077 6974 6820 2520 6173 2069 tart with % as i\n-00003cb0: 6e20 5465 5820 616e 6420 6578 7465 6e64 n TeX and extend\n-00003cc0: 2074 6f20 7468 6520 656e 640a 6f66 2074 to the end.of t\n-00003cd0: 6865 206c 696e 6520 2874 686f 7567 6820 he line (though \n-00003ce0: 2320 7769 6c6c 2077 6f72 6b20 6173 2077 # will work as w\n-00003cf0: 656c 6c29 2e20 2057 6869 7465 2073 7061 ell). White spa\n-00003d00: 6365 2c20 696e 636c 7564 696e 670a 7370 ce, including.sp\n-00003d10: 6163 6573 2c20 7461 6273 2061 6e64 2062 aces, tabs and b\n-00003d20: 6c61 6e6b 206c 696e 6573 2c20 6861 7320 lank lines, has \n-00003d30: 6e6f 2065 6666 6563 7420 696e 2074 6865 no effect in the\n-00003d40: 20e2 8098 736b 6574 6368 e280 9920 6c61 ...sketch... la\n-00003d50: 6e67 7561 6765 2e0a 0a00 085b 696d 6167 nguage.....[imag\n-00003d60: 6520 7372 633d 2265 7830 3430 2e70 6e67 e src=\"ex040.png\n-00003d70: 2220 7465 7874 3d22 496d 6167 6520 6578 \" text=\"Image ex\n-00003d80: 3034 3020 6f6d 6974 7465 6420 696e 2074 040 omitted in t\n-00003d90: 6578 7420 7665 7273 696f 6e20 6f66 2074 ext version of t\n-00003da0: 6869 7320 646f 6375 6d65 6e74 2e22 0008 his document.\"..\n-00003db0: 5d0a 0a49 6620 7765 206c 6f6f 6b20 696e ]..If we look in\n-00003dc0: 7369 6465 2074 6865 2054 6558 2066 696c side the TeX fil\n-00003dd0: 6520 7072 6f64 7563 6564 2062 7920 e280 e produced by ..\n-00003de0: 9873 6b65 7463 68e2 8099 2c20 7468 6572 .sketch..., ther\n-00003df0: 6520 7769 6c6c 2062 6520 6f6e 6c79 0a74 e will be only.t\n-00003e00: 6872 6565 2070 6f6c 7967 6f6e 732e 2020 hree polygons. \n-00003e10: 5468 6520 666f 7572 7468 2068 6173 2062 The fourth has b\n-00003e20: 6565 6e20 6375 6c6c 6564 2062 6563 6175 een culled becau\n-00003e30: 7365 2069 7420 6973 2061 2022 6261 636b se it is a \"back\n-00003e40: 2066 6163 6522 0a6f 6620 7468 6520 7465 face\".of the te\n-00003e50: 7472 6168 6564 726f 6e2c 2069 6e76 6973 trahedron, invis\n-00003e60: 6962 6c65 2074 6f20 6f75 7220 7669 6577 ible to our view\n-00003e70: 2e20 2049 7420 6973 2075 6e6e 6563 6573 . 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For zero-thi\n-00003f50: 636b 6e65 7373 2073 6f6c 6964 732c 2063 ckness solids, c\n-00003f60: 756c 6c69 6e67 2069 7320 6120 7072 6f62 ulling is a prob\n-00003f70: 6c65 6d2e 0a4f 6e65 2073 6f6c 7574 696f lem..One solutio\n-00003f80: 6e20 6973 2074 6f20 7573 6520 6120 7061 n is to use a pa\n-00003f90: 6972 206f 6620 e280 9873 6b65 7463 68e2 ir of ...sketch.\n-00003fa0: 8099 2070 6f6c 7967 6f6e 7320 666f 7220 .. polygons for \n-00003fb0: 6561 6368 0a7a 6572 6f2d 7468 6963 6b6e each.zero-thickn\n-00003fc0: 6573 7320 6661 6365 2c20 6964 656e 7469 ess face, identi\n-00003fd0: 6361 6c20 6578 6365 7074 2077 6974 6820 cal except with \n-00003fe0: 6f70 706f 7369 7465 2076 6572 7465 7820 opposite vertex \n-00003ff0: 6f72 6465 7273 2e20 2054 6869 730a 6973 orders. This.is\n-00004000: 2075 6e77 6965 6c64 7920 616e 6420 6578 unwieldy and ex\n-00004010: 7065 6e73 6976 652e 2020 4120 6265 7474 pensive. A bett\n-00004020: 6572 2077 6179 2069 7320 746f 2073 6574 er way is to set\n-00004030: 2074 6865 20e2 8098 736b 6574 6368 e280 the ...sketch..\n-00004040: 9920 696e 7465 726e 616c 0a6f 7074 696f . internal.optio\n-00004050: 6e20 e280 9863 756c 6ce2 8099 2074 6f20 n ...cull... to \n-00004060: e280 9866 616c 7365 e280 9920 696e 2074 ...false... in t\n-00004070: 6865 2075 7375 616c 20e2 8098 5053 5472 he usual ...PSTr\n-00004080: 6963 6b73 e280 9920 6d61 6e6e 6572 2e0a icks... manner..\n-00004090: 2020 706f 6c79 676f 6e5b 6375 6c6c 3d66 polygon[cull=f\n-000040a0: 616c 7365 5d28 7031 2928 7032 2928 7033 alse](p1)(p2)(p3\n-000040b0: 290a 5468 6520 666f 6c6c 6f77 696e 6720 ).The following \n-000040c0: 7368 6f77 7320 7468 6520 7361 6d65 2068 shows the same h\n-000040d0: 656c 6978 2073 6861 7065 2064 7261 776e elix shape drawn\n-000040e0: 2066 6972 7374 2077 6974 6820 6375 6c6c first with cull\n-000040f0: 3d74 7275 6520 2874 6865 0a64 6566 6175 =true (the.defau\n-00004100: 6c74 2920 616e 6420 7468 656e 2063 756c lt) and then cul\n-00004110: 6c3d 6661 6c73 652e 0a0a 0008 5b69 6d61 l=false.....[ima\n-00004120: 6765 2073 7263 3d22 6578 3034 352e 706e ge src=\"ex045.pn\n-00004130: 6722 2074 6578 743d 2249 6d61 6765 2065 g\" text=\"Image e\n-00004140: 7830 3435 206f 6d69 7474 6564 2069 6e20 x045 omitted in \n-00004150: 7465 7874 2076 6572 7369 6f6e 206f 6620 text version of \n-00004160: 7468 6973 2064 6f63 756d 656e 742e 2200 this document.\".\n-00004170: 085d 0a0a 5765 276c 6c20 736f 6f6e 2073 .]..We'll soon s\n-00004180: 6565 2068 6f77 2074 6f20 7072 6f64 7563 ee how to produc\n-00004190: 6520 7468 6573 6520 6865 6c69 7865 7320 e these helixes \n-000041a0: 7769 7468 2061 2066 6577 206c 696e 6573 with a few lines\n-000041b0: 206f 6620 e280 9873 6b65 7463 68e2 8099 of ...sketch...\n-000041c0: 0a6c 616e 6775 6167 6520 636f 6465 2e0a .language code..\n-000041d0: 0a20 2020 4974 206d 6179 2062 6520 7465 . It may be te\n-000041e0: 6d70 7469 6e67 2074 6f20 7475 726e 2063 mpting to turn c\n-000041f0: 756c 6c69 6e67 206f 6666 2067 7261 7475 ulling off gratu\n-00004200: 6974 6f75 736c 7920 736f 2074 6861 7420 itously so that \n-00004210: 7665 7274 6578 0a6f 7264 6572 2063 616e vertex.order can\n-00004220: 2062 6520 6967 6e6f 7265 642e 2020 5468 be ignored. Th\n-00004230: 6973 2069 7320 6e6f 7420 6120 676f 6f64 is is not a good\n-00004240: 2069 6465 6120 6265 6361 7573 6520 6f75 idea because ou\n-00004250: 7470 7574 2066 696c 6520 7369 7a65 0a61 tput file size.a\n-00004260: 6e64 2054 6558 2061 6e64 2050 6f73 7473 nd TeX and Posts\n-00004270: 6372 6970 7420 7072 6f63 6573 7369 6e67 cript processing\n-00004280: 2074 696d 6520 626f 7468 2064 6570 656e time both depen\n-00004290: 6420 6f6e 2074 6865 206e 756d 6265 7220 d on the number \n-000042a0: 6f66 0a6f 7574 7075 7420 706f 6c79 676f of.output polygo\n-000042b0: 6e73 2e20 2043 756c 6c69 6e67 2075 7375 ns. 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See *note Gl\n-00004350: 6f62 616c 206f 7074 696f 6e73 3a3a 2061 obal options:: a\n-00004360: 6e64 202a 6e6f 7465 204c 696d 6974 7320 nd *note Limits \n-00004370: 6f6e 2065 7272 6f72 0a64 6574 6563 7469 on error.detecti\n-00004380: 6f6e 3a3a 2e0a 0a1f 0a46 696c 653a 2073 on::.....File: s\n-00004390: 6b65 7463 682e 696e 666f 2c20 204e 6f64 ketch.info, Nod\n-000043a0: 653a 2053 7065 6369 616c 206f 626a 6563 e: Special objec\n-000043b0: 7473 2c20 204e 6578 743a 204f 626a 6563 ts, Next: Objec\n-000043c0: 7420 7472 616e 7366 6f72 6d73 2c20 2050 t transforms, P\n-000043d0: 7265 763a 2044 7261 7769 6e67 2061 2073 rev: Drawing a s\n-000043e0: 6f6c 6964 2c20 2055 703a 2049 6e74 726f olid, Up: Intro\n-000043f0: 6475 6374 696f 6e20 6279 2065 7861 6d70 duction by examp\n-00004400: 6c65 0a0a 322e 3420 5370 6563 6961 6c20 le..2.4 Special \n-00004410: 6f62 6a65 6374 730a 3d3d 3d3d 3d3d 3d3d objects.========\n-00004420: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a57 6520 ===========..We \n-00004430: 6361 6e20 6164 6420 6c61 6265 6c73 2074 can add labels t\n-00004440: 6f20 6120 6472 6177 696e 6720 6279 2075 o a drawing by u\n-00004450: 7369 6e67 2073 7065 6369 616c 206f 626a sing special obj\n-00004460: 6563 7473 2c20 7768 6963 6820 7072 6f76 ects, which prov\n-00004470: 6964 6520 610a 7761 7920 746f 2065 6d62 ide a.way to emb\n-00004480: 6564 2072 6177 204c 6154 6558 2061 6e64 ed raw LaTeX and\n-00004490: 20e2 8098 5053 5472 6963 6b73 e280 9920 ...PSTricks... \n-000044a0: 636f 6465 2e20 2041 6464 696e 6720 7468 code. Adding th\n-000044b0: 6973 2074 6f20 7468 650a 7465 7472 6168 is to the.tetrah\n-000044c0: 6564 726f 6e20 646f 6573 2074 6865 2074 edron does the t\n-000044d0: 7269 636b 2e0a 2020 7370 6563 6961 6c20 rick.. special \n-000044e0: 7c5c 666f 6f74 6e6f 7465 7369 7a65 0a20 |\\footnotesize. \n-000044f0: 2020 2020 2020 2020 2020 5c75 7075 747b \\uput{\n-00004500: 3270 747d 5b75 725d 2331 7b24 5031 247d 2pt}[ur]#1{$P1$}\n-00004510: 0a20 2020 2020 2020 2020 2020 5c75 7075 . \\upu\n-00004520: 745b 725d 2332 7b24 5032 247d 0a20 2020 t[r]#2{$P2$}. \n-00004530: 2020 2020 2020 2020 5c75 7075 745b 755d \\uput[u]\n-00004540: 2333 7b24 5033 247d 0a20 2020 2020 2020 #3{$P3$}. \n-00004550: 2020 2020 5c75 7075 745b 645d 2334 7b24 \\uput[d]#4{$\n-00004560: 5034 247d 7c0a 2020 2020 2870 3129 2870 P4$}|. (p1)(p\n-00004570: 3229 2870 3329 2870 3429 0a48 6572 6520 2)(p3)(p4).Here \n-00004580: 6973 2074 6865 2072 6573 756c 742e 0a0a is the result...\n-00004590: 0008 5b69 6d61 6765 2073 7263 3d22 6578 ..[image src=\"ex\n-000045a0: 3034 322e 706e 6722 2074 6578 743d 2249 042.png\" text=\"I\n-000045b0: 6d61 6765 2065 7830 3432 206f 6d69 7474 mage ex042 omitt\n-000045c0: 6564 2069 6e20 7465 7874 2076 6572 7369 ed in text versi\n-000045d0: 6f6e 206f 6620 7468 6973 2064 6f63 756d on of this docum\n-000045e0: 656e 742e 2200 085d 0a0a 2020 2054 6865 ent.\"..].. The\n-000045f0: 7265 2061 7265 2073 6576 6572 616c 2064 re are several d\n-00004600: 6574 6169 6c73 2074 6f20 6e6f 7465 2068 etails to note h\n-00004610: 6572 652e 2020 4669 7273 742c 2074 6865 ere. First, the\n-00004620: 2071 756f 7469 6e67 0a63 6f6e 7665 6e74 quoting.convent\n-00004630: 696f 6e20 666f 7220 7468 6520 7261 7720 ion for the raw \n-00004640: 636f 6465 2069 7320 7369 6d69 6c61 7220 code is similar \n-00004650: 746f 2074 6865 204c 6154 6558 205c 7665 to the LaTeX \\ve\n-00004660: 7262 2063 6f6d 6d61 6e64 2e20 2054 6865 rb command. The\n-00004670: 0a66 6972 7374 206e 6f6e 2d77 6869 7465 .first non-white\n-00004680: 2073 7061 6365 2063 6861 7261 6374 6572 space character\n-00004690: 2066 6f6c 6c6f 7769 6e67 2073 7065 6369 following speci\n-000046a0: 616c 2069 7320 756e 6465 7273 746f 6f64 al is understood\n-000046b0: 2074 6f20 6265 0a74 6865 2071 756f 7465 to be.the quote\n-000046c0: 2063 6861 7261 6374 6572 2c20 696e 2074 character, in t\n-000046d0: 6869 7320 6361 7365 20e2 8098 7ce2 8099 his case ...|...\n-000046e0: 2e20 2054 6865 2072 6177 2074 6578 7420 . The raw text \n-000046f0: 636f 6e74 696e 7565 7320 756e 7469 6c0a continues until.\n-00004700: 7468 6973 2063 6861 7261 6374 6572 2072 this character r\n-00004710: 6563 7572 732e 0a0a 2020 2053 6563 6f6e ecurs... Secon\n-00004720: 642c 2074 6865 2061 7267 756d 656e 7420 d, the argument \n-00004730: 7265 6665 7265 6e63 6573 2023 312c 2023 references #1, #\n-00004740: 322c 2023 332c 2061 6e64 2023 3420 7265 2, #3, and #4 re\n-00004750: 6665 7220 746f 2070 6f69 6e74 2c0a 7665 fer to point,.ve\n-00004760: 6374 6f72 2c20 6f72 2073 6361 6c61 7220 ctor, or scalar \n-00004770: 7661 6c75 6573 2069 6e20 7468 6520 6c69 values in the li\n-00004780: 7374 2074 6861 7420 666f 6c6c 6f77 2e20 st that follow. \n-00004790: 2054 6869 7320 6973 2073 696d 696c 6172 This is similar\n-000047a0: 2074 6f0a 5465 5820 6d61 6372 6f20 7379 to.TeX macro sy\n-000047b0: 6e74 6178 2e20 2054 6865 2074 7261 6e73 ntax. The trans\n-000047c0: 666f 726d 6564 2061 6e64 2074 776f 2d64 formed and two-d\n-000047d0: 696d 656e 7369 6f6e 616c 2070 726f 6a65 imensional proje\n-000047e0: 6374 696f 6e73 206f 660a 7468 6573 6520 ctions of.these \n-000047f0: 7468 7265 652d 6469 6d65 6e73 696f 6e61 three-dimensiona\n-00004800: 6c20 706f 696e 7473 2061 7265 2073 7562 l points are sub\n-00004810: 7374 6974 7574 6564 2069 6e20 7468 6520 stituted in the \n-00004820: 6669 6e61 6c20 6f75 7470 7574 2e20 2041 final output. A\n-00004830: 6e0a 6172 6775 6d65 6e74 2072 6566 6572 n.argument refer\n-00004840: 656e 6365 206f 6620 7468 6520 666f 726d ence of the form\n-00004850: 2023 312d 3220 6973 2072 6570 6c61 6365 #1-2 is replace\n-00004860: 6420 7769 7468 2074 6865 2061 6e67 6c65 d with the angle\n-00004870: 2069 6e0a 6465 6772 6565 7320 6f66 2074 in.degrees of t\n-00004880: 6865 2074 776f 2d64 696d 656e 7369 6f6e he two-dimension\n-00004890: 616c 2076 6563 746f 7220 7468 6174 2063 al vector that c\n-000048a0: 6f6e 6e65 6374 7320 7468 6520 7072 6f6a onnects the proj\n-000048b0: 6563 7469 6f6e 7320 6f66 0a74 6865 2074 ections of.the t\n-000048c0: 776f 2072 6573 7065 6374 6976 6520 6172 wo respective ar\n-000048d0: 6775 6d65 6e74 2070 6f69 6e74 732c 2068 gument points, h\n-000048e0: 6572 6520 2331 2061 6e64 2023 322e 2020 ere #1 and #2. \n-000048f0: 5468 6520 7375 6273 7469 7475 7465 640a The substituted.\n-00004900: 616e 676c 6520 6973 2065 6e63 6c6f 7365 angle is enclose\n-00004910: 6420 696e 2063 7572 6c79 2062 7261 6365 d in curly brace\n-00004920: 7320 e280 987b 207d e280 992e 2020 5768 s ...{ }.... Wh\n-00004930: 656e 20e2 8098 5469 6b5a 2f50 4746 e280 en ...TikZ/PGF..\n-00004940: 9920 6f75 7470 7574 2069 730a 6265 696e . output is.bein\n-00004950: 6720 6765 6e65 7261 7465 642c 2074 6865 g generated, the\n-00004960: 2061 6e67 6c65 2069 7320 726f 756e 6465 angle is rounde\n-00004970: 6420 746f 2074 6865 206e 6561 7265 7374 d to the nearest\n-00004980: 2064 6567 7265 6520 6265 6361 7573 650a degree because.\n-00004990: 6e6f 6e2d 696e 7465 6765 7220 616e 676c non-integer angl\n-000049a0: 6573 2061 7265 206e 6f74 2061 6c6c 6f77 es are not allow\n-000049b0: 6564 2062 7920 e280 9854 696b 5a2f 5047 ed by ...TikZ/PG\n-000049c0: 46e2 8099 2070 7269 6d69 7469 7665 732e F... primitives.\n-000049d0: 0a0a 2020 2041 7320 6f66 2056 6572 7369 .. As of Versi\n-000049e0: 6f6e 2030 2e33 206f 6620 e280 9873 6b65 on 0.3 of ...ske\n-000049f0: 7463 68e2 8099 2c20 7370 6563 6961 6c20 tch..., special \n-00004a00: 6172 6775 6d65 6e74 7320 6d61 7920 6265 arguments may be\n-00004a10: 2073 6361 6c61 7273 206f 720a 7665 6374 scalars or.vect\n-00004a20: 6f72 7320 696e 2061 6464 6974 696f 6e20 ors in addition \n-00004a30: 746f 2070 6f69 6e74 732e 2020 5265 6665 to points. Refe\n-00004a40: 7265 6e63 6573 2074 6f20 7363 616c 6172 rences to scalar\n-00004a50: 2061 7267 756d 656e 7473 2061 7265 0a6d arguments are.m\n-00004a60: 6572 656c 7920 7265 706c 6163 6564 2077 erely replaced w\n-00004a70: 6974 6820 6120 6e75 6d62 6572 2066 6f72 ith a number for\n-00004a80: 6d61 7474 6564 206a 7573 7420 6173 2061 matted just as a\n-00004a90: 6e79 2070 6f69 6e74 2063 6f6f 7264 696e ny point coordin\n-00004aa0: 6174 652e 0a52 6566 6572 656e 6365 7320 ate..References \n-00004ab0: 746f 2076 6563 746f 7273 2062 6563 6f6d to vectors becom\n-00004ac0: 6520 7477 6f2d 6469 6d65 6e73 696f 6e61 e two-dimensiona\n-00004ad0: 6c20 706f 696e 7473 2e20 2054 6865 2074 l points. The t\n-00004ae0: 6963 6b20 6f70 6572 6174 6f72 0a74 6861 ick operator.tha\n-00004af0: 7420 7365 6c65 6374 7320 696e 6469 7669 t selects indivi\n-00004b00: 6475 616c 2063 6f6d 706f 6e65 6e74 7320 dual components \n-00004b10: 6f66 2070 6f69 6e74 7320 616e 6420 7665 of points and ve\n-00004b20: 6374 6f72 7320 656c 7365 7768 6572 6520 ctors elsewhere \n-00004b30: 696e 0ae2 8098 736b 6574 6368 e280 9920 in....sketch... \n-00004b40: 2873 6565 2066 6f72 2065 7861 6d70 6c65 (see for example\n-00004b50: 202a 6e6f 7465 2041 6666 696e 6520 6172 *note Affine ar\n-00004b60: 6974 686d 6574 6963 3a3a 2920 6361 6e20 ithmetic::) can \n-00004b70: 616c 736f 2062 6520 6170 706c 6965 640a also be applied.\n-00004b80: 746f 2070 6f69 6e74 2061 6e64 2076 6563 to point and vec\n-00004b90: 746f 7220 6172 6775 6d65 6e74 2072 6566 tor argument ref\n-00004ba0: 6572 656e 6365 732e 2020 416c 6c20 7468 erences. All th\n-00004bb0: 7265 6520 6469 6d65 6e73 696f 6e73 206f ree dimensions o\n-00004bc0: 6620 610a 7472 616e 7366 6f72 6d65 6420 f a.transformed \n-00004bd0: 706f 696e 7420 6f72 2076 6563 746f 7220 point or vector \n-00004be0: 6361 6e20 616c 736f 2062 6520 7375 6273 can also be subs\n-00004bf0: 7469 7475 6564 2077 6974 6820 e280 9827 titued with ...'\n-00004c00: 33e2 8099 2e20 2053 6565 202a 6e6f 7465 3.... See *note\n-00004c10: 0a53 7065 6369 616c 733a 3a20 666f 7220 .Specials:: for \n-00004c20: 6465 7461 696c 732e 0a0a 2020 2042 7920 details... By \n-00004c30: 6465 6661 756c 742c 20e2 8098 7370 6563 default, ...spec\n-00004c40: 6961 6ce2 8099 206f 626a 6563 7473 2061 ial... objects a\n-00004c50: 7265 2070 7269 6e74 6564 206c 6173 742c re printed last,\n-00004c60: 206f 7665 726c 6179 696e 6720 616c 6c20 overlaying all \n-00004c70: 6f74 6865 720a 6f62 6a65 6374 7320 696e other.objects in\n-00004c80: 2074 6865 2073 6365 6e65 2e20 2049 6620 the scene. If \n-00004c90: 796f 7520 7370 6563 6966 7920 7468 6520 you specify the \n-00004ca0: 696e 7465 726e 616c 206f 7074 696f 6e20 internal option \n-00004cb0: e280 986c 6179 3d69 6ee2 8099 2c20 7468 ...lay=in..., th\n-00004cc0: 650a 6869 6464 656e 2073 7572 6661 6365 e.hidden surface\n-00004cd0: 2061 6c67 6f72 6974 686d 2063 6f6e 7369 algorithm consi\n-00004ce0: 6465 7273 2074 6865 2065 6e74 6972 6520 ders the entire \n-00004cf0: 7370 6563 6961 6c20 6f62 6a65 6374 2074 special object t\n-00004d00: 6f20 6265 2074 6865 0a66 6972 7374 2070 o be the.first p\n-00004d10: 6f69 6e74 2028 2331 2920 696e 2074 6865 oint (#1) in the\n-00004d20: 2061 7267 756d 656e 7420 6c69 7374 2e20 argument list. \n-00004d30: 2049 6620 7468 6174 2070 6f69 6e74 2069 If that point i\n-00004d40: 7320 6265 6869 6e64 2028 6f66 0a73 6d61 s behind (of.sma\n-00004d50: 6c6c 6572 207a 2d63 6f6d 706f 6e65 6e74 ller z-component\n-00004d60: 2074 6861 6e29 2061 6e79 2064 7261 7761 than) any drawa\n-00004d70: 626c 652c 2074 6865 6e20 7468 6520 656e ble, then the en\n-00004d80: 7469 7265 2073 7065 6369 616c 206f 626a tire special obj\n-00004d90: 6563 740a 6973 2064 7261 776e 2062 6566 ect.is drawn bef\n-00004da0: 6f72 6520 7468 6174 2064 7261 7761 626c ore that drawabl\n-00004db0: 652c 2073 6f20 7468 6520 6472 6177 6162 e, so the drawab\n-00004dc0: 6c65 206f 6273 6375 7265 7320 7061 7274 le obscures part\n-00004dd0: 7320 6f66 2074 6865 0a73 7065 6369 616c s of the.special\n-00004de0: 206f 626a 6563 7420 7468 6174 206f 7665 object that ove\n-00004df0: 726c 6170 7320 6974 2e20 2049 6e20 6f75 rlaps it. In ou\n-00004e00: 7220 6578 616d 706c 652c 2070 3120 6973 r example, p1 is\n-00004e10: 2074 6865 2066 726f 6e74 2d6d 6f73 740a the front-most.\n-00004e20: 706f 696e 7420 696e 2074 6865 2073 6365 point in the sce\n-00004e30: 6e65 2028 6861 7320 7468 6520 6c61 7267 ne (has the larg\n-00004e40: 6573 7420 7a2d 636f 6d70 6f6e 656e 7429 est z-component)\n-00004e50: 2c20 736f 2061 6464 696e 6720 e280 986c , so adding ...l\n-00004e60: 6179 3d69 6ee2 8099 2068 6173 0a6e 6f20 ay=in... has.no \n-00004e70: 6566 6665 6374 2e0a 0a20 2020 5769 7468 effect... With\n-00004e80: 206f 7074 696f 6e20 e280 986c 6179 3d75 option ...lay=u\n-00004e90: 6e64 6572 e280 992c 2061 2073 7065 6369 nder..., a speci\n-00004ea0: 616c 2069 7320 6472 6177 6e20 5f62 6566 al is drawn _bef\n-00004eb0: 6f72 655f 2c20 6865 6e63 6520 6170 7065 ore_, hence appe\n-00004ec0: 6172 730a 5f75 6e64 6572 5f20 616e 7920 ars._under_ any \n-00004ed0: 6f66 2074 6865 206f 626a 6563 7473 2068 of the objects h\n-00004ee0: 616e 646c 6564 2062 7920 7468 6520 6869 andled by the hi\n-00004ef0: 6464 656e 2073 7572 6661 6365 2061 6c67 dden surface alg\n-00004f00: 6f72 6974 686d 2e0a 5468 6973 2069 7320 orithm..This is \n-00004f10: 686f 7720 7468 6520 6c69 6768 7420 6772 how the light gr\n-00004f20: 6179 2061 7865 7320 7765 7265 2061 6464 ay axes were add\n-00004f30: 6564 2074 6f20 7468 6520 2268 656c 6c6f ed to the \"hello\n-00004f40: 2077 6f72 6c64 2220 6578 616d 706c 650a world\" example.\n-00004f50: 2a6e 6f74 6520 4865 6c6c 6f20 776f 726c *note Hello worl\n-00004f60: 643a 3a2e 0a0a 2020 2053 7065 6369 616c d::... Special\n-00004f70: 206f 626a 6563 7473 2061 7265 2070 6f77 objects are pow\n-00004f80: 6572 6675 6c2c 2077 6974 6820 6d61 6e79 erful, with many\n-00004f90: 2070 6f73 7369 626c 6520 7573 6573 2e0a possible uses..\n-00004fa0: 0a1f 0a46 696c 653a 2073 6b65 7463 682e ...File: sketch.\n-00004fb0: 696e 666f 2c20 204e 6f64 653a 204f 626a info, Node: Obj\n-00004fc0: 6563 7420 7472 616e 7366 6f72 6d73 2c20 ect transforms, \n-00004fd0: 204e 6578 743a 2052 6570 6561 7465 6420 Next: Repeated \n-00004fe0: 6f62 6a65 6374 732c 2020 5072 6576 3a20 objects, Prev: \n-00004ff0: 5370 6563 6961 6c20 6f62 6a65 6374 732c Special objects,\n-00005000: 2020 5570 3a20 496e 7472 6f64 7563 7469 Up: Introducti\n-00005010: 6f6e 2062 7920 6578 616d 706c 650a 0a32 on by example..2\n-00005020: 2e35 2054 7261 6e73 666f 726d 730a 3d3d .5 Transforms.==\n-00005030: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4e6f ============..No\n-00005040: 7720 6c65 7427 7320 6164 6420 6120 7365 w let's add a se\n-00005050: 636f 6e64 2063 6f70 7920 6f66 2074 6865 cond copy of the\n-00005060: 2070 6965 7263 6564 2074 6574 7261 6865 pierced tetrahe\n-00005070: 6472 6f6e 2e20 2057 6527 6c6c 2072 6f74 dron. We'll rot\n-00005080: 6174 650a 7468 6520 636f 7079 2039 3020 ate.the copy 90 \n-00005090: 6465 6772 6565 7320 6162 6f75 7420 7468 degrees about th\n-000050a0: 6520 782d 6178 6973 2077 6974 6820 7468 e x-axis with th\n-000050b0: 6520 6f72 6967 696e 2061 7320 e280 9c63 e origin as ...c\n-000050c0: 656e 7465 7220 6f66 0a72 6f74 6174 696f enter of.rotatio\n-000050d0: 6ee2 809d 2073 6f20 7765 2063 616e 2073 n... so we can s\n-000050e0: 6565 2074 6865 2062 6163 6b2c 2074 6865 ee the back, the\n-000050f0: 6e20 7472 616e 736c 6174 6520 6974 2074 n translate it t\n-00005100: 6f20 7468 6520 7269 6768 742d 2d69 6e20 o the right--in \n-00005110: 7468 650a 706f 7369 7469 7665 2078 2d64 the.positive x-d\n-00005120: 6972 6563 7469 6f6e 2d2d 736f 2069 7420 irection--so it \n-00005130: 646f 6573 6e27 7420 636f 6c6c 6964 6520 doesn't collide \n-00005140: 7769 7468 2074 6865 206f 7269 6769 6e61 with the origina\n-00005150: 6c2e 2020 546f 2068 656c 700a 7573 2073 l. To help.us s\n-00005160: 6565 2077 6861 7427 7320 676f 696e 6720 ee what's going \n-00005170: 6f6e 2c20 6d61 6b65 2074 6865 2062 6163 on, make the bac\n-00005180: 6b20 7369 6465 2067 7261 792e 0a20 2064 k side gray.. d\n-00005190: 6566 2070 6965 7263 6564 5f74 6574 7261 ef pierced_tetra\n-000051a0: 6865 6472 6f6e 207b 0a20 2020 2064 6566 hedron {. def\n-000051b0: 2070 3120 2830 2c30 2c31 2920 6465 6620 p1 (0,0,1) def \n-000051c0: 7032 2028 312c 302c 3029 0a20 2020 2064 p2 (1,0,0). d\n-000051d0: 6566 2070 3320 2830 2c31 2c30 2920 6465 ef p3 (0,1,0) de\n-000051e0: 6620 7034 2028 2d2e 332c 2d2e 352c 2d2e f p4 (-.3,-.5,-.\n-000051f0: 3829 0a20 2020 2070 6f6c 7967 6f6e 2870 8). polygon(p\n-00005200: 3129 2870 3229 2870 3329 2020 2020 2020 1)(p2)(p3) \n+00002750: 2a2a 2a2a 2a2a 2a2a 0a0a 5468 6520 e280 ********..The ..\n+00002760: 9873 6b65 7463 68e2 8099 2069 6e70 7574 .sketch... input\n+00002770: 206c 616e 6775 6167 6520 7769 6c6c 2073 language will s\n+00002780: 6565 6d20 6661 6d69 6c69 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To prod\n+00003570: 7563 6520 e280 9854 696b 5a2f 5047 46e2 uce ...TikZ/PGF.\n+00003580: 8099 2c20 7468 6520 636f 7272 6573 706f .., the correspo\n+00003590: 6e64 696e 6720 636f 6465 2069 730a 2020 nding code is. \n+000035a0: 706f 6c79 676f 6e5b 6669 6c6c 3d6c 6967 polygon[fill=lig\n+000035b0: 6874 6772 6179 2c6c 696e 6520 7769 6474 htgray,line widt\n+000035c0: 683d 3370 745d 2830 2c30 2c31 2928 312c h=3pt](0,0,1)(1,\n+000035d0: 302c 3029 2830 2c31 2c30 290a 2020 6c69 0,0)(0,1,0). li\n+000035e0: 6e65 5b73 7479 6c65 3d64 6f74 7465 645d ne[style=dotted]\n+000035f0: 282d 312c 2d31 2c2d 3129 2832 2c32 2c32 (-1,-1,-1)(2,2,2\n+00003600: 290a 2020 676c 6f62 616c 207b 206c 616e ). global { lan\n+00003610: 6775 6167 6520 7469 6b7a 207d 0a54 6865 guage tikz }.The\n+00003620: 2066 696e 616c 20e2 8098 676c 6f62 616c final ...global\n+00003630: e280 9920 696e 7374 7275 6374 7320 e280 ... instructs ..\n+00003640: 9873 6b65 7463 68e2 8099 2074 6f20 7072 .sketch... to pr\n+00003650: 6f64 7563 6520 e280 9854 696b 5a2f 5047 oduce ...TikZ/PG\n+00003660: 46e2 8099 2063 6f64 6520 6173 0a6f 7574 F... code as.out\n+00003670: 7075 7420 7261 7468 6572 2074 6861 6e20 put rather than \n+00003680: 7468 6520 6465 6661 756c 742c 20e2 8098 the default, ...\n+00003690: 5053 5472 6963 6b73 e280 992e 2020 4e6f PSTricks.... 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The.resu\n+00003710: 6c74 696e 6720 e280 9854 696b 5a2f 5047 lting ...TikZ/PG\n+00003720: 46e2 8099 206f 7574 7075 7420 6973 0a20 F... output is. \n+00003730: 205c 6265 6769 6e7b 7469 6b7a 7069 6374 \\begin{tikzpict\n+00003740: 7572 657d 5b6a 6f69 6e3d 726f 756e 645d ure}[join=round]\n+00003750: 0a20 205c 6472 6177 5b64 6f74 7465 645d . \\draw[dotted]\n+00003760: 282d 312c 2d31 292d 2d28 2e33 3333 2c2e (-1,-1)--(.333,.\n+00003770: 3333 3329 3b0a 2020 5c66 696c 6c64 7261 333);. \\filldra\n+00003780: 775b 6669 6c6c 3d6c 6967 6874 6772 6179 w[fill=lightgray\n+00003790: 2c6c 696e 6520 7769 6474 683d 3370 745d ,line width=3pt]\n+000037a0: 2830 2c30 292d 2d28 312c 3029 2d2d 2830 (0,0)--(1,0)--(0\n+000037b0: 2c31 292d 2d63 7963 6c65 3b0a 2020 5c64 ,1)--cycle;. \\d\n+000037c0: 7261 775b 646f 7474 6564 5d28 2e33 3333 raw[dotted](.333\n+000037d0: 2c2e 3333 3329 2d2d 2832 2c32 293b 0a20 ,.333)--(2,2);. \n+000037e0: 205c 656e 647b 7469 6b7a 7069 6374 7572 \\end{tikzpictur\n+000037f0: 657d 0a54 6865 2072 656d 6169 6e69 6e67 e}.The remaining\n+00003800: 2065 7861 6d70 6c65 7320 6f66 2074 6869 examples of thi\n+00003810: 7320 6d61 6e75 616c 2061 7265 2069 6e20 s manual are in \n+00003820: 5053 5472 6963 6b73 2073 7479 6c65 2e0a PSTricks style..\n+00003830: 0a1f 0a46 696c 653a 2073 6b65 7463 682e ...File: sketch.\n+00003840: 696e 666f 2c20 204e 6f64 653a 2044 7261 info, Node: Dra\n+00003850: 7769 6e67 2061 2073 6f6c 6964 2c20 204e wing a solid, N\n+00003860: 6578 743a 2053 7065 6369 616c 206f 626a ext: Special obj\n+00003870: 6563 7473 2c20 2050 7265 763a 2044 7261 ects, Prev: Dra\n+00003880: 7769 6e67 206f 7074 696f 6e73 2c20 2055 wing options, U\n+00003890: 703a 2049 6e74 726f 6475 6374 696f 6e20 p: Introduction \n+000038a0: 6279 2065 7861 6d70 6c65 0a0a 322e 3320 by example..2.3 \n+000038b0: 4472 6177 696e 6720 6120 736f 6c69 640a Drawing a solid.\n+000038c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================\n+000038d0: 3d3d 3d0a 0a4c 6574 2773 2074 7279 2073 ===..Let's try s\n+000038e0: 6f6d 6574 6869 6e67 206d 6f72 6520 6578 omething more ex\n+000038f0: 6369 7469 6e67 2e20 20e2 8098 536b 6574 citing. ...Sket\n+00003900: 6368 e280 9920 6861 7320 6e6f 206e 6f74 ch... has no not\n+00003910: 696f 6e20 6f66 2061 2073 6f6c 6964 2c0a ion of a solid,.\n+00003920: 6275 7420 706f 6c79 676f 6e61 6c20 e280 but polygonal ..\n+00003930: 9c66 6163 6573 e280 9d20 6361 6e20 6265 .faces... can be\n+00003940: 2075 7365 6420 746f 2072 6570 7265 7365 used to represe\n+00003950: 6e74 2074 6865 2062 6f75 6e64 6172 7920 nt the boundary \n+00003960: 6f66 2061 2073 6f6c 6964 2e0a 546f 2074 of a solid..To t\n+00003970: 6865 2070 7265 7669 6f75 7320 6578 616d he previous exam\n+00003980: 706c 652c 206c 6574 2773 2061 6464 2074 ple, let's add t\n+00003990: 6872 6565 206d 6f72 6520 7472 6961 6e67 hree more triang\n+000039a0: 756c 6172 2070 6f6c 7967 6f6e 7320 746f ular polygons to\n+000039b0: 0a6d 616b 6520 7468 6520 6661 6365 7320 .make the faces \n+000039c0: 6f66 2061 6e20 6972 7265 6775 6c61 7220 of an irregular \n+000039d0: 7465 7472 6168 6564 726f 6e2e 0a20 2025 tetrahedron.. %\n+000039e0: 2076 6572 7469 6365 7320 6f66 2074 6865 vertices of the\n+000039f0: 2074 6574 7261 6865 6472 6f6e 0a20 2064 tetrahedron. d\n+00003a00: 6566 2070 3120 2830 2c30 2c31 2920 6465 ef p1 (0,0,1) de\n+00003a10: 6620 7032 2028 312c 302c 3029 0a20 2064 f p2 (1,0,0). d\n+00003a20: 6566 2070 3320 2830 2c31 2c30 2920 6465 ef p3 (0,1,0) de\n+00003a30: 6620 7034 2028 2d2e 332c 2d2e 352c 2d2e f p4 (-.3,-.5,-.\n+00003a40: 3829 0a0a 2020 2520 6661 6365 7320 6f66 8).. % faces of\n+00003a50: 2074 6865 2074 6574 7261 6865 6472 6f6e the tetrahedron\n+00003a60: 2e0a 2020 706f 6c79 676f 6e28 7031 2928 .. polygon(p1)(\n+00003a70: 7032 2928 7033 2920 2520 6f72 6967 696e p2)(p3) % origin\n+00003a80: 616c 2066 726f 6e74 2070 6f6c 7967 6f6e al front polygon\n+00003a90: 0a20 2070 6f6c 7967 6f6e 2870 3129 2870 . polygon(p1)(p\n+00003aa0: 3429 2870 3229 2025 2062 6f74 746f 6d0a 4)(p2) % bottom.\n+00003ab0: 2020 706f 6c79 676f 6e28 7031 2928 7033 polygon(p1)(p3\n+00003ac0: 2928 7034 2920 2520 6c65 6674 0a20 2070 )(p4) % left. p\n+00003ad0: 6f6c 7967 6f6e 2870 3329 2870 3229 2870 olygon(p3)(p2)(p\n+00003ae0: 3429 2025 2072 6561 720a 0a20 2025 206c 4) % rear.. % l\n+00003af0: 696e 6520 746f 2070 6965 7263 6520 7468 ine to pierce th\n+00003b00: 6520 7465 7472 6168 6564 726f 6e0a 2020 e tetrahedron. \n+00003b10: 6c69 6e65 5b6c 696e 6563 6f6c 6f72 3d72 line[linecolor=r\n+00003b20: 6564 5d28 2d31 2c2d 312c 2d31 2928 322c ed](-1,-1,-1)(2,\n+00003b30: 322c 3229 0a54 6869 7320 6578 616d 706c 2,2).This exampl\n+00003b40: 6520 7573 6573 20e2 809c 6465 6669 6e69 e uses ...defini\n+00003b50: 7469 6f6e 73e2 809d 2c20 7768 6963 6820 tions..., which \n+00003b60: 6265 6769 6e20 7769 7468 20e2 8098 6465 begin with ...de\n+00003b70: 66e2 8099 2e20 2054 6865 7365 20e2 809c f.... These ...\n+00003b80: 6465 6669 6e65 e280 9d0a 6f72 2067 6976 define....or giv\n+00003b90: 6520 6e61 6d65 7320 746f 2070 6f69 6e74 e names to point\n+00003ba0: 732c 2077 6869 6368 2061 7265 2074 6865 s, which are the\n+00003bb0: 6e20 6176 6169 6c61 626c 6520 6173 20e2 n available as .\n+00003bc0: 809c 7265 6665 7265 6e63 6573 e280 9d20 ..references... \n+00003bd0: 6279 0a65 6e63 6c6f 7369 6e67 2074 6865 by.enclosing the\n+00003be0: 206e 616d 6573 2069 6e20 7061 7265 6e74 names in parent\n+00003bf0: 6865 7365 732c 2065 2e67 2e20 2866 6f6f heses, e.g. (foo\n+00003c00: 292e 2020 5468 6520 7061 7265 6e74 6865 ). The parenthe\n+00003c10: 7365 7320 6465 6e6f 7465 0a74 6861 7420 ses denote.that \n+00003c20: 7468 6520 6e61 6d65 7320 7265 6665 7220 the names refer \n+00003c30: 746f 2070 6f69 6e74 733b 2074 6865 7920 to points; they \n+00003c40: 6172 6520 7265 7175 6972 6564 2e20 2054 are required. T\n+00003c50: 6865 7265 2063 616e 2062 6520 6e6f 0a77 here can be no.w\n+00003c60: 6869 7465 2073 7061 6365 2062 6574 7765 hite space betwe\n+00003c70: 656e 2074 6865 6d20 616e 6420 7468 6520 en them and the \n+00003c80: 6e61 6d65 2e0a 0a20 2020 4173 2079 6f75 name... As you\n+00003c90: 2063 616e 2073 6565 2c20 636f 6d6d 656e can see, commen\n+00003ca0: 7473 2073 7461 7274 2077 6974 6820 2520 ts start with % \n+00003cb0: 6173 2069 6e20 5465 5820 616e 6420 6578 as in TeX and ex\n+00003cc0: 7465 6e64 2074 6f20 7468 6520 656e 640a tend to the end.\n+00003cd0: 6f66 2074 6865 206c 696e 6520 2874 686f of the line (tho\n+00003ce0: 7567 6820 2320 7769 6c6c 2077 6f72 6b20 ugh # will work \n+00003cf0: 6173 2077 656c 6c29 2e20 2057 6869 7465 as well). White\n+00003d00: 2073 7061 6365 2c20 696e 636c 7564 696e space, includin\n+00003d10: 670a 7370 6163 6573 2c20 7461 6273 2061 g.spaces, tabs a\n+00003d20: 6e64 2062 6c61 6e6b 206c 696e 6573 2c20 nd blank lines, \n+00003d30: 6861 7320 6e6f 2065 6666 6563 7420 696e has no effect in\n+00003d40: 2074 6865 20e2 8098 736b 6574 6368 e280 the ...sketch..\n+00003d50: 9920 6c61 6e67 7561 6765 2e0a 0a00 085b . language.....[\n+00003d60: 696d 6167 6520 7372 633d 2265 7830 3430 image src=\"ex040\n+00003d70: 2e70 6e67 2220 7465 7874 3d22 496d 6167 .png\" text=\"Imag\n+00003d80: 6520 6578 3034 3020 6f6d 6974 7465 6420 e ex040 omitted \n+00003d90: 696e 2074 6578 7420 7665 7273 696f 6e20 in text version \n+00003da0: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document\n+00003db0: 2e22 0008 5d0a 0a49 6620 7765 206c 6f6f .\"..]..If we loo\n+00003dc0: 6b20 696e 7369 6465 2074 6865 2054 6558 k inside the TeX\n+00003dd0: 2066 696c 6520 7072 6f64 7563 6564 2062 file produced b\n+00003de0: 7920 e280 9873 6b65 7463 68e2 8099 2c20 y ...sketch..., \n+00003df0: 7468 6572 6520 7769 6c6c 2062 6520 6f6e there will be on\n+00003e00: 6c79 0a74 6872 6565 2070 6f6c 7967 6f6e ly.three polygon\n+00003e10: 732e 2020 5468 6520 666f 7572 7468 2068 s. The fourth h\n+00003e20: 6173 2062 6565 6e20 6375 6c6c 6564 2062 as been culled b\n+00003e30: 6563 6175 7365 2069 7420 6973 2061 2022 ecause it is a \"\n+00003e40: 6261 636b 2066 6163 6522 0a6f 6620 7468 back face\".of th\n+00003e50: 6520 7465 7472 6168 6564 726f 6e2c 2069 e tetrahedron, i\n+00003e60: 6e76 6973 6962 6c65 2074 6f20 6f75 7220 nvisible to our \n+00003e70: 7669 6577 2e20 2049 7420 6973 2075 6e6e view. It is unn\n+00003e80: 6563 6573 7361 7279 2c20 616e 6420 736f ecessary, and so\n+00003e90: 2069 740a 6973 2072 656d 6f76 6564 2e0a it.is removed..\n+00003ea0: 0a20 2020 496e 2073 6f6d 6520 6472 6177 . In some draw\n+00003eb0: 696e 6773 2c20 706f 6c79 676f 6e73 2061 ings, polygons a\n+00003ec0: 6374 2061 7320 7a65 726f 2d74 6869 636b ct as zero-thick\n+00003ed0: 6e65 7373 2073 6f6c 6964 2073 7572 6661 ness solid surfa\n+00003ee0: 6365 7320 7769 7468 0a62 6f74 6820 7369 ces with.both si\n+00003ef0: 6465 7320 7669 7369 626c 6520 7261 7468 des visible rath\n+00003f00: 6572 2074 6861 6e20 6173 2074 6865 2066 er than as the f\n+00003f10: 6163 6573 206f 6620 736f 6c69 6420 6f62 aces of solid ob\n+00003f20: 6a65 6374 732c 2077 6865 7265 2062 6163 jects, where bac\n+00003f30: 6b0a 6661 6365 7320 6361 6e20 6265 2063 k.faces can be c\n+00003f40: 756c 6c65 642e 2020 466f 7220 7a65 726f ulled. For zero\n+00003f50: 2d74 6869 636b 6e65 7373 2073 6f6c 6964 -thickness solid\n+00003f60: 732c 2063 756c 6c69 6e67 2069 7320 6120 s, culling is a \n+00003f70: 7072 6f62 6c65 6d2e 0a4f 6e65 2073 6f6c problem..One sol\n+00003f80: 7574 696f 6e20 6973 2074 6f20 7573 6520 ution is to use \n+00003f90: 6120 7061 6972 206f 6620 e280 9873 6b65 a pair of ...ske\n+00003fa0: 7463 68e2 8099 2070 6f6c 7967 6f6e 7320 tch... polygons \n+00003fb0: 666f 7220 6561 6368 0a7a 6572 6f2d 7468 for each.zero-th\n+00003fc0: 6963 6b6e 6573 7320 6661 6365 2c20 6964 ickness face, id\n+00003fd0: 656e 7469 6361 6c20 6578 6365 7074 2077 entical except w\n+00003fe0: 6974 6820 6f70 706f 7369 7465 2076 6572 ith opposite ver\n+00003ff0: 7465 7820 6f72 6465 7273 2e20 2054 6869 tex orders. Thi\n+00004000: 730a 6973 2075 6e77 6965 6c64 7920 616e s.is unwieldy an\n+00004010: 6420 6578 7065 6e73 6976 652e 2020 4120 d expensive. A \n+00004020: 6265 7474 6572 2077 6179 2069 7320 746f better way is to\n+00004030: 2073 6574 2074 6865 20e2 8098 736b 6574 set the ...sket\n+00004040: 6368 e280 9920 696e 7465 726e 616c 0a6f ch... internal.o\n+00004050: 7074 696f 6e20 e280 9863 756c 6ce2 8099 ption ...cull...\n+00004060: 2074 6f20 e280 9866 616c 7365 e280 9920 to ...false... \n+00004070: 696e 2074 6865 2075 7375 616c 20e2 8098 in the usual ...\n+00004080: 5053 5472 6963 6b73 e280 9920 6d61 6e6e PSTricks... mann\n+00004090: 6572 2e0a 2020 706f 6c79 676f 6e5b 6375 er.. polygon[cu\n+000040a0: 6c6c 3d66 616c 7365 5d28 7031 2928 7032 ll=false](p1)(p2\n+000040b0: 2928 7033 290a 5468 6520 666f 6c6c 6f77 )(p3).The follow\n+000040c0: 696e 6720 7368 6f77 7320 7468 6520 7361 ing shows the sa\n+000040d0: 6d65 2068 656c 6978 2073 6861 7065 2064 me helix shape d\n+000040e0: 7261 776e 2066 6972 7374 2077 6974 6820 rawn first with \n+000040f0: 6375 6c6c 3d74 7275 6520 2874 6865 0a64 cull=true (the.d\n+00004100: 6566 6175 6c74 2920 616e 6420 7468 656e efault) and then\n+00004110: 2063 756c 6c3d 6661 6c73 652e 0a0a 0008 cull=false.....\n+00004120: 5b69 6d61 6765 2073 7263 3d22 6578 3034 [image src=\"ex04\n+00004130: 352e 706e 6722 2074 6578 743d 2249 6d61 5.png\" text=\"Ima\n+00004140: 6765 2065 7830 3435 206f 6d69 7474 6564 ge ex045 omitted\n+00004150: 2069 6e20 7465 7874 2076 6572 7369 6f6e in text version\n+00004160: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen\n+00004170: 742e 2200 085d 0a0a 5765 276c 6c20 736f t.\"..]..We'll so\n+00004180: 6f6e 2073 6565 2068 6f77 2074 6f20 7072 on see how to pr\n+00004190: 6f64 7563 6520 7468 6573 6520 6865 6c69 oduce these heli\n+000041a0: 7865 7320 7769 7468 2061 2066 6577 206c xes with a few l\n+000041b0: 696e 6573 206f 6620 e280 9873 6b65 7463 ines of ...sketc\n+000041c0: 68e2 8099 0a6c 616e 6775 6167 6520 636f h....language co\n+000041d0: 6465 2e0a 0a20 2020 4974 206d 6179 2062 de... It may b\n+000041e0: 6520 7465 6d70 7469 6e67 2074 6f20 7475 e tempting to tu\n+000041f0: 726e 2063 756c 6c69 6e67 206f 6666 2067 rn culling off g\n+00004200: 7261 7475 6974 6f75 736c 7920 736f 2074 ratuitously so t\n+00004210: 6861 7420 7665 7274 6578 0a6f 7264 6572 hat vertex.order\n+00004220: 2063 616e 2062 6520 6967 6e6f 7265 642e can be ignored.\n+00004230: 2020 5468 6973 2069 7320 6e6f 7420 6120 This is not a \n+00004240: 676f 6f64 2069 6465 6120 6265 6361 7573 good idea becaus\n+00004250: 6520 6f75 7470 7574 2066 696c 6520 7369 e output file si\n+00004260: 7a65 0a61 6e64 2054 6558 2061 6e64 2050 ze.and TeX and P\n+00004270: 6f73 7473 6372 6970 7420 7072 6f63 6573 ostscript proces\n+00004280: 7369 6e67 2074 696d 6520 626f 7468 2064 sing time both d\n+00004290: 6570 656e 6420 6f6e 2074 6865 206e 756d epend on the num\n+000042a0: 6265 7220 6f66 0a6f 7574 7075 7420 706f ber of.output po\n+000042b0: 6c79 676f 6e73 2e20 2043 756c 6c69 6e67 lygons. Culling\n+000042c0: 2075 7375 616c 6c79 2069 6d70 726f 7665 usually improve\n+000042d0: 7320 7065 7266 6f72 6d61 6e63 6520 6279 s performance by\n+000042e0: 2061 2066 6163 746f 7220 6f66 0a74 776f a factor of.two\n+000042f0: 2e20 204f 6e20 7468 6520 6f74 6865 7220 . On the other \n+00004300: 6861 6e64 2c20 676c 6f62 616c 6c79 2073 hand, globally s\n+00004310: 6574 7469 6e67 20e2 8098 6375 6c6c 3d66 etting ...cull=f\n+00004320: 616c 7365 e280 9920 6973 2072 6561 736f alse... is reaso\n+00004330: 6e61 626c 650a 7768 696c 6520 6465 6275 nable.while debu\n+00004340: 6767 696e 672e 2020 5365 6520 2a6e 6f74 gging. See *not\n+00004350: 6520 476c 6f62 616c 206f 7074 696f 6e73 e Global options\n+00004360: 3a3a 2061 6e64 202a 6e6f 7465 204c 696d :: and *note Lim\n+00004370: 6974 7320 6f6e 2065 7272 6f72 0a64 6574 its on error.det\n+00004380: 6563 7469 6f6e 3a3a 2e0a 0a1f 0a46 696c ection::.....Fil\n+00004390: 653a 2073 6b65 7463 682e 696e 666f 2c20 e: sketch.info, \n+000043a0: 204e 6f64 653a 2053 7065 6369 616c 206f Node: Special o\n+000043b0: 626a 6563 7473 2c20 204e 6578 743a 204f bjects, Next: O\n+000043c0: 626a 6563 7420 7472 616e 7366 6f72 6d73 bject transforms\n+000043d0: 2c20 2050 7265 763a 2044 7261 7769 6e67 , Prev: Drawing\n+000043e0: 2061 2073 6f6c 6964 2c20 2055 703a 2049 a solid, Up: I\n+000043f0: 6e74 726f 6475 6374 696f 6e20 6279 2065 ntroduction by e\n+00004400: 7861 6d70 6c65 0a0a 322e 3420 5370 6563 xample..2.4 Spec\n+00004410: 6961 6c20 6f62 6a65 6374 730a 3d3d 3d3d ial objects.====\n+00004420: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============.\n+00004430: 0a57 6520 6361 6e20 6164 6420 6c61 6265 .We can add labe\n+00004440: 6c73 2074 6f20 6120 6472 6177 696e 6720 ls to a drawing \n+00004450: 6279 2075 7369 6e67 2073 7065 6369 616c by using special\n+00004460: 206f 626a 6563 7473 2c20 7768 6963 6820 objects, which \n+00004470: 7072 6f76 6964 6520 610a 7761 7920 746f provide a.way to\n+00004480: 2065 6d62 6564 2072 6177 204c 6154 6558 embed raw LaTeX\n+00004490: 2061 6e64 20e2 8098 5053 5472 6963 6b73 and ...PSTricks\n+000044a0: e280 9920 636f 6465 2e20 2041 6464 696e ... code. Addin\n+000044b0: 6720 7468 6973 2074 6f20 7468 650a 7465 g this to the.te\n+000044c0: 7472 6168 6564 726f 6e20 646f 6573 2074 trahedron does t\n+000044d0: 6865 2074 7269 636b 2e0a 2020 7370 6563 he trick.. spec\n+000044e0: 6961 6c20 7c5c 666f 6f74 6e6f 7465 7369 ial |\\footnotesi\n+000044f0: 7a65 0a20 2020 2020 2020 2020 2020 5c75 ze. \\u\n+00004500: 7075 747b 3270 747d 5b75 725d 2331 7b24 put{2pt}[ur]#1{$\n+00004510: 5031 247d 0a20 2020 2020 2020 2020 2020 P1$}. \n+00004520: 5c75 7075 745b 725d 2332 7b24 5032 247d \\uput[r]#2{$P2$}\n+00004530: 0a20 2020 2020 2020 2020 2020 5c75 7075 . \\upu\n+00004540: 745b 755d 2333 7b24 5033 247d 0a20 2020 t[u]#3{$P3$}. \n+00004550: 2020 2020 2020 2020 5c75 7075 745b 645d \\uput[d]\n+00004560: 2334 7b24 5034 247d 7c0a 2020 2020 2870 #4{$P4$}|. (p\n+00004570: 3129 2870 3229 2870 3329 2870 3429 0a48 1)(p2)(p3)(p4).H\n+00004580: 6572 6520 6973 2074 6865 2072 6573 756c ere is the resul\n+00004590: 742e 0a0a 0008 5b69 6d61 6765 2073 7263 t.....[image src\n+000045a0: 3d22 6578 3034 322e 706e 6722 2074 6578 =\"ex042.png\" tex\n+000045b0: 743d 2249 6d61 6765 2065 7830 3432 206f t=\"Image ex042 o\n+000045c0: 6d69 7474 6564 2069 6e20 7465 7874 2076 mitted in text v\n+000045d0: 6572 7369 6f6e 206f 6620 7468 6973 2064 ersion of this d\n+000045e0: 6f63 756d 656e 742e 2200 085d 0a0a 2020 ocument.\"..].. \n+000045f0: 2054 6865 7265 2061 7265 2073 6576 6572 There are sever\n+00004600: 616c 2064 6574 6169 6c73 2074 6f20 6e6f al details to no\n+00004610: 7465 2068 6572 652e 2020 4669 7273 742c te here. First,\n+00004620: 2074 6865 2071 756f 7469 6e67 0a63 6f6e the quoting.con\n+00004630: 7665 6e74 696f 6e20 666f 7220 7468 6520 vention for the \n+00004640: 7261 7720 636f 6465 2069 7320 7369 6d69 raw code is simi\n+00004650: 6c61 7220 746f 2074 6865 204c 6154 6558 lar to the LaTeX\n+00004660: 205c 7665 7262 2063 6f6d 6d61 6e64 2e20 \\verb command. \n+00004670: 2054 6865 0a66 6972 7374 206e 6f6e 2d77 The.first non-w\n+00004680: 6869 7465 2073 7061 6365 2063 6861 7261 hite space chara\n+00004690: 6374 6572 2066 6f6c 6c6f 7769 6e67 2073 cter following s\n+000046a0: 7065 6369 616c 2069 7320 756e 6465 7273 pecial is unders\n+000046b0: 746f 6f64 2074 6f20 6265 0a74 6865 2071 tood to be.the q\n+000046c0: 756f 7465 2063 6861 7261 6374 6572 2c20 uote character, \n+000046d0: 696e 2074 6869 7320 6361 7365 20e2 8098 in this case ...\n+000046e0: 7ce2 8099 2e20 2054 6865 2072 6177 2074 |.... The raw t\n+000046f0: 6578 7420 636f 6e74 696e 7565 7320 756e ext continues un\n+00004700: 7469 6c0a 7468 6973 2063 6861 7261 6374 til.this charact\n+00004710: 6572 2072 6563 7572 732e 0a0a 2020 2053 er recurs... S\n+00004720: 6563 6f6e 642c 2074 6865 2061 7267 756d econd, the argum\n+00004730: 656e 7420 7265 6665 7265 6e63 6573 2023 ent references #\n+00004740: 312c 2023 322c 2023 332c 2061 6e64 2023 1, #2, #3, and #\n+00004750: 3420 7265 6665 7220 746f 2070 6f69 6e74 4 refer to point\n+00004760: 2c0a 7665 6374 6f72 2c20 6f72 2073 6361 ,.vector, or sca\n+00004770: 6c61 7220 7661 6c75 6573 2069 6e20 7468 lar values in th\n+00004780: 6520 6c69 7374 2074 6861 7420 666f 6c6c e list that foll\n+00004790: 6f77 2e20 2054 6869 7320 6973 2073 696d ow. This is sim\n+000047a0: 696c 6172 2074 6f0a 5465 5820 6d61 6372 ilar to.TeX macr\n+000047b0: 6f20 7379 6e74 6178 2e20 2054 6865 2074 o syntax. The t\n+000047c0: 7261 6e73 666f 726d 6564 2061 6e64 2074 ransformed and t\n+000047d0: 776f 2d64 696d 656e 7369 6f6e 616c 2070 wo-dimensional p\n+000047e0: 726f 6a65 6374 696f 6e73 206f 660a 7468 rojections of.th\n+000047f0: 6573 6520 7468 7265 652d 6469 6d65 6e73 ese three-dimens\n+00004800: 696f 6e61 6c20 706f 696e 7473 2061 7265 ional points are\n+00004810: 2073 7562 7374 6974 7574 6564 2069 6e20 substituted in \n+00004820: 7468 6520 6669 6e61 6c20 6f75 7470 7574 the final output\n+00004830: 2e20 2041 6e0a 6172 6775 6d65 6e74 2072 . An.argument r\n+00004840: 6566 6572 656e 6365 206f 6620 7468 6520 eference of the \n+00004850: 666f 726d 2023 312d 3220 6973 2072 6570 form #1-2 is rep\n+00004860: 6c61 6365 6420 7769 7468 2074 6865 2061 laced with the a\n+00004870: 6e67 6c65 2069 6e0a 6465 6772 6565 7320 ngle in.degrees \n+00004880: 6f66 2074 6865 2074 776f 2d64 696d 656e of the two-dimen\n+00004890: 7369 6f6e 616c 2076 6563 746f 7220 7468 sional vector th\n+000048a0: 6174 2063 6f6e 6e65 6374 7320 7468 6520 at connects the \n+000048b0: 7072 6f6a 6563 7469 6f6e 7320 6f66 0a74 projections of.t\n+000048c0: 6865 2074 776f 2072 6573 7065 6374 6976 he two respectiv\n+000048d0: 6520 6172 6775 6d65 6e74 2070 6f69 6e74 e argument point\n+000048e0: 732c 2068 6572 6520 2331 2061 6e64 2023 s, here #1 and #\n+000048f0: 322e 2020 5468 6520 7375 6273 7469 7475 2. The substitu\n+00004900: 7465 640a 616e 676c 6520 6973 2065 6e63 ted.angle is enc\n+00004910: 6c6f 7365 6420 696e 2063 7572 6c79 2062 losed in curly b\n+00004920: 7261 6365 7320 e280 987b 207d e280 992e races ...{ }....\n+00004930: 2020 5768 656e 20e2 8098 5469 6b5a 2f50 When ...TikZ/P\n+00004940: 4746 e280 9920 6f75 7470 7574 2069 730a GF... output is.\n+00004950: 6265 696e 6720 6765 6e65 7261 7465 642c being generated,\n+00004960: 2074 6865 2061 6e67 6c65 2069 7320 726f the angle is ro\n+00004970: 756e 6465 6420 746f 2074 6865 206e 6561 unded to the nea\n+00004980: 7265 7374 2064 6567 7265 6520 6265 6361 rest degree beca\n+00004990: 7573 650a 6e6f 6e2d 696e 7465 6765 7220 use.non-integer \n+000049a0: 616e 676c 6573 2061 7265 206e 6f74 2061 angles are not a\n+000049b0: 6c6c 6f77 6564 2062 7920 e280 9854 696b llowed by ...Tik\n+000049c0: 5a2f 5047 46e2 8099 2070 7269 6d69 7469 Z/PGF... primiti\n+000049d0: 7665 732e 0a0a 2020 2041 7320 6f66 2056 ves... As of V\n+000049e0: 6572 7369 6f6e 2030 2e33 206f 6620 e280 ersion 0.3 of ..\n+000049f0: 9873 6b65 7463 68e2 8099 2c20 7370 6563 .sketch..., spec\n+00004a00: 6961 6c20 6172 6775 6d65 6e74 7320 6d61 ial arguments ma\n+00004a10: 7920 6265 2073 6361 6c61 7273 206f 720a y be scalars or.\n+00004a20: 7665 6374 6f72 7320 696e 2061 6464 6974 vectors in addit\n+00004a30: 696f 6e20 746f 2070 6f69 6e74 732e 2020 ion to points. \n+00004a40: 5265 6665 7265 6e63 6573 2074 6f20 7363 References to sc\n+00004a50: 616c 6172 2061 7267 756d 656e 7473 2061 alar arguments a\n+00004a60: 7265 0a6d 6572 656c 7920 7265 706c 6163 re.merely replac\n+00004a70: 6564 2077 6974 6820 6120 6e75 6d62 6572 ed with a number\n+00004a80: 2066 6f72 6d61 7474 6564 206a 7573 7420 formatted just \n+00004a90: 6173 2061 6e79 2070 6f69 6e74 2063 6f6f as any point coo\n+00004aa0: 7264 696e 6174 652e 0a52 6566 6572 656e rdinate..Referen\n+00004ab0: 6365 7320 746f 2076 6563 746f 7273 2062 ces to vectors b\n+00004ac0: 6563 6f6d 6520 7477 6f2d 6469 6d65 6e73 ecome two-dimens\n+00004ad0: 696f 6e61 6c20 706f 696e 7473 2e20 2054 ional points. T\n+00004ae0: 6865 2074 6963 6b20 6f70 6572 6174 6f72 he tick operator\n+00004af0: 0a74 6861 7420 7365 6c65 6374 7320 696e .that selects in\n+00004b00: 6469 7669 6475 616c 2063 6f6d 706f 6e65 dividual compone\n+00004b10: 6e74 7320 6f66 2070 6f69 6e74 7320 616e nts of points an\n+00004b20: 6420 7665 6374 6f72 7320 656c 7365 7768 d vectors elsewh\n+00004b30: 6572 6520 696e 0ae2 8098 736b 6574 6368 ere in....sketch\n+00004b40: e280 9920 2873 6565 2066 6f72 2065 7861 ... (see for exa\n+00004b50: 6d70 6c65 202a 6e6f 7465 2041 6666 696e mple *note Affin\n+00004b60: 6520 6172 6974 686d 6574 6963 3a3a 2920 e arithmetic::) \n+00004b70: 6361 6e20 616c 736f 2062 6520 6170 706c can also be appl\n+00004b80: 6965 640a 746f 2070 6f69 6e74 2061 6e64 ied.to point and\n+00004b90: 2076 6563 746f 7220 6172 6775 6d65 6e74 vector argument\n+00004ba0: 2072 6566 6572 656e 6365 732e 2020 416c references. Al\n+00004bb0: 6c20 7468 7265 6520 6469 6d65 6e73 696f l three dimensio\n+00004bc0: 6e73 206f 6620 610a 7472 616e 7366 6f72 ns of a.transfor\n+00004bd0: 6d65 6420 706f 696e 7420 6f72 2076 6563 med point or vec\n+00004be0: 746f 7220 6361 6e20 616c 736f 2062 6520 tor can also be \n+00004bf0: 7375 6273 7469 7475 6564 2077 6974 6820 substitued with \n+00004c00: e280 9827 33e2 8099 2e20 2053 6565 202a ...'3.... See *\n+00004c10: 6e6f 7465 0a53 7065 6369 616c 733a 3a20 note.Specials:: \n+00004c20: 666f 7220 6465 7461 696c 732e 0a0a 2020 for details... \n+00004c30: 2042 7920 6465 6661 756c 742c 20e2 8098 By default, ...\n+00004c40: 7370 6563 6961 6ce2 8099 206f 626a 6563 special... objec\n+00004c50: 7473 2061 7265 2070 7269 6e74 6564 206c ts are printed l\n+00004c60: 6173 742c 206f 7665 726c 6179 696e 6720 ast, overlaying \n+00004c70: 616c 6c20 6f74 6865 720a 6f62 6a65 6374 all other.object\n+00004c80: 7320 696e 2074 6865 2073 6365 6e65 2e20 s in the scene. \n+00004c90: 2049 6620 796f 7520 7370 6563 6966 7920 If you specify \n+00004ca0: 7468 6520 696e 7465 726e 616c 206f 7074 the internal opt\n+00004cb0: 696f 6e20 e280 986c 6179 3d69 6ee2 8099 ion ...lay=in...\n+00004cc0: 2c20 7468 650a 6869 6464 656e 2073 7572 , the.hidden sur\n+00004cd0: 6661 6365 2061 6c67 6f72 6974 686d 2063 face algorithm c\n+00004ce0: 6f6e 7369 6465 7273 2074 6865 2065 6e74 onsiders the ent\n+00004cf0: 6972 6520 7370 6563 6961 6c20 6f62 6a65 ire special obje\n+00004d00: 6374 2074 6f20 6265 2074 6865 0a66 6972 ct to be the.fir\n+00004d10: 7374 2070 6f69 6e74 2028 2331 2920 696e st point (#1) in\n+00004d20: 2074 6865 2061 7267 756d 656e 7420 6c69 the argument li\n+00004d30: 7374 2e20 2049 6620 7468 6174 2070 6f69 st. If that poi\n+00004d40: 6e74 2069 7320 6265 6869 6e64 2028 6f66 nt is behind (of\n+00004d50: 0a73 6d61 6c6c 6572 207a 2d63 6f6d 706f .smaller z-compo\n+00004d60: 6e65 6e74 2074 6861 6e29 2061 6e79 2064 nent than) any d\n+00004d70: 7261 7761 626c 652c 2074 6865 6e20 7468 rawable, then th\n+00004d80: 6520 656e 7469 7265 2073 7065 6369 616c e entire special\n+00004d90: 206f 626a 6563 740a 6973 2064 7261 776e object.is drawn\n+00004da0: 2062 6566 6f72 6520 7468 6174 2064 7261 before that dra\n+00004db0: 7761 626c 652c 2073 6f20 7468 6520 6472 wable, so the dr\n+00004dc0: 6177 6162 6c65 206f 6273 6375 7265 7320 awable obscures \n+00004dd0: 7061 7274 7320 6f66 2074 6865 0a73 7065 parts of the.spe\n+00004de0: 6369 616c 206f 626a 6563 7420 7468 6174 cial object that\n+00004df0: 206f 7665 726c 6170 7320 6974 2e20 2049 overlaps it. I\n+00004e00: 6e20 6f75 7220 6578 616d 706c 652c 2070 n our example, p\n+00004e10: 3120 6973 2074 6865 2066 726f 6e74 2d6d 1 is the front-m\n+00004e20: 6f73 740a 706f 696e 7420 696e 2074 6865 ost.point in the\n+00004e30: 2073 6365 6e65 2028 6861 7320 7468 6520 scene (has the \n+00004e40: 6c61 7267 6573 7420 7a2d 636f 6d70 6f6e largest z-compon\n+00004e50: 656e 7429 2c20 736f 2061 6464 696e 6720 ent), so adding \n+00004e60: e280 986c 6179 3d69 6ee2 8099 2068 6173 ...lay=in... has\n+00004e70: 0a6e 6f20 6566 6665 6374 2e0a 0a20 2020 .no effect... \n+00004e80: 5769 7468 206f 7074 696f 6e20 e280 986c With option ...l\n+00004e90: 6179 3d75 6e64 6572 e280 992c 2061 2073 ay=under..., a s\n+00004ea0: 7065 6369 616c 2069 7320 6472 6177 6e20 pecial is drawn \n+00004eb0: 5f62 6566 6f72 655f 2c20 6865 6e63 6520 _before_, hence \n+00004ec0: 6170 7065 6172 730a 5f75 6e64 6572 5f20 appears._under_ \n+00004ed0: 616e 7920 6f66 2074 6865 206f 626a 6563 any of the objec\n+00004ee0: 7473 2068 616e 646c 6564 2062 7920 7468 ts handled by th\n+00004ef0: 6520 6869 6464 656e 2073 7572 6661 6365 e hidden surface\n+00004f00: 2061 6c67 6f72 6974 686d 2e0a 5468 6973 algorithm..This\n+00004f10: 2069 7320 686f 7720 7468 6520 6c69 6768 is how the ligh\n+00004f20: 7420 6772 6179 2061 7865 7320 7765 7265 t gray axes were\n+00004f30: 2061 6464 6564 2074 6f20 7468 6520 2268 added to the \"h\n+00004f40: 656c 6c6f 2077 6f72 6c64 2220 6578 616d ello world\" exam\n+00004f50: 706c 650a 2a6e 6f74 6520 4865 6c6c 6f20 ple.*note Hello \n+00004f60: 776f 726c 643a 3a2e 0a0a 2020 2053 7065 world::... Spe\n+00004f70: 6369 616c 206f 626a 6563 7473 2061 7265 cial objects are\n+00004f80: 2070 6f77 6572 6675 6c2c 2077 6974 6820 powerful, with \n+00004f90: 6d61 6e79 2070 6f73 7369 626c 6520 7573 many possible us\n+00004fa0: 6573 2e0a 0a1f 0a46 696c 653a 2073 6b65 es.....File: ske\n+00004fb0: 7463 682e 696e 666f 2c20 204e 6f64 653a tch.info, Node:\n+00004fc0: 204f 626a 6563 7420 7472 616e 7366 6f72 Object transfor\n+00004fd0: 6d73 2c20 204e 6578 743a 2052 6570 6561 ms, Next: Repea\n+00004fe0: 7465 6420 6f62 6a65 6374 732c 2020 5072 ted objects, Pr\n+00004ff0: 6576 3a20 5370 6563 6961 6c20 6f62 6a65 ev: Special obje\n+00005000: 6374 732c 2020 5570 3a20 496e 7472 6f64 cts, Up: Introd\n+00005010: 7563 7469 6f6e 2062 7920 6578 616d 706c uction by exampl\n+00005020: 650a 0a32 2e35 2054 7261 6e73 666f 726d e..2.5 Transform\n+00005030: 730a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d s.==============\n+00005040: 0a0a 4e6f 7720 6c65 7427 7320 6164 6420 ..Now let's add \n+00005050: 6120 7365 636f 6e64 2063 6f70 7920 6f66 a second copy of\n+00005060: 2074 6865 2070 6965 7263 6564 2074 6574 the pierced tet\n+00005070: 7261 6865 6472 6f6e 2e20 2057 6527 6c6c rahedron. We'll\n+00005080: 2072 6f74 6174 650a 7468 6520 636f 7079 rotate.the copy\n+00005090: 2039 3020 6465 6772 6565 7320 6162 6f75 90 degrees abou\n+000050a0: 7420 7468 6520 782d 6178 6973 2077 6974 t the x-axis wit\n+000050b0: 6820 7468 6520 6f72 6967 696e 2061 7320 h the origin as \n+000050c0: e280 9c63 656e 7465 7220 6f66 0a72 6f74 ...center of.rot\n+000050d0: 6174 696f 6ee2 809d 2073 6f20 7765 2063 ation... so we c\n+000050e0: 616e 2073 6565 2074 6865 2062 6163 6b2c an see the back,\n+000050f0: 2074 6865 6e20 7472 616e 736c 6174 6520 then translate \n+00005100: 6974 2074 6f20 7468 6520 7269 6768 742d it to the right-\n+00005110: 2d69 6e20 7468 650a 706f 7369 7469 7665 -in the.positive\n+00005120: 2078 2d64 6972 6563 7469 6f6e 2d2d 736f x-direction--so\n+00005130: 2069 7420 646f 6573 6e27 7420 636f 6c6c it doesn't coll\n+00005140: 6964 6520 7769 7468 2074 6865 206f 7269 ide with the ori\n+00005150: 6769 6e61 6c2e 2020 546f 2068 656c 700a ginal. To help.\n+00005160: 7573 2073 6565 2077 6861 7427 7320 676f us see what's go\n+00005170: 696e 6720 6f6e 2c20 6d61 6b65 2074 6865 ing on, make the\n+00005180: 2062 6163 6b20 7369 6465 2067 7261 792e back side gray.\n+00005190: 0a20 2064 6566 2070 6965 7263 6564 5f74 . def pierced_t\n+000051a0: 6574 7261 6865 6472 6f6e 207b 0a20 2020 etrahedron {. \n+000051b0: 2064 6566 2070 3120 2830 2c30 2c31 2920 def p1 (0,0,1) \n+000051c0: 6465 6620 7032 2028 312c 302c 3029 0a20 def p2 (1,0,0). \n+000051d0: 2020 2064 6566 2070 3320 2830 2c31 2c30 def p3 (0,1,0\n+000051e0: 2920 6465 6620 7034 2028 2d2e 332c 2d2e ) def p4 (-.3,-.\n+000051f0: 352c 2d2e 3829 0a20 2020 2070 6f6c 7967 5,-.8). polyg\n+00005200: 6f6e 2870 3129 2870 3229 2870 3329 2020 on(p1)(p2)(p3) \n 00005210: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00005220: 2520 6f72 6967 696e 616c 0a20 2020 2070 % original. p\n-00005230: 6f6c 7967 6f6e 2870 3129 2870 3429 2870 olygon(p1)(p4)(p\n-00005240: 3229 2020 2020 2020 2020 2020 2020 2020 2) \n-00005250: 2020 2020 2020 2020 2520 626f 7474 6f6d % bottom\n-00005260: 0a20 2020 2070 6f6c 7967 6f6e 2870 3129 . polygon(p1)\n-00005270: 2870 3329 2870 3429 2020 2020 2020 2020 (p3)(p4) \n-00005280: 2020 2020 2020 2020 2020 2020 2020 2520 % \n-00005290: 6c65 6674 0a20 2020 2070 6f6c 7967 6f6e left. polygon\n-000052a0: 5b66 696c 6c63 6f6c 6f72 3d6c 6967 6874 [fillcolor=light\n-000052b0: 6772 6179 5d28 7033 2928 7032 2928 7034 gray](p3)(p2)(p4\n-000052c0: 2920 2520 7265 6172 0a20 2020 206c 696e ) % rear. lin\n-000052d0: 655b 6c69 6e65 636f 6c6f 723d 7265 645d e[linecolor=red]\n-000052e0: 282d 312c 2d31 2c2d 3129 2832 2c32 2c32 (-1,-1,-1)(2,2,2\n-000052f0: 290a 2020 7d0a 2020 7b70 6965 7263 6564 ). }. {pierced\n-00005300: 5f74 6574 7261 6865 6472 6f6e 7d20 2025 _tetrahedron} %\n-00005310: 2074 6574 7261 6865 6472 6f6e 2069 6e20 tetrahedron in \n-00005320: 6f72 6967 696e 616c 2070 6f73 6974 696f original positio\n-00005330: 6e0a 2020 7075 7420 7b20 726f 7461 7465 n. put { rotate\n-00005340: 2839 302c 2028 302c 302c 3029 2c20 5b31 (90, (0,0,0), [1\n-00005350: 2c30 2c30 5d29 2025 2063 6f70 7920 696e ,0,0]) % copy in\n-00005360: 206e 6577 2070 6f73 6974 696f 6e0a 2020 new position. \n-00005370: 2020 2020 2020 7468 656e 2074 7261 6e73 then trans\n-00005380: 6c61 7465 285b 322e 352c 302c 305d 2920 late([2.5,0,0]) \n-00005390: 7d20 7b70 6965 7263 6564 5f74 6574 7261 } {pierced_tetra\n-000053a0: 6865 6472 6f6e 7d0a 4865 7265 2074 6865 hedron}.Here the\n-000053b0: 2065 6e74 6972 6520 636f 6465 206f 6620 entire code of \n-000053c0: 7468 6520 7072 6576 696f 7573 2065 7861 the previous exa\n-000053d0: 6d70 6c65 2068 6173 2062 6565 6e20 7772 mple has been wr\n-000053e0: 6170 7065 6420 696e 2061 0a64 6566 696e apped in a.defin\n-000053f0: 6974 696f 6e20 6279 2066 6f72 6d69 6e67 ition by forming\n-00005400: 2061 20e2 809c 626c 6f63 6be2 809d 2077 a ...block... w\n-00005410: 6974 6820 6272 6163 6573 2028 6120 7369 ith braces (a si\n-00005420: 6e67 6c65 2069 7465 6d20 776f 756c 6420 ngle item would \n-00005430: 6e6f 740a 6e65 6564 2074 6865 6d29 2e20 not.need them). \n-00005440: 2054 6865 2070 6f69 6e74 2064 6566 696e The point defin\n-00005450: 6974 696f 6e73 206e 6573 7465 6420 696e itions nested in\n-00005460: 7369 6465 2074 6865 2062 7261 6365 7320 side the braces \n-00005470: 6172 650a e280 9c6c 6578 6963 616c 6c79 are....lexically\n-00005480: 2073 636f 7065 64e2 809d 2e20 2054 6865 scoped.... The\n-00005490: 6972 206d 6561 6e69 6e67 2065 7874 656e ir meaning exten\n-000054a0: 6473 206f 6e6c 7920 746f 2074 6865 2065 ds only to the e\n-000054b0: 6e64 206f 6620 7468 6520 626c 6f63 6b2e nd of the block.\n-000054c0: 0a54 6865 206f 7574 6572 2064 6566 2069 .The outer def i\n-000054d0: 7320 6361 6c6c 6564 2061 20e2 809c 6472 s called a ...dr\n-000054e0: 6177 6162 6c65 e280 9d20 6465 6669 6e69 awable... defini\n-000054f0: 7469 6f6e 2062 6563 6175 7365 2069 7420 tion because it \n-00005500: 6465 7363 7269 6265 730a 736f 6d65 7468 describes.someth\n-00005510: 696e 6720 7468 6174 2063 616e 2062 6520 ing that can be \n-00005520: 6472 6177 6e2e 0a0a 2020 2041 2064 7261 drawn... A dra\n-00005530: 7761 626c 6520 6465 6669 6e69 7469 6f6e wable definition\n-00005540: 2062 7920 6974 7365 6c66 2063 6175 7365 by itself cause\n-00005550: 7320 6e6f 7468 696e 6720 746f 2068 6170 s nothing to hap\n-00005560: 7065 6e20 756e 7469 6c20 6974 730a 6e61 pen until its.na\n-00005570: 6d65 2069 7320 7265 6665 7265 6e63 6564 me is referenced\n-00005580: 2e20 2044 7261 7761 626c 6520 7265 6665 . Drawable refe\n-00005590: 7265 6e63 6573 206d 7573 7420 6265 2065 rences must be e\n-000055a0: 6e63 6c6f 7365 6420 696e 2063 7572 6c79 nclosed in curly\n-000055b0: 0a62 7261 6365 732c 2065 2e67 2e20 7b66 .braces, e.g. {f\n-000055c0: 6f6f 7d2c 2077 6974 6820 6e6f 2069 6e74 oo}, with no int\n-000055d0: 6572 7665 6e69 6e67 2077 6869 7465 2073 ervening white s\n-000055e0: 7061 6365 2e20 2049 6e20 7468 6520 636f pace. In the co\n-000055f0: 6465 2061 626f 7665 2c0a 7468 6520 6669 de above,.the fi\n-00005600: 7273 7420 7265 6665 7265 6e63 6520 7b70 rst reference {p\n-00005610: 6965 7263 6564 5f74 6574 7261 6865 6472 ierced_tetrahedr\n-00005620: 6f6e 7d20 6973 2061 2070 6c61 696e 206f on} is a plain o\n-00005630: 6e65 2e20 2049 7473 2065 6666 6563 7420 ne. Its effect \n-00005640: 6973 0a6d 6572 656c 7920 746f 2064 7570 is.merely to dup\n-00005650: 6c69 6361 7465 2074 6865 2065 6172 6c69 licate the earli\n-00005660: 6572 2064 7261 7769 6e67 2e20 2041 6c6d er drawing. Alm\n-00005670: 6f73 7420 616e 7920 7365 7269 6573 206f ost any series o\n-00005680: 6620 e280 9873 6b65 7463 68e2 8099 0a63 f ...sketch....c\n-00005690: 6f6d 6d61 6e64 7320 7374 7566 6620 6d61 ommands stuff ma\n-000056a0: 7920 6265 2072 6570 6c61 6365 6420 7769 y be replaced wi\n-000056b0: 7468 2064 6566 2066 6f6f 207b 2073 7475 th def foo { stu\n-000056c0: 6666 207d 207b 666f 6f7d 2077 6974 686f ff } {foo} witho\n-000056d0: 7574 0a63 6861 6e67 696e 6720 6974 7320 ut.changing its \n-000056e0: 6d65 616e 696e 672e 0a0a 2020 2054 6865 meaning... The\n-000056f0: 2070 7574 2063 6f6d 6d61 6e64 2073 7570 put command sup\n-00005700: 706c 6965 7320 6120 7365 636f 6e64 2072 plies a second r\n-00005710: 6566 6572 656e 6365 2c20 7468 6973 2074 eference, this t\n-00005720: 696d 6520 7769 7468 2061 0ae2 809c 7472 ime with a....tr\n-00005730: 616e 7366 6f72 6de2 809d 2061 7070 6c69 ansform... appli\n-00005740: 6564 2066 6972 7374 2e20 2054 6865 2072 ed first. The r\n-00005750: 6f74 6174 6520 7472 616e 7366 6f72 6d20 otate transform \n-00005760: 7475 726e 7320 7468 6520 7465 7472 6168 turns the tetrah\n-00005770: 6564 726f 6e0a 3930 2064 6567 7265 6573 edron.90 degrees\n-00005780: 2061 626f 7574 2074 6865 206f 7269 6769 about the origi\n-00005790: 6e2e 2020 5468 6520 6178 6973 206f 6620 n. The axis of \n-000057a0: 726f 7461 7469 6f6e 2069 7320 7468 6520 rotation is the \n-000057b0: 7665 6374 6f72 0a5b 312c 302c 305d 2e20 vector.[1,0,0]. \n-000057c0: 2042 7920 7468 6520 e280 9c72 6967 6874 By the ...right\n-000057d0: 2068 616e 6420 7275 6c65 e280 9d2c 2074 hand rule..., t\n-000057e0: 6869 7320 6361 7573 6573 2074 6865 2074 his causes the t\n-000057f0: 6f70 206f 6620 7468 650a 7465 7472 6168 op of the.tetrah\n-00005800: 6564 726f 6e20 746f 2072 6f74 6174 6520 edron to rotate \n-00005810: 746f 7761 7264 2074 6865 2076 6965 7765 toward the viewe\n-00005820: 7220 616e 6420 7468 6520 626f 7474 6f6d r and the bottom\n-00005830: 2061 7761 792e 2020 5468 6520 7275 6c65 away. The rule\n-00005840: 0a72 6563 6569 7665 7320 6974 7320 6e61 .receives its na\n-00005850: 6d65 2066 726f 6d20 7468 6520 666f 6c6c me from the foll\n-00005860: 6f77 696e 6720 6465 6669 6e69 7469 6f6e owing definition\n-00005870: 3a0a 2020 2020 202a 5269 6768 7420 6861 :. *Right ha\n-00005880: 6e64 2072 756c 652e 2a20 2049 6620 7468 nd rule.* If th\n-00005890: 6520 7269 6768 7420 6861 6e64 2069 7320 e right hand is \n-000058a0: 7772 6170 7065 6420 6172 6f75 6e64 2061 wrapped around a\n-000058b0: 6e79 2061 7869 730a 2020 2020 2077 6974 ny axis. wit\n-000058c0: 6820 7468 6520 7468 756d 6220 706f 696e h the thumb poin\n-000058d0: 7469 6e67 2069 6e20 7468 6520 6178 6973 ting in the axis\n-000058e0: 2064 6972 6563 7469 6f6e 2c20 7468 656e direction, then\n-000058f0: 2074 6865 2066 696e 6765 7273 0a20 2020 the fingers. \n-00005900: 2020 6375 726c 2069 6e20 7468 6520 6469 curl in the di\n-00005910: 7265 6374 696f 6e20 6f66 2070 6f73 6974 rection of posit\n-00005920: 6976 6520 726f 7461 7469 6f6e 2061 626f ive rotation abo\n-00005930: 7574 2074 6861 7420 6178 6973 2e0a 2020 ut that axis.. \n-00005940: 2054 6865 2074 7261 6e73 6c61 7465 2074 The translate t\n-00005950: 7261 6e73 666f 726d 206d 6f76 6573 2074 ransform moves t\n-00005960: 6865 2070 7972 616d 6964 206c 6174 6572 he pyramid later\n-00005970: 616c 6c79 2074 6f20 7468 6520 7269 6768 ally to the righ\n-00005980: 7420 6279 0a61 6464 696e 6720 7468 6520 t by.adding the \n-00005990: 7665 6374 6f72 205b 322e 352c 302c 305d vector [2.5,0,0]\n-000059a0: 2074 6f20 6561 6368 2076 6572 7465 7820 to each vertex \n-000059b0: 636f 6f72 6469 6e61 7465 2e20 2054 6865 coordinate. The\n-000059c0: 2072 6573 756c 7420 6973 0a73 686f 776e result is.shown\n-000059d0: 2068 6572 652e 0a0a 0008 5b69 6d61 6765 here.....[image\n-000059e0: 2073 7263 3d22 6578 3035 302e 706e 6722 src=\"ex050.png\"\n-000059f0: 2074 6578 743d 2249 6d61 6765 2065 7830 text=\"Image ex0\n-00005a00: 3530 206f 6d69 7474 6564 2069 6e20 7465 50 omitted in te\n-00005a10: 7874 2076 6572 7369 6f6e 206f 6620 7468 xt version of th\n-00005a20: 6973 2064 6f63 756d 656e 742e 2200 085d is document.\"..]\n-00005a30: 0a0a 1f0a 4669 6c65 3a20 736b 6574 6368 ....File: sketch\n-00005a40: 2e69 6e66 6f2c 2020 4e6f 6465 3a20 5265 .info, Node: Re\n-00005a50: 7065 6174 6564 206f 626a 6563 7473 2c20 peated objects, \n-00005a60: 204e 6578 743a 2053 7765 7074 206f 626a Next: Swept obj\n-00005a70: 6563 7473 2c20 2050 7265 763a 204f 626a ects, Prev: Obj\n-00005a80: 6563 7420 7472 616e 7366 6f72 6d73 2c20 ect transforms, \n-00005a90: 2055 703a 2049 6e74 726f 6475 6374 696f Up: Introductio\n-00005aa0: 6e20 6279 2065 7861 6d70 6c65 0a0a 322e n by example..2.\n-00005ab0: 3620 5265 7065 6174 6564 206f 626a 6563 6 Repeated objec\n-00005ac0: 7473 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ts.=============\n-00005ad0: 3d3d 3d3d 3d3d 3d0a 0a54 6f20 6472 6177 =======..To draw\n-00005ae0: 2073 6576 656e 2069 6e73 7461 6e63 6573 seven instances\n-00005af0: 206f 6620 7468 6520 7465 7472 6168 6564 of the tetrahed\n-00005b00: 726f 6e2c 2065 6163 6820 6469 6666 6572 ron, each differ\n-00005b10: 696e 6720 6672 6f6d 2074 6865 206c 6173 ing from the las\n-00005b20: 740a 6279 2074 6865 2073 616d 6520 7472 t.by the same tr\n-00005b30: 616e 7366 6f72 6d2c 2072 6570 6c61 6365 ansform, replace\n-00005b40: 2074 6865 206c 6173 7420 7477 6f20 636f the last two co\n-00005b50: 6d6d 616e 6473 206f 6620 7468 6520 7072 mmands of the pr\n-00005b60: 6576 696f 7573 0a65 7861 6d70 6c65 2077 evious.example w\n-00005b70: 6974 680a 2020 7265 7065 6174 207b 2037 ith. repeat { 7\n-00005b80: 2c20 726f 7461 7465 2831 352c 2028 302c , rotate(15, (0,\n-00005b90: 302c 3029 2c20 5b31 2c30 2c30 5d29 2025 0,0), [1,0,0]) %\n-00005ba0: 2063 6f70 7920 696e 206e 6577 2070 6f73 copy in new pos\n-00005bb0: 6974 696f 6e0a 2020 2020 2020 2020 2020 ition. \n-00005bc0: 2020 2020 7468 656e 2074 7261 6e73 6c61 then transla\n-00005bd0: 7465 285b 322c 302c 305d 2920 7d20 7b70 te([2,0,0]) } {p\n-00005be0: 6965 7263 6564 5f74 6574 7261 6865 6472 ierced_tetrahedr\n-00005bf0: 6f6e 7d0a 416e 6420 7468 6520 7265 7375 on}.And the resu\n-00005c00: 6c74 2e2e 2e0a 0a00 085b 696d 6167 6520 lt.......[image \n-00005c10: 7372 633d 2265 7830 3630 2e70 6e67 2220 src=\"ex060.png\" \n-00005c20: 7465 7874 3d22 496d 6167 6520 6578 3036 text=\"Image ex06\n-00005c30: 3020 6f6d 6974 7465 6420 696e 2074 6578 0 omitted in tex\n-00005c40: 7420 7665 7273 696f 6e20 6f66 2074 6869 t version of thi\n-00005c50: 7320 646f 6375 6d65 6e74 2e22 0008 5d0a s document.\"..].\n-00005c60: 0a1f 0a46 696c 653a 2073 6b65 7463 682e ...File: sketch.\n-00005c70: 696e 666f 2c20 204e 6f64 653a 2053 7765 info, Node: Swe\n-00005c80: 7074 206f 626a 6563 7473 2c20 2050 7265 pt objects, Pre\n-00005c90: 763a 2052 6570 6561 7465 6420 6f62 6a65 v: Repeated obje\n-00005ca0: 6374 732c 2020 5570 3a20 496e 7472 6f64 cts, Up: Introd\n-00005cb0: 7563 7469 6f6e 2062 7920 6578 616d 706c uction by exampl\n-00005cc0: 650a 0a32 2e37 2053 7765 7074 206f 626a e..2.7 Swept obj\n-00005cd0: 6563 7473 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d ects.===========\n-00005ce0: 3d3d 3d3d 3d3d 0a0a 4d61 6e79 2066 616d ======..Many fam\n-00005cf0: 696c 6961 7220 7368 6170 6573 2063 616e iliar shapes can\n-00005d00: 2062 6520 6765 6e65 7261 7465 6420 6279 be generated by\n-00005d10: 2073 7765 6570 696e 6720 7369 6d70 6c65 sweeping simple\n-00005d20: 7220 6f6e 6573 2074 6872 6f75 6768 0a73 r ones through.s\n-00005d30: 7061 6365 2061 6e64 2063 6f6e 7369 6465 pace and conside\n-00005d40: 7269 6e67 2074 6865 2072 6573 756c 7469 ring the resulti\n-00005d50: 6e67 2070 6174 682c 2073 7572 6661 6365 ng path, surface\n-00005d60: 2c20 6f72 2076 6f6c 756d 652e 2020 e280 , or volume. ..\n-00005d70: 9853 6b65 7463 68e2 8099 0a69 6d70 6c65 .Sketch....imple\n-00005d80: 6d65 6e74 7320 7468 6973 2069 6465 6120 ments this idea \n-00005d90: 696e 2074 6865 2073 7765 6570 2063 6f6d in the sweep com\n-00005da0: 6d61 6e64 2e0a 2020 6465 6620 6e5f 7365 mand.. def n_se\n-00005db0: 6773 2038 0a20 2073 7765 6570 207b 206e gs 8. sweep { n\n-00005dc0: 5f73 6567 732c 2072 6f74 6174 6528 3138 _segs, rotate(18\n-00005dd0: 3020 2f20 6e5f 7365 6773 2c20 2830 2c30 0 / n_segs, (0,0\n-00005de0: 2c30 292c 205b 302c 302c 315d 2920 7d20 ,0), [0,0,1]) } \n-00005df0: 2831 2c30 2c30 290a 5468 6973 2063 6f64 (1,0,0).This cod\n-00005e00: 6520 7377 6565 7073 2074 6865 2070 6f69 e sweeps the poi\n-00005e10: 6e74 2028 312c 302c 3029 2065 6967 6874 nt (1,0,0) eight\n-00005e20: 2074 696d 6573 2062 7920 726f 7461 7469 times by rotati\n-00005e30: 6e67 2069 7420 3138 302f 3820 3d0a 3232 ng it 180/8 =.22\n-00005e40: 2e35 2064 6567 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Here.is wha\n-000061e0: 7420 6974 206c 6f6f 6b73 206c 696b 652e t it looks like.\n-000061f0: 0a0a 0008 5b69 6d61 6765 2073 7263 3d22 ....[image src=\"\n-00006200: 6578 3037 302e 706e 6722 2074 6578 743d ex070.png\" text=\n-00006210: 2249 6d61 6765 2065 7830 3730 206f 6d69 \"Image ex070 omi\n-00006220: 7474 6564 2069 6e20 7465 7874 2076 6572 tted in text ver\n-00006230: 7369 6f6e 206f 6620 7468 6973 2064 6f63 sion of this doc\n-00006240: 756d 656e 742e 2200 085d 0a0a 2020 2054 ument.\"..].. T\n-00006250: 6869 7320 6973 2074 6865 2066 6972 7374 his is the first\n-00006260: 2065 7861 6d70 6c65 2077 6520 6861 7665 example we have\n-00006270: 2073 6565 6e20 6f66 20e2 8098 736b 6574 seen of ...sket\n-00006280: 6368 e280 9920 6172 6974 686d 6574 6963 ch... arithmetic\n-00006290: 2e20 2054 6865 0a65 7870 7265 7373 696f . The.expressio\n-000062a0: 6e20 3138 3020 2f20 6e5f 7365 6773 2063 n 180 / n_segs c\n-000062b0: 6175 7365 7320 7468 6520 6569 6768 7420 auses the eight \n-000062c0: 726f 7461 7469 6f6e 7320 746f 2061 6464 rotations to add\n-000062d0: 2074 6f20 3138 302e 2020 4966 0a79 6f75 to 180. 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Code an\n-000063f0: 6420 7265 7375 6c74 2066 6f6c 6c6f 770a d result follow.\n-00006400: 2020 6465 6620 6e5f 7365 6773 2038 0a20 def n_segs 8. \n-00006410: 2073 7765 6570 207b 206e 5f73 6567 733c sweep { n_segs<\n-00006420: 3e2c 2072 6f74 6174 6528 3138 3020 2f20 >, rotate(180 / \n-00006430: 6e5f 7365 6773 2c20 2830 2c30 2c30 292c n_segs, (0,0,0),\n-00006440: 205b 302c 302c 315d 2920 7d20 2831 2c30 [0,0,1]) } (1,0\n-00006450: 2c30 290a 0008 5b69 6d61 6765 2073 7263 ,0)...[image src\n-00006460: 3d22 6578 3038 302e 706e 6722 2074 6578 =\"ex080.png\" tex\n-00006470: 743d 2249 6d61 6765 2065 7830 3830 206f t=\"Image ex080 o\n-00006480: 6d69 7474 6564 2069 6e20 7465 7874 2076 mitted in text v\n-00006490: 6572 7369 6f6e 206f 6620 7468 6973 2064 ersion of this d\n-000064a0: 6f63 756d 656e 742e 2200 085d 0a0a 1f0a ocument.\"..]....\n-000064b0: 4669 6c65 3a20 736b 6574 6368 2e69 6e66 File: sketch.inf\n-000064c0: 6f2c 2020 4e6f 6465 3a20 506f 6c79 6c69 o, Node: Polyli\n-000064d0: 6e65 2073 7765 6570 732c 2020 4e65 7874 ne sweeps, Next\n-000064e0: 3a20 4e65 7374 6564 2073 7765 6570 732c : Nested sweeps,\n-000064f0: 2020 5072 6576 3a20 506f 696e 7420 7377 Prev: Point sw\n-00006500: 6565 7073 2c20 2055 703a 2053 7765 7074 eeps, Up: Swept\n-00006510: 206f 626a 6563 7473 0a0a 322e 372e 3220 objects..2.7.2 \n-00006520: 506f 6c79 6c69 6e65 2073 7765 6570 730a Polyline sweeps.\n-00006530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ----------------\n-00006540: 2d2d 2d2d 2d0a 0a53 7765 6570 696e 6720 -----..Sweeping \n-00006550: 6120 706f 6c79 6c69 6e65 2070 726f 6475 a polyline produ\n-00006560: 6365 7320 6120 7375 7266 6163 6520 636f ces a surface co\n-00006570: 6d70 6f73 6564 206f 6620 6d61 6e79 2066 mposed of many f\n-00006580: 6163 6573 2e20 2054 6865 0a75 6e62 726f aces. 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This def is a\n-00006800: 20e2 809c 7665 6374 6f72 e280 9d20 6465 ...vector... de\n-00006810: 6669 6e69 7469 6f6e 2c20 7768 6963 6820 finition, which \n-00006820: 6d75 7374 2062 6520 7265 6665 7265 6e63 must be referenc\n-00006830: 6564 0a77 6974 6820 7371 7561 7265 2062 ed.with square b\n-00006840: 7261 636b 6574 732c 2065 2e67 2e20 5b66 rackets, e.g. [f\n-00006850: 6f6f 5d2e 0a0a 1f0a 4669 6c65 3a20 736b oo].....File: sk\n-00006860: 6574 6368 2e69 6e66 6f2c 2020 4e6f 6465 etch.info, Node\n-00006870: 3a20 4e65 7374 6564 2073 7765 6570 732c : Nested sweeps,\n-00006880: 2020 4e65 7874 3a20 506f 6c79 676f 6e20 Next: Polygon \n-00006890: 7377 6565 7073 2c20 2050 7265 763a 2050 sweeps, Prev: P\n-000068a0: 6f6c 796c 696e 6520 7377 6565 7073 2c20 olyline sweeps, \n-000068b0: 2055 703a 2053 7765 7074 206f 626a 6563 Up: Swept objec\n-000068c0: 7473 0a0a 322e 372e 3320 4e65 7374 6564 ts..2.7.3 Nested\n-000068d0: 2073 7765 6570 730a 2d2d 2d2d 2d2d 2d2d sweeps.--------\n-000068e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a57 6865 -----------..Whe\n-000068f0: 6e20 7468 6520 6365 6e74 6572 2070 6f69 n the center poi\n-00006900: 6e74 206f 6620 726f 7461 7469 6f6e 2069 nt of rotation i\n-00006910: 7320 6f6d 6974 7465 642c 2074 6865 206f s omitted, the o\n-00006920: 7269 6769 6e20 6973 2061 7373 756d 6564 rigin is assumed\n-00006930: 2e0a 5768 656e 2061 2070 6f69 6e74 2068 ..When a point h\n-00006940: 6173 206f 6e6c 7920 7477 6f20 636f 6f72 as only two coor\n-00006950: 6469 6e61 7465 732c 2074 6865 7920 6172 dinates, they ar\n-00006960: 6520 7461 6b65 6e20 6173 2078 2061 6e64 e taken as x and\n-00006970: 2079 2c20 7769 7468 0a7a 3d30 2061 7373 y, with.z=0 ass\n-00006980: 756d 6564 2e20 2041 2074 6f72 6f69 6420 umed. A toroid \n-00006990: 6973 2074 6865 7265 666f 7265 206f 6274 is therefore obt\n-000069a0: 6169 6e65 6420 7769 7468 2074 6869 7320 ained with this \n-000069b0: 636f 6465 2e0a 2020 6465 6620 6e5f 746f code.. def n_to\n-000069c0: 726f 6964 5f73 6567 7320 3230 2020 2064 roid_segs 20 d\n-000069d0: 6566 206e 5f63 6972 636c 655f 7365 6773 ef n_circle_segs\n-000069e0: 2031 360a 2020 6465 6620 725f 6d69 6e6f 16. def r_mino\n-000069f0: 7220 3120 2020 2020 2020 2020 2064 6566 r 1 def\n-00006a00: 2072 5f6d 616a 6f72 2031 2e35 0a20 2073 r_major 1.5. s\n-00006a10: 7765 6570 207b 206e 5f74 6f72 6f69 645f weep { n_toroid_\n-00006a20: 7365 6773 2c20 726f 7461 7465 2833 3630 segs, rotate(360\n-00006a30: 202f 206e 5f74 6f72 6f69 645f 7365 6773 / n_toroid_segs\n-00006a40: 2c20 5b30 2c31 2c30 5d29 207d 0a20 2020 , [0,1,0]) }. \n-00006a50: 2073 7765 6570 207b 206e 5f63 6972 636c sweep { n_circl\n-00006a60: 655f 7365 6773 2c20 726f 7461 7465 2833 e_segs, rotate(3\n-00006a70: 3630 202f 206e 5f63 6972 636c 655f 7365 60 / n_circle_se\n-00006a80: 6773 2c20 2872 5f6d 616a 6f72 2c30 2c30 gs, (r_major,0,0\n-00006a90: 2929 207d 200a 2020 2020 2020 2872 5f6d )) } . (r_m\n-00006aa0: 616a 6f72 202b 2072 5f6d 696e 6f72 2c20 ajor + r_minor, \n-00006ab0: 3029 0a20 2020 466f 7220 696e 7475 6974 0). For intuit\n-00006ac0: 696f 6e2c 2074 6865 2069 6465 6120 6f66 ion, the idea of\n-00006ad0: 2074 6865 2063 6f64 6520 6973 2074 6f20 the code is to \n-00006ae0: 736b 6574 6368 2061 2063 6972 636c 6520 sketch a circle \n-00006af0: 746f 2074 6865 0a72 6967 6874 206f 6620 to the.right of \n-00006b00: 7468 6520 6f72 6967 696e 2069 6e20 7468 the origin in th\n-00006b10: 6520 7879 2d70 6c61 6e65 2c20 7468 656e e xy-plane, then\n-00006b20: 2072 6f74 6174 6520 7468 6174 2063 6972 rotate that cir\n-00006b30: 636c 6520 226f 7574 206f 6620 7468 650a cle \"out of the.\n-00006b40: 706c 616e 6522 2061 626f 7574 2074 6865 plane\" about the\n-00006b50: 2079 2d61 7869 7320 746f 206d 616b 6520 y-axis to make \n-00006b60: 7468 6520 6669 6e61 6c20 6669 6775 7265 the final figure\n-00006b70: 2e20 2054 6869 7320 7072 6f64 7563 6573 . This produces\n-00006b80: 2074 6865 0a66 6f6c 6c6f 7769 6e67 2e20 the.following. \n-00006b90: 2028 4120 7669 6577 2072 6f74 6174 696f (A view rotatio\n-00006ba0: 6e20 616e 6420 736f 6d65 2061 7865 7320 n and some axes \n-00006bb0: 6861 7665 2062 6565 6e20 6164 6465 642e have been added.\n-00006bc0: 290a 0a00 085b 696d 6167 6520 7372 633d )....[image src=\n-00006bd0: 2265 7830 3930 2e70 6e67 2220 7465 7874 \"ex090.png\" text\n-00006be0: 3d22 496d 6167 6520 6578 3039 3020 6f6d =\"Image ex090 om\n-00006bf0: 6974 7465 6420 696e 2074 6578 7420 7665 itted in text ve\n-00006c00: 7273 696f 6e20 6f66 2074 6869 7320 646f rsion of this do\n-00006c10: 6375 6d65 6e74 2e22 0008 5d0a 0a20 2020 cument.\"..].. \n-00006c20: 5468 6973 2065 7861 6d70 6c65 2061 6c73 This example als\n-00006c30: 6f20 7368 6f77 7320 7468 6174 2074 6865 o shows that the\n-00006c40: 2073 7765 7074 206f 626a 6563 7420 6d61 swept object ma\n-00006c50: 7920 6974 7365 6c66 2062 6520 616e 6f74 y itself be anot\n-00006c60: 6865 720a e280 9873 7765 6570 e280 992e her....sweep....\n-00006c70: 2020 496e 2066 6163 742c 2069 7420 6d61 In fact, it ma\n-00006c80: 7920 6265 205f 616e 795f 20e2 8098 736b y be _any_ ...sk\n-00006c90: 6574 6368 e280 9920 6578 7072 6573 7369 etch... expressi\n-00006ca0: 6f6e 2074 6861 7420 7265 7375 6c74 7320 on that results \n-00006cb0: 696e 2061 0a6c 6973 7420 6f66 206f 6e65 in a.list of one\n-00006cc0: 206f 7220 6d6f 7265 2070 6f69 6e74 7320 or more points \n-00006cd0: 6f72 2c20 616c 7465 726e 6174 656c 792c or, alternately,\n-00006ce0: 2061 206c 6973 7420 6f66 206f 6e65 206f a list of one o\n-00006cf0: 7220 6d6f 7265 0a70 6f6c 796c 696e 6573 r more.polylines\n-00006d00: 2061 6e64 2070 6f6c 7967 6f6e 732e 2020 and polygons. \n-00006d10: 5468 6520 6c61 7474 6572 206b 696e 6420 The latter kind \n-00006d20: 6f66 206c 6973 7420 6361 6e20 6265 2063 of list can be c\n-00006d30: 7265 6174 6564 2077 6974 6820 610a 7b20 reated with a.{ \n-00006d40: 7d2d 656e 636c 6f73 6564 2062 6c6f 636b }-enclosed block\n-00006d50: 2c20 7065 7268 6170 7320 666f 6c6c 6f77 , perhaps follow\n-00006d60: 696e 6720 6120 7075 7420 6f72 2072 6570 ing a put or rep\n-00006d70: 6561 742e 0a0a 1f0a 4669 6c65 3a20 736b eat.....File: sk\n-00006d80: 6574 6368 2e69 6e66 6f2c 2020 4e6f 6465 etch.info, Node\n-00006d90: 3a20 506f 6c79 676f 6e20 7377 6565 7073 : Polygon sweeps\n-00006da0: 2c20 204e 6578 743a 2050 6f6c 796c 696e , Next: Polylin\n-00006db0: 6520 7377 6565 7073 2077 6974 6820 636c e sweeps with cl\n-00006dc0: 6f73 7572 652c 2020 5072 6576 3a20 4e65 osure, Prev: Ne\n-00006dd0: 7374 6564 2073 7765 6570 732c 2020 5570 sted sweeps, Up\n-00006de0: 3a20 5377 6570 7420 6f62 6a65 6374 730a : Swept objects.\n-00006df0: 0a32 2e37 2e34 2050 6f6c 7967 6f6e 2073 .2.7.4 Polygon s\n-00006e00: 7765 6570 730a 2d2d 2d2d 2d2d 2d2d 2d2d weeps.----------\n-00006e10: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5377 6565 ----------..Swee\n-00006e20: 7069 6e67 2061 2070 6f6c 7967 6f6e 2063 ping a polygon c\n-00006e30: 7265 6174 6573 2061 2063 6c6f 7365 6420 reates a closed \n-00006e40: 7375 7266 6163 6520 7769 7468 2070 6f6c surface with pol\n-00006e50: 7967 6f6e 7320 6174 2074 6865 2065 6e64 ygons at the end\n-00006e60: 732c 0a77 6869 6368 2061 7265 206a 7573 s,.which are jus\n-00006e70: 7420 636f 7069 6573 206f 6620 7468 6520 t copies of the \n-00006e80: 6f72 6967 696e 616c 2c20 6170 7072 6f70 original, approp\n-00006e90: 7269 6174 656c 7920 706f 7369 7469 6f6e riately position\n-00006ea0: 6564 2e20 202a 4e6f 7465 0a53 6f6c 6964 ed. *Note.Solid\n-00006eb0: 2063 6f69 6c20 6578 616d 706c 653a 3a2e coil example::.\n-00006ec0: 2020 4f70 7469 6f6e 7320 6f6e 2074 6865 Options on the\n-00006ed0: 2073 7765 7074 2070 6f6c 7967 6f6e 2c20 swept polygon, \n-00006ee0: 6966 2074 6865 7920 6578 6973 742c 2061 if they exist, a\n-00006ef0: 7265 0a61 7070 6c69 6564 2074 6f20 7468 re.applied to th\n-00006f00: 6520 656e 6473 2e20 204f 7468 6572 7769 e ends. Otherwi\n-00006f10: 7365 2074 6865 2073 7765 6570 206f 7074 se the sweep opt\n-00006f20: 696f 6e73 2061 7265 2075 7365 6420 7468 ions are used th\n-00006f30: 726f 7567 686f 7574 2e0a 0a1f 0a46 696c roughout.....Fil\n-00006f40: 653a 2073 6b65 7463 682e 696e 666f 2c20 e: sketch.info, \n-00006f50: 204e 6f64 653a 2050 6f6c 796c 696e 6520 Node: Polyline \n-00006f60: 7377 6565 7073 2077 6974 6820 636c 6f73 sweeps with clos\n-00006f70: 7572 652c 2020 4e65 7874 3a20 4166 6669 ure, Next: Affi\n-00006f80: 6e65 2061 7269 7468 6d65 7469 632c 2020 ne arithmetic, \n-00006f90: 5072 6576 3a20 506f 6c79 676f 6e20 7377 Prev: Polygon sw\n-00006fa0: 6565 7073 2c20 2055 703a 2053 7765 7074 eeps, Up: Swept\n-00006fb0: 206f 626a 6563 7473 0a0a 322e 372e 3520 objects..2.7.5 \n-00006fc0: 506f 6c79 6c69 6e65 2073 7765 6570 7320 Polyline sweeps \n-00006fd0: 7769 7468 2063 6c6f 7375 7265 0a2d 2d2d with closure.---\n-00006fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ----------------\n-00006ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------.\n-00007000: 0a41 2070 6f6c 796c 696e 6520 7377 6565 .A polyline swee\n-00007010: 7020 7769 7468 2061 2063 6c6f 7375 7265 p with a closure\n-00007020: 2074 6167 2063 7265 6174 6573 2061 6e6f tag creates ano\n-00007030: 7468 6572 206b 696e 6420 6f66 2063 6c6f ther kind of clo\n-00007040: 7365 640a 7375 7266 6163 652e 2020 4669 sed.surface. Fi\n-00007050: 7273 742c 2074 6865 2070 6f6c 796c 696e rst, the polylin\n-00007060: 6520 7365 676d 656e 7473 2061 7265 2063 e segments are c\n-00007070: 6f6e 6e65 6374 6564 2062 7920 6661 6365 onnected by face\n-00007080: 732c 206a 7573 7420 6173 0a77 6974 686f s, just as.witho\n-00007090: 7574 2074 6865 2063 6c6f 7375 7265 2074 ut the closure t\n-000070a0: 6167 2e20 2054 6865 6e2c 2065 6163 6820 ag. Then, each \n-000070b0: 7365 7420 6f66 2065 6e64 2070 6f69 6e74 set of end point\n-000070c0: 7320 6973 206a 6f69 6e65 6420 746f 206d s is joined to m\n-000070d0: 616b 650a 6120 706f 6c79 676f 6e2c 206f ake.a polygon, o\n-000070e0: 6e65 2066 6f72 2065 6163 6820 656e 642e ne for each end.\n-000070f0: 2020 4120 636f 6465 2066 6f72 2073 6576 A code for sev\n-00007100: 6572 616c 2076 6965 7773 206f 6620 6120 eral views of a \n-00007110: 6379 6c69 6e64 7269 6361 6c0a 7072 6973 cylindrical.pris\n-00007120: 6d20 666f 6c6c 6f77 732e 0a20 2064 6566 m follows.. def\n-00007130: 206e 5f63 796c 5f73 6567 7320 3230 2020 n_cyl_segs 20 \n-00007140: 6465 6620 6e5f 7669 6577 7320 3520 2064 def n_views 5 d\n-00007150: 6566 2049 205b 312c 302c 305d 0a20 2064 ef I [1,0,0]. d\n-00007160: 6566 2065 6e64 6f70 7473 205b 6669 6c6c ef endopts [fill\n-00007170: 636f 6c6f 723d 6c69 6768 7467 7261 795d color=lightgray]\n-00007180: 0a20 2072 6570 6561 7420 7b20 6e5f 7669 . repeat { n_vi\n-00007190: 6577 732c 2072 6f74 6174 6528 3138 302f ews, rotate(180/\n-000071a0: 6e5f 7669 6577 732c 205b 495d 2920 7468 n_views, [I]) th\n-000071b0: 656e 2074 7261 6e73 6c61 7465 285b 495d en translate([I]\n-000071c0: 202a 2032 2e31 2920 7d20 0a20 2020 2073 * 2.1) } . s\n-000071d0: 7765 6570 5b65 6e64 6f70 7473 5d7b 206e weep[endopts]{ n\n-000071e0: 5f63 796c 5f73 6567 733c 3e2c 2072 6f74 _cyl_segs<>, rot\n-000071f0: 6174 6528 3336 302f 6e5f 6379 6c5f 7365 ate(360/n_cyl_se\n-00007200: 6773 2c20 5b30 2c31 2c30 5d29 207d 200a gs, [0,1,0]) } .\n-00007210: 2020 2020 2020 6c69 6e65 5b66 696c 6c63 line[fillc\n-00007220: 6f6c 6f72 3d77 6869 7465 5d28 312c 2d31 olor=white](1,-1\n-00007230: 2928 312c 3129 0a49 7420 7072 6f64 7563 )(1,1).It produc\n-00007240: 6573 2074 6869 7320 6472 6177 696e 672e es this drawing.\n-00007250: 0a0a 0008 5b69 6d61 6765 2073 7263 3d22 ....[image src=\"\n-00007260: 6578 3131 302e 706e 6722 2074 6578 743d ex110.png\" text=\n-00007270: 2249 6d61 6765 2065 7831 3130 206f 6d69 \"Image ex110 omi\n-00007280: 7474 6564 2069 6e20 7465 7874 2076 6572 tted in text ver\n-00007290: 7369 6f6e 206f 6620 7468 6973 2064 6f63 sion of this doc\n-000072a0: 756d 656e 742e 2200 085d 0a0a 5468 6520 ument.\"..]..The \n-000072b0: 6f70 7469 6f6e 7320 6f66 2074 6865 2073 options of the s\n-000072c0: 7765 7074 206c 696e 652c 2069 6620 616e wept line, if an\n-000072d0: 792c 2061 7265 2061 7070 6c69 6564 2074 y, are applied t\n-000072e0: 6f20 7468 6520 6661 6365 7320 7072 6f64 o the faces prod\n-000072f0: 7563 6564 0a62 7920 7377 6565 7069 6e67 uced.by sweeping\n-00007300: 2074 6865 206c 696e 652c 2062 7574 206e the line, but n\n-00007310: 6f74 2074 6865 2065 6e64 2070 6f6c 7967 ot the end polyg\n-00007320: 6f6e 732e 2020 4f74 6865 7277 6973 652c ons. Otherwise,\n-00007330: 2074 6865 2073 7765 6570 0a6f 7074 696f the sweep.optio\n-00007340: 6e73 2061 7265 2061 7070 6c69 6564 2074 ns are applied t\n-00007350: 6872 6f75 6768 6f75 742e 2020 5468 6520 hroughout. The \n-00007360: 6465 6620 696e 2074 6869 7320 6578 616d def in this exam\n-00007370: 706c 6520 6973 2061 6e20 e280 9c6f 7074 ple is an ...opt\n-00007380: 696f 6ee2 809d 0a64 6566 696e 6974 696f ion....definitio\n-00007390: 6e2e 2020 5265 6665 7265 6e63 6573 2074 n. References t\n-000073a0: 6f20 6f70 7469 6f6e 7320 6d75 7374 2062 o options must b\n-000073b0: 6520 656e 636c 6f73 6564 2069 6e20 7371 e enclosed in sq\n-000073c0: 7561 7265 2062 7261 636b 6574 732c 0a65 uare brackets,.e\n-000073d0: 2e67 2e20 2020 5b66 6f6f 5d2e 2020 596f .g. [foo]. Yo\n-000073e0: 7520 6361 6e20 636f 6e63 6174 656e 6174 u can concatenat\n-000073f0: 6520 7365 7665 7261 6c20 7365 7473 206f e several sets o\n-00007400: 6620 6f70 7469 6f6e 7320 7769 7468 2061 f options with a\n-00007410: 2073 696e 676c 650a 7265 6665 7265 6e63 single.referenc\n-00007420: 652c 2065 2e67 2e20 2020 5b63 6f6c 6f72 e, e.g. [color\n-00007430: 2c20 7369 7a65 2c20 7374 796c 655d 2020 , size, style] \n-00007440: 776f 756c 6420 6361 7573 6520 7468 6520 would cause the \n-00007450: 6f70 7469 6f6e 0a64 6566 696e 6974 696f option.definitio\n-00007460: 6e73 2066 6f72 20e2 8098 636f 6c6f 72e2 ns for ...color.\n-00007470: 8099 2c20 e280 9873 697a 65e2 8099 2c20 .., ...size..., \n-00007480: 616e 6420 e280 9873 7479 6c65 e280 9920 and ...style... \n-00007490: 746f 2061 7070 6561 7220 696e 2073 6571 to appear in seq\n-000074a0: 7565 6e63 6520 696e 0a74 6865 206f 7574 uence in.the out\n-000074b0: 7075 7420 6372 6561 7465 6420 6279 2074 put created by t\n-000074c0: 6865 20e2 8098 736b 6574 6368 e280 9920 he ...sketch... \n-000074d0: 636f 6d6d 616e 6420 636f 6e74 6169 6e69 command containi\n-000074e0: 6e67 2074 6865 2072 6566 6572 656e 6365 ng the reference\n-000074f0: 2e0a 4861 7070 696c 792c 2074 6865 2073 ..Happily, the s\n-00007500: 796e 7461 7820 6f66 20e2 8098 736b 6574 yntax of ...sket\n-00007510: 6368 e280 9920 6973 2073 7563 6820 7468 ch... is such th\n-00007520: 6174 206f 7074 696f 6e73 2072 6566 6572 at options refer\n-00007530: 656e 6365 7320 6361 6e0a 6e65 7665 7220 ences can.never \n-00007540: 6265 2063 6f6e 6675 7365 6420 7769 7468 be confused with\n-00007550: 2076 6563 746f 7220 7265 6665 7265 6e63 vector referenc\n-00007560: 6573 2e20 2057 6869 6c65 206e 6f74 2061 es. While not a\n-00007570: 7070 6172 656e 7420 696e 2074 6869 730a pparent in this.\n-00007580: 6578 616d 706c 652c 206f 7074 696f 6e73 example, options\n-00007590: 2072 6566 6572 656e 6365 7320 6172 6520 references are \n-000075a0: 7573 6566 756c 2077 6865 6e20 6465 6669 useful when defi\n-000075b0: 6e69 6e67 206d 616e 7920 6f62 6a65 6374 ning many object\n-000075c0: 7320 7769 7468 2061 0a73 696d 696c 6172 s with a.similar\n-000075d0: 2061 7070 6561 7261 6e63 652e 0a0a 1f0a appearance.....\n-000075e0: 4669 6c65 3a20 736b 6574 6368 2e69 6e66 File: sketch.inf\n-000075f0: 6f2c 2020 4e6f 6465 3a20 4166 6669 6e65 o, Node: Affine\n-00007600: 2061 7269 7468 6d65 7469 632c 2020 4e65 arithmetic, Ne\n-00007610: 7874 3a20 4d6f 7265 2074 6f20 6c65 6172 xt: More to lear\n-00007620: 6e2c 2020 5072 6576 3a20 506f 6c79 6c69 n, Prev: Polyli\n-00007630: 6e65 2073 7765 6570 7320 7769 7468 2063 ne sweeps with c\n-00007640: 6c6f 7375 7265 2c20 2055 703a 2053 7765 losure, Up: Swe\n-00007650: 7074 206f 626a 6563 7473 0a0a 322e 372e pt objects..2.7.\n-00007660: 3620 4166 6669 6e65 2061 7269 7468 6d65 6 Affine arithme\n-00007670: 7469 630a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d tic.------------\n-00007680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The\n-00007690: 2061 7269 7468 6d65 7469 6320 5b49 5d20 arithmetic [I] \n-000076a0: 2a20 322e 3120 6162 6f76 6520 6869 6e74 * 2.1 above hint\n-000076b0: 7320 6174 2061 206c 6172 6765 7220 7472 s at a larger tr\n-000076c0: 7574 682e 2020 e280 9853 6b65 7463 68e2 uth. ...Sketch.\n-000076d0: 8099 0a6f 7065 7261 746f 7273 2077 6f72 ...operators wor\n-000076e0: 6b20 6f6e 2073 6361 6c61 7273 2c20 7665 k on scalars, ve\n-000076f0: 6374 6f72 732c 2070 6f69 6e74 732c 2061 ctors, points, a\n-00007700: 6e64 2074 7261 6e73 666f 726d 7320 6163 nd transforms ac\n-00007710: 636f 7264 696e 6720 746f 0a74 6865 2067 cording to.the g\n-00007720: 656e 6572 616c 2072 756c 6573 206f 6620 eneral rules of \n-00007730: e280 9c61 6666 696e 6520 616c 6765 6272 ...affine algebr\n-00007740: 61e2 809d 2e20 2054 6869 7320 6361 6e20 a.... This can \n-00007750: 6265 2068 656c 7066 756c 2066 6f72 2073 be helpful for s\n-00007760: 6574 7469 6e67 0a75 7020 6469 6167 7261 etting.up diagra\n-00007770: 6d73 2077 6974 6820 636f 6d70 7574 6564 ms with computed\n-00007780: 2067 656f 6d65 7472 792e 2020 466f 7220 geometry. For \n-00007790: 6578 616d 706c 652c 2069 6620 796f 7520 example, if you \n-000077a0: 6861 7665 2074 7269 616e 676c 650a 7665 have triangle.ve\n-000077b0: 7274 6963 6573 2028 7031 2920 7468 726f rtices (p1) thro\n-000077c0: 7567 6820 2870 3329 2061 6e64 206e 6565 ugh (p3) and nee\n-000077d0: 6420 746f 2064 7261 7720 6120 756e 6974 d to draw a unit\n-000077e0: 206e 6f72 6d61 6c20 7665 6374 6f72 0a70 normal vector.p\n-000077f0: 6f69 6e74 696e 6720 6f75 7420 6f66 2074 ointing out of t\n-00007800: 6865 2063 656e 7465 7220 6f66 2074 6865 he center of the\n-00007810: 2074 7269 616e 676c 652c 2074 6869 7320 triangle, this \n-00007820: 636f 6465 2064 6f65 7320 7468 6520 7472 code does the tr\n-00007830: 6963 6b2e 0a20 2064 6566 2070 3120 2831 ick.. def p1 (1\n-00007840: 2c30 2c30 2920 2064 6566 2070 3220 2830 ,0,0) def p2 (0\n-00007850: 2c30 2e35 2c30 2920 2064 6566 2070 3320 ,0.5,0) def p3 \n-00007860: 282d 302e 352c 2d31 2c32 290a 2020 6465 (-0.5,-1,2). de\n-00007870: 6620 4f20 2830 2c30 2c30 290a 2020 6465 f O (0,0,0). de\n-00007880: 6620 4e20 756e 6974 2820 2828 7033 2920 f N unit( ((p3) \n-00007890: 2d20 2870 3229 2920 2a20 2828 7031 2920 - (p2)) * ((p1) \n-000078a0: 2d20 2870 3229 2920 290a 2020 6465 6620 - (p2)) ). def \n-000078b0: 6e31 2028 2870 3129 2d28 4f29 202b 2028 n1 ((p1)-(O) + (\n-000078c0: 7032 292d 284f 2920 2b20 2870 3329 2d28 p2)-(O) + (p3)-(\n-000078d0: 4f29 2920 2f20 3320 2b20 284f 290a 2020 O)) / 3 + (O). \n-000078e0: 6465 6620 6e32 2028 6e31 292b 5b4e 5d0a def n2 (n1)+[N].\n-000078f0: 2020 706f 6c79 676f 6e28 7031 2928 7032 polygon(p1)(p2\n-00007900: 2928 7033 290a 2020 6c69 6e65 5b61 7272 )(p3). line[arr\n-00007910: 6f77 733d 2a2d 3e5d 286e 3129 286e 3229 ows=*->](n1)(n2)\n-00007920: 0a54 6865 2066 6972 7374 206c 696e 6520 .The first line \n-00007930: 636f 6d70 7574 6573 2074 6865 2063 726f computes the cro\n-00007940: 7373 2070 726f 6475 6374 206f 6620 7477 ss product of tw\n-00007950: 6f20 6564 6765 2076 6563 746f 7273 206f o edge vectors o\n-00007960: 6620 7468 650a 7472 6961 6e67 6c65 2061 f the.triangle a\n-00007970: 6e64 2073 6361 6c65 7320 6974 2074 6f20 nd scales it to \n-00007980: 756e 6974 206c 656e 6774 682e 2020 5468 unit length. Th\n-00007990: 6520 7365 636f 6e64 2063 6f6d 7075 7465 e second compute\n-000079a0: 7320 7468 6520 6176 6572 6167 650a 6f66 s the average.of\n-000079b0: 2074 6865 2076 6572 7469 6365 732e 2020 the vertices. \n-000079c0: 4e6f 7465 2074 6861 7420 7375 6274 7261 Note that subtra\n-000079d0: 6374 696f 6e20 616e 6420 6164 6469 7469 ction and additi\n-000079e0: 6f6e 206f 6620 7468 6520 6f72 6967 696e on of the origin\n-000079f0: 0a65 6666 6563 7469 7665 6c79 2063 6f6e .effectively con\n-00007a00: 7665 7274 2076 6563 746f 7273 2074 6f20 vert vectors to \n-00007a10: 706f 696e 7473 2061 6e64 205f 7669 6365 points and _vice\n-00007a20: 2076 6572 7361 5f2e 2020 5468 6520 6c69 versa_. The li\n-00007a30: 6e65 0a63 6f6d 6d61 6e64 2064 7261 7773 ne.command draws\n-00007a40: 2074 6865 206e 6f72 6d61 6c20 6174 2074 the normal at t\n-00007a50: 6865 2063 6f72 7265 6374 2073 706f 742e he correct spot.\n-00007a60: 0a0a 0008 5b69 6d61 6765 2073 7263 3d22 ....[image src=\"\n-00007a70: 6578 3130 302e 706e 6722 2074 6578 743d ex100.png\" text=\n-00007a80: 2249 6d61 6765 2065 7831 3030 206f 6d69 \"Image ex100 omi\n-00007a90: 7474 6564 2069 6e20 7465 7874 2076 6572 tted in text ver\n-00007aa0: 7369 6f6e 206f 6620 7468 6973 2064 6f63 sion of this doc\n-00007ab0: 756d 656e 742e 2200 085d 0a0a 2020 2054 ument.\"..].. T\n-00007ac0: 776f 2063 6176 6561 7473 2072 6567 6172 wo caveats regar\n-00007ad0: 6469 6e67 2074 6869 7320 6578 616d 706c ding this exampl\n-00007ae0: 6520 7265 6d61 696e 2e20 2046 6972 7374 e remain. First\n-00007af0: 2c20 7468 6520 6f6e 6c79 2077 6179 2074 , the only way t\n-00007b00: 6f0a 7573 6520 e280 9850 5354 7269 636b o.use ...PSTrick\n-00007b10: 73e2 8099 2d73 7479 6c65 2061 7272 6f77 s...-style arrow\n-00007b20: 7320 6973 2077 6974 6820 6172 726f 7773 s is with arrows\n-00007b30: 3d2e 2020 5468 6520 616c 7465 726e 6174 =. The alternat\n-00007b40: 6976 6520 7379 6e74 6178 2066 6f72 0ae2 ive syntax for..\n-00007b50: 8098 5053 5472 6963 6b73 e280 9920 6172 ..PSTricks... ar\n-00007b60: 726f 7773 2069 7320 6e6f 7420 616c 6c6f rows is not allo\n-00007b70: 7765 6420 696e 2073 6b65 7463 682e 2020 wed in sketch. \n-00007b80: 5365 636f 6e64 2c20 796f 7520 6d69 6768 Second, you migh\n-00007b90: 7420 6c69 6b65 2074 6f0a 656c 696d 696e t like to.elimin\n-00007ba0: 6174 6520 7468 6520 7468 6972 6420 6465 ate the third de\n-00007bb0: 6620 616e 6420 7772 6974 6520 696e 7374 f and write inst\n-00007bc0: 6561 6420 7468 6520 666f 6c6c 6f77 696e ead the followin\n-00007bd0: 672e 0a20 206c 696e 655b 6172 726f 7773 g.. line[arrows\n-00007be0: 3d2a 2d3e 5d28 6e31 2920 286e 3129 2b5b =*->](n1) (n1)+[\n-00007bf0: 4e5d 0a54 6869 7320 6973 206e 6f74 2061 N].This is not a\n-00007c00: 6c6c 6f77 6564 2e20 2054 6865 2070 6f69 llowed. The poi\n-00007c10: 6e74 206c 6973 7473 2069 6e20 6472 6177 nt lists in draw\n-00007c20: 6162 6c65 7320 6d61 7920 636f 6e73 6973 ables may consis\n-00007c30: 7420 6f6e 6c79 206f 660a 6578 706c 6963 t only of.explic\n-00007c40: 6974 2070 6f69 6e74 7320 6f72 2070 6f69 it points or poi\n-00007c50: 6e74 2072 6566 6572 656e 6365 732e 2020 nt references. \n-00007c60: 596f 7520 6d61 792c 2068 6f77 6576 6572 You may, however\n-00007c70: 2c20 7573 6520 6172 6974 686d 6574 6963 , use arithmetic\n-00007c80: 0a74 6f20 6361 6c63 756c 6174 6520 706f .to calculate po\n-00007c90: 696e 7420 636f 6d70 6f6e 656e 7473 2e20 int components. \n-00007ca0: 2054 6865 2066 6f6c 6c6f 7769 6e67 2077 The following w\n-00007cb0: 6f72 6b73 2c20 7468 6f75 6768 2069 7427 orks, though it'\n-00007cc0: 7320 610a 6c69 7474 6c65 2063 756d 6265 s a.little cumbe\n-00007cd0: 7273 6f6d 652e 0a20 206c 696e 655b 6172 rsome.. line[ar\n-00007ce0: 726f 7773 3d2a 2d3e 5d28 6e31 2928 286e rows=*->](n1)((n\n-00007cf0: 3129 2778 2b28 4e29 2778 2c20 286e 3129 1)'x+(N)'x, (n1)\n-00007d00: 2779 2b28 4e29 2779 2c20 286e 3129 277a 'y+(N)'y, (n1)'z\n-00007d10: 2b28 4e29 277a 290a 4f62 7669 6f75 736c +(N)'z).Obviousl\n-00007d20: 792c 2074 6865 20e2 809c 7469 636b 206f y, the ...tick o\n-00007d30: 7065 7261 746f 72e2 809d 20e2 8098 2778 perator... ...'x\n-00007d40: e280 9920 6578 7472 6163 7473 2063 6f6d ... extracts com\n-00007d50: 706f 6e65 6e74 7320 6f66 2070 6f69 6e74 ponents of point\n-00007d60: 7320 616e 640a 7665 6374 6f72 732e 0a0a s and.vectors...\n-00007d70: 1f0a 4669 6c65 3a20 736b 6574 6368 2e69 ..File: sketch.i\n-00007d80: 6e66 6f2c 2020 4e6f 6465 3a20 4d6f 7265 nfo, Node: More\n-00007d90: 2074 6f20 6c65 6172 6e2c 2020 5072 6576 to learn, Prev\n-00007da0: 3a20 4166 6669 6e65 2061 7269 7468 6d65 : Affine arithme\n-00007db0: 7469 632c 2020 5570 3a20 5377 6570 7420 tic, Up: Swept \n-00007dc0: 6f62 6a65 6374 730a 0a32 2e37 2e37 204d objects..2.7.7 M\n-00007dd0: 6f72 6520 746f 206c 6561 726e 0a2d 2d2d ore to learn.---\n-00007de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ----------------\n-00007df0: 0a0a 5468 6973 2069 7320 6e6f 7420 7468 ..This is not th\n-00007e00: 6520 656e 6420 6f66 2074 6865 2073 746f e end of the sto\n-00007e10: 7279 206f 6e20 7377 6565 7073 2120 2057 ry on sweeps! W\n-00007e20: 6520 696e 7669 7465 2074 6865 2072 6561 e invite the rea\n-00007e30: 6465 7220 696e 746f 0a74 6865 206d 6169 der into.the mai\n-00007e40: 6e20 626f 6479 206f 6620 7468 6973 2064 n body of this d\n-00007e50: 6f63 756d 656e 7461 7469 6f6e 202a 6e6f ocumentation *no\n-00007e60: 7465 2053 7765 6570 733a 3a20 746f 206c te Sweeps:: to l\n-00007e70: 6561 726e 206d 6f72 652e 0a0a 0008 5b69 earn more.....[i\n-00007e80: 6d61 6765 2073 7263 3d22 6578 3132 302e mage src=\"ex120.\n-00007e90: 706e 6722 2074 6578 743d 2249 6d61 6765 png\" text=\"Image\n-00007ea0: 2065 7831 3230 206f 6d69 7474 6564 2069 ex120 omitted i\n-00007eb0: 6e20 7465 7874 2076 6572 7369 6f6e 206f n text version o\n-00007ec0: 6620 7468 6973 2064 6f63 756d 656e 742e f this document.\n-00007ed0: 2200 085d 0a0a 5768 6f20 6b6e 6f77 7320 \"..]..Who knows \n-00007ee0: 7768 6572 6520 796f 7527 6c6c 2066 696e where you'll fin\n-00007ef0: 6973 683f 0a0a 1f0a 4669 6c65 3a20 736b ish?....File: sk\n-00007f00: 6574 6368 2e69 6e66 6f2c 2020 4e6f 6465 etch.info, Node\n-00007f10: 3a20 496e 7075 7420 6c61 6e67 7561 6765 : Input language\n-00007f20: 2c20 204e 6578 743a 2042 7569 6c64 696e , Next: Buildin\n-00007f30: 6720 6120 6472 6177 696e 672c 2020 5072 g a drawing, Pr\n-00007f40: 6576 3a20 496e 7472 6f64 7563 7469 6f6e ev: Introduction\n-00007f50: 2062 7920 6578 616d 706c 652c 2020 5570 by example, Up\n-00007f60: 3a20 546f 700a 0a33 2049 6e70 7574 206c : Top..3 Input l\n-00007f70: 616e 6775 6167 650a 2a2a 2a2a 2a2a 2a2a anguage.********\n-00007f80: 2a2a 2a2a 2a2a 2a2a 0a0a 5468 6973 2063 ********..This c\n-00007f90: 6861 7074 6572 2064 6573 6372 6962 6573 hapter describes\n-00007fa0: 2074 6865 20e2 8098 736b 6574 6368 e280 the ...sketch..\n-00007fb0: 9920 696e 7075 7420 6c61 6e67 7561 6765 . input language\n-00007fc0: 2069 6e20 6465 7461 696c 2e0a 0a2a 204d in detail...* M\n-00007fd0: 656e 753a 0a0a 2a20 4c61 6e67 7561 6765 enu:..* Language\n-00007fe0: 2062 6173 6963 733a 3a20 2020 2020 2020 basics:: \n-00007ff0: 2020 2020 2020 4361 7365 2c20 7370 6163 Case, spac\n-00008000: 652c 2063 6f6d 6d65 6e74 732c 2069 6e63 e, comments, inc\n-00008010: 6c75 6465 2066 696c 6573 2e0a 2a20 4472 lude files..* Dr\n-00008020: 6177 6162 6c65 733a 3a20 2020 2020 2020 awables:: \n-00008030: 2020 2020 2020 2020 2020 2020 5468 696e Thin\n-00008040: 6773 2074 6861 7420 6361 6e20 6265 2064 gs that can be d\n-00008050: 7261 776e 2e0a 2a20 4465 6669 6e69 7469 rawn..* Definiti\n-00008060: 6f6e 733a 3a20 2020 2020 2020 2020 2020 ons:: \n-00008070: 2020 2020 2020 4769 7669 6e67 2074 6869 Giving thi\n-00008080: 6e67 7320 6e61 6d65 732e 0a2a 2047 6c6f ngs names..* Glo\n-00008090: 6261 6c20 656e 7669 726f 6e6d 656e 743a bal environment:\n-000080a0: 3a20 2020 2020 2020 2020 2041 6666 6563 : Affec\n-000080b0: 7420 7468 6520 656e 7469 7265 2064 7261 t the entire dra\n-000080c0: 7769 6e67 2e0a 0a1f 0a46 696c 653a 2073 wing.....File: s\n-000080d0: 6b65 7463 682e 696e 666f 2c20 204e 6f64 ketch.info, Nod\n-000080e0: 653a 204c 616e 6775 6167 6520 6261 7369 e: Language basi\n-000080f0: 6373 2c20 204e 6578 743a 2044 7261 7761 cs, Next: Drawa\n-00008100: 626c 6573 2c20 2050 7265 763a 2049 6e70 bles, Prev: Inp\n-00008110: 7574 206c 616e 6775 6167 652c 2020 5570 ut language, Up\n-00008120: 3a20 496e 7075 7420 6c61 6e67 7561 6765 : Input language\n-00008130: 0a0a 332e 3120 4261 7369 6373 0a3d 3d3d ..3.1 Basics.===\n-00008140: 3d3d 3d3d 3d3d 3d0a 0ae2 8098 536b 6574 =======.....Sket\n-00008150: 6368 e280 9920 696e 7075 7420 6973 2070 ch... input is p\n-00008160: 6c61 696e 2041 5343 4949 2074 6578 742c lain ASCII text,\n-00008170: 2075 7375 616c 6c79 2073 746f 7265 6420 usually stored \n-00008180: 696e 2061 6e20 696e 7075 7420 6669 6c65 in an input file\n-00008190: 2e20 2049 740a 6465 7363 7269 6265 7320 . 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They mere\n-00008310: 6c79 0a63 6f6e 7472 6962 7574 6520 746f ly.contribute to\n-00008320: 2074 6861 7420 6465 636c 6172 6174 696f that declaratio\n-00008330: 6e2e 0a0a 2020 2041 2066 6577 2073 796e n... A few syn\n-00008340: 7461 6374 6963 2064 6574 6169 6c73 2061 tactic details a\n-00008350: 7265 2069 6d70 6f72 7461 6e74 2e20 2043 re important. C\n-00008360: 6173 6520 6973 2073 6967 6e69 6669 6361 ase is significa\n-00008370: 6e74 2069 6e20 7468 650a e280 9873 6b65 nt in the....ske\n-00008380: 7463 68e2 8099 206c 616e 6775 6167 652e tch... language.\n-00008390: 2020 5769 7468 2061 2066 6577 2065 7863 With a few exc\n-000083a0: 6570 7469 6f6e 732c 2077 6869 7465 2073 eptions, white s\n-000083b0: 7061 6365 2069 7320 6e6f 742e 2020 5468 pace is not. Th\n-000083c0: 6973 0a69 6e63 6c75 6465 7320 6c69 6e65 is.includes line\n-000083d0: 2062 7265 616b 732e 2020 436f 6d6d 656e breaks. Commen\n-000083e0: 7473 2062 6567 696e 2077 6974 6820 e280 ts begin with ..\n-000083f0: 9825 e280 9920 6f72 20e2 8098 23e2 8099 .%... or ...#...\n-00008400: 2061 6e64 2065 7874 656e 6420 746f 2074 and extend to t\n-00008410: 6865 0a65 6e64 206f 6620 7468 6520 6c69 he.end of the li\n-00008420: 6e65 2e20 2059 6f75 2063 616e 2064 6973 ne. You can dis\n-00008430: 6162 6c65 2061 2063 6875 6e6b 206f 6620 able a chunk of \n-00008440: 7379 6e74 6163 7469 6361 6c6c 7920 636f syntactically co\n-00008450: 7272 6563 740a e280 9873 6b65 7463 68e2 rrect....sketch.\n-00008460: 8099 2063 6f64 6520 6279 2065 6e63 6c6f .. code by enclo\n-00008470: 7369 6e67 2069 7420 696e 2061 20e2 8098 sing it in a ...\n-00008480: 6465 66e2 8099 2e20 2054 6865 7265 2069 def.... There i\n-00008490: 7320 6120 7369 6d70 6c65 2022 696e 636c s a simple \"incl\n-000084a0: 7564 650a 6669 6c65 2220 6d65 6368 616e ude.file\" mechan\n-000084b0: 6973 6d2e 2020 5468 6520 636f 6d6d 616e ism. The comman\n-000084c0: 640a 2020 696e 7075 747b 6f74 6865 7266 d. input{otherf\n-000084d0: 696c 652e 736b 7d0a 6361 7573 6573 2074 ile.sk}.causes t\n-000084e0: 6865 2063 6f6e 7465 6e74 7320 6f66 20e2 he contents of .\n-000084f0: 8098 6f74 6865 7266 696c 652e 736b e280 ..otherfile.sk..\n-00008500: 9920 746f 2062 6520 696e 7365 7274 6564 . to be inserted\n-00008510: 2061 7320 7468 6f75 6768 2074 6865 7920 as though they \n-00008520: 7765 7265 0a70 6172 7420 6f66 2074 6865 were.part of the\n-00008530: 2063 7572 7265 6e74 2066 696c 652e 0a0a current file...\n-00008540: 2a20 4d65 6e75 3a0a 0a2a 2049 6465 6e74 * Menu:..* Ident\n-00008550: 6966 6965 7273 3a3a 2020 2020 2020 2020 ifiers:: \n-00008560: 2020 2020 2020 2020 204e 616d 6573 2066 Names f\n-00008570: 6f72 2074 6869 6e67 732e 0a2a 204b 6579 or things..* Key\n-00008580: 2061 6e64 2072 6573 6572 7665 6420 776f and reserved wo\n-00008590: 7264 733a 3a20 2020 2020 204e 616d 6573 rds:: Names\n-000085a0: 2079 6f75 2073 686f 756c 646e 2774 2075 you shouldn't u\n-000085b0: 7365 2e0a 2a20 4c69 7465 7261 6c73 3a3a se..* Literals::\n-000085c0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000085d0: 2020 2020 436f 6e73 7461 6e74 7320 616e Constants an\n-000085e0: 6420 636f 6e73 7472 7563 746f 7273 2e0a d constructors..\n-000085f0: 2a20 4172 6974 686d 6574 6963 3a3a 2020 * Arithmetic:: \n-00008600: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00008610: 5275 6c65 7320 666f 7220 6578 7072 6573 Rules for expres\n-00008620: 7369 6f6e 732e 0a2a 204f 7074 696f 6e73 sions..* Options\n-00008630: 3a3a 2020 2020 2020 2020 2020 2020 2020 :: \n-00008640: 2020 2020 2020 204d 6f64 6966 7969 6e67 Modifying\n-00008650: 206f 626a 6563 7420 6170 7065 6172 616e object appearan\n-00008660: 6365 2e0a 0a1f 0a46 696c 653a 2073 6b65 ce.....File: ske\n-00008670: 7463 682e 696e 666f 2c20 204e 6f64 653a tch.info, Node:\n-00008680: 2049 6465 6e74 6966 6965 7273 2c20 204e Identifiers, N\n-00008690: 6578 743a 204b 6579 2061 6e64 2072 6573 ext: Key and res\n-000086a0: 6572 7665 6420 776f 7264 732c 2020 5072 erved words, Pr\n-000086b0: 6576 3a20 4c61 6e67 7561 6765 2062 6173 ev: Language bas\n-000086c0: 6963 732c 2020 5570 3a20 4c61 6e67 7561 ics, Up: Langua\n-000086d0: 6765 2062 6173 6963 730a 0a33 2e31 2e31 ge basics..3.1.1\n-000086e0: 2049 6465 6e74 6966 6965 7273 0a2d 2d2d Identifiers.---\n-000086f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------..\n-00008700: 4964 656e 7469 6669 6572 7320 696e 20e2 Identifiers in .\n-00008710: 8098 736b 6574 6368 e280 9920 6172 6520 ..sketch... are \n-00008720: 7265 6665 7265 6e63 6573 2074 6f20 6561 references to ea\n-00008730: 726c 6965 722d 6465 6669 6e65 6420 6f70 rlier-defined op\n-00008740: 7469 6f6e 732c 0a73 6361 6c61 7273 2c20 tions,.scalars, \n-00008750: 706f 696e 7473 2c20 7665 6374 6f72 732c points, vectors,\n-00008760: 2074 7261 6e73 666f 726d 732c 2064 7261 transforms, dra\n-00008770: 7761 626c 6573 2c20 616e 6420 7461 6773 wables, and tags\n-00008780: 2e20 2044 6566 696e 6974 696f 6e73 0a61 . Definitions.a\n-00008790: 7265 2065 7870 6c61 696e 6564 2069 6e20 re explained in \n-000087a0: 2a6e 6f74 6520 4465 6669 6e69 7469 6f6e *note Definition\n-000087b0: 733a 3a2e 0a0a 2020 2041 6e20 6964 656e s::... An iden\n-000087c0: 7469 6669 6572 2063 6f6e 7369 7374 7320 tifier consists \n-000087d0: 6f66 2061 206c 6561 6469 6e67 206c 6574 of a leading let\n-000087e0: 7465 7220 666f 6c6c 6f77 6564 2062 7920 ter followed by \n-000087f0: 6c65 7474 6572 732c 0a6e 756d 6265 7273 letters,.numbers\n-00008800: 2061 6e64 2075 6e64 6572 7363 6f72 6573 and underscores\n-00008810: 2e20 2054 6865 206c 6173 7420 6368 6172 . The last char\n-00008820: 6163 7465 7220 6d61 7920 5f6e 6f74 5f20 acter may _not_ \n-00008830: 6265 2061 6e20 756e 6465 7273 636f 7265 be an underscore\n-00008840: 2e0a 4b65 7977 6f72 6473 2063 616e 6e6f ..Keywords canno\n-00008850: 7420 6265 2075 7365 6420 6173 2069 6465 t be used as ide\n-00008860: 6e74 6966 6965 7273 2c20 616e 6420 7265 ntifiers, and re\n-00008870: 7365 7276 6564 2077 6f72 6473 206f 7567 served words oug\n-00008880: 6874 2074 6f20 6265 0a61 766f 6964 6564 ht to be.avoided\n-00008890: 2e20 202a 4e6f 7465 204b 6579 2061 6e64 . *Note Key and\n-000088a0: 2072 6573 6572 7665 6420 776f 7264 733a reserved words:\n-000088b0: 3a2e 0a0a 1f0a 4669 6c65 3a20 736b 6574 :.....File: sket\n-000088c0: 6368 2e69 6e66 6f2c 2020 4e6f 6465 3a20 ch.info, Node: \n-000088d0: 4b65 7920 616e 6420 7265 7365 7276 6564 Key and reserved\n-000088e0: 2077 6f72 6473 2c20 204e 6578 743a 204c words, Next: L\n-000088f0: 6974 6572 616c 732c 2020 5072 6576 3a20 iterals, Prev: \n-00008900: 4964 656e 7469 6669 6572 732c 2020 5570 Identifiers, Up\n-00008910: 3a20 4c61 6e67 7561 6765 2062 6173 6963 : Language basic\n-00008920: 730a 0a33 2e31 2e32 204b 6579 2061 6e64 s..3.1.2 Key and\n-00008930: 2072 6573 6572 7665 6420 776f 7264 730a reserved words.\n-00008940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ----------------\n-00008950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th\n-00008960: 6520 6b65 7977 6f72 6473 206f 6620 e280 e keywords of ..\n-00008970: 9873 6b65 7463 68e2 8099 2061 7265 20e2 .sketch... are .\n-00008980: 8098 7069 6374 7572 6562 6f78 e280 9920 ..picturebox... \n-00008990: e280 9863 7572 7665 e280 9920 e280 9864 ...curve... ...d\n-000089a0: 6566 e280 9920 e280 9864 6f74 73e2 8099 ef... ...dots...\n-000089b0: 20e2 8098 6672 616d 65e2 8099 0ae2 8098 ...frame.......\n-000089c0: 676c 6f62 616c e280 9920 e280 9869 6e70 global... ...inp\n-000089d0: 7574 e280 9920 e280 986c 696e 65e2 8099 ut... ...line...\n-000089e0: 20e2 8098 706f 6c79 676f 6ee2 8099 20e2 ...polygon... .\n-000089f0: 8098 7075 74e2 8099 20e2 8098 7265 7065 ..put... ...repe\n-00008a00: 6174 e280 9920 e280 9873 6574 e280 9920 at... ...set... \n-00008a10: e280 9873 7765 6570 e280 9920 616e 640a ...sweep... and.\n-00008a20: e280 9874 6865 6ee2 8099 2e20 2054 6865 ...then.... The\n-00008a30: 20e2 8098 736b 6574 6368 e280 9920 7061 ...sketch... pa\n-00008a40: 7273 6572 2077 696c 6c20 6e6f 7465 2061 rser will note a\n-00008a50: 2073 796e 7461 7820 6572 726f 7220 6966 syntax error if\n-00008a60: 2061 6e79 206f 6620 7468 6573 650a 6172 any of these.ar\n-00008a70: 6520 7573 6564 2069 6e20 706c 6163 6520 e used in place \n-00008a80: 6f66 2061 2070 726f 7065 7220 6964 656e of a proper iden\n-00008a90: 7469 6669 6572 2e0a 0a20 2020 496e 2061 tifier... In a\n-00008aa0: 6464 6974 696f 6e2c 2074 6865 7265 2061 ddition, there a\n-00008ab0: 7265 2072 6573 6572 7665 6420 776f 7264 re reserved word\n-00008ac0: 7320 7468 6174 2063 616e 2063 7572 7265 s that can curre\n-00008ad0: 6e74 6c79 2062 6520 6465 6669 6e65 640a ntly be defined.\n-00008ae0: 6279 2074 6865 2075 7365 722c 2062 7574 by the user, but\n-00008af0: 2077 6974 6820 7468 6520 7269 736b 2074 with the risk t\n-00008b00: 6861 7420 6675 7475 7265 2076 6572 7369 hat future versi\n-00008b10: 6f6e 7320 6f66 20e2 8098 736b 6574 6368 ons of ...sketch\n-00008b20: e280 9920 7769 6c6c 0a72 656a 6563 7420 ... will.reject \n-00008b30: 7468 6f73 6520 6465 6669 6e69 7469 6f6e those definition\n-00008b40: 732e 2020 5468 6520 7265 7365 7276 6564 s. The reserved\n-00008b50: 2077 6f72 6473 2061 7265 20e2 8098 6163 words are ...ac\n-00008b60: 6f73 e280 9920 e280 9861 7369 6ee2 8099 os... ...asin...\n-00008b70: 20e2 8098 6174 616e 32e2 8099 0ae2 8098 ...atan2.......\n-00008b80: 636f 73e2 8099 20e2 8098 696e 7665 7273 cos... ...invers\n-00008b90: 65e2 8099 20e2 8098 7065 7273 7065 6374 e... ...perspect\n-00008ba0: 6976 65e2 8099 20e2 8098 7072 6f6a 6563 ive... ...projec\n-00008bb0: 74e2 8099 20e2 8098 726f 7461 7465 e280 t... ...rotate..\n-00008bc0: 9920 e280 9873 6361 6c65 e280 9920 e280 . ...scale... ..\n-00008bd0: 9873 696e e280 9920 e280 9873 7065 6369 .sin... ...speci\n-00008be0: 616c e280 990a e280 9873 7172 74e2 8099 al.......sqrt...\n-00008bf0: 20e2 8098 7472 616e 736c 6174 65e2 8099 ...translate...\n-00008c00: 20e2 8098 756e 6974 e280 9920 616e 6420 ...unit... and \n-00008c10: e280 9876 6965 77e2 8099 2e0a 0a1f 0a46 ...view........F\n-00008c20: 696c 653a 2073 6b65 7463 682e 696e 666f ile: sketch.info\n-00008c30: 2c20 204e 6f64 653a 204c 6974 6572 616c , Node: Literal\n-00008c40: 732c 2020 4e65 7874 3a20 4172 6974 686d s, Next: Arithm\n-00008c50: 6574 6963 2c20 2050 7265 763a 204b 6579 etic, Prev: Key\n-00008c60: 2061 6e64 2072 6573 6572 7665 6420 776f and reserved wo\n-00008c70: 7264 732c 2020 5570 3a20 4c61 6e67 7561 rds, Up: Langua\n-00008c80: 6765 2062 6173 6963 730a 0a33 2e31 2e33 ge basics..3.1.3\n-00008c90: 204c 6974 6572 616c 730a 2d2d 2d2d 2d2d Literals.------\n-00008ca0: 2d2d 2d2d 2d2d 2d2d 0a0a 4c69 7465 7261 --------..Litera\n-00008cb0: 6c73 2069 6e20 e280 9873 6b65 7463 68e2 ls in ...sketch.\n-00008cc0: 8099 2069 6e63 6c75 6465 2073 6361 6c61 .. include scala\n-00008cd0: 7273 2c20 706f 696e 7473 2c20 7665 6374 rs, points, vect\n-00008ce0: 6f72 732c 2061 6e64 2074 7261 6e73 666f ors, and transfo\n-00008cf0: 726d 732e 0a4c 6974 6572 616c 732c 2061 rms..Literals, a\n-00008d00: 6c6f 6e67 2077 6974 6820 6465 6669 6e65 long with define\n-00008d10: 6420 6f62 6a65 6374 2072 6566 6572 656e d object referen\n-00008d20: 6365 732c 2061 7265 2075 7365 6420 696e ces, are used in\n-00008d30: 2061 7269 7468 6d65 7469 630a 6578 7072 arithmetic.expr\n-00008d40: 6573 7369 6f6e 732e 2020 2a4e 6f74 6520 essions. *Note \n-00008d50: 4172 6974 686d 6574 6963 3a3a 2e0a 0a2a Arithmetic::...*\n-00008d60: 204d 656e 753a 0a0a 2a20 5363 616c 6172 Menu:..* Scalar\n-00008d70: 206c 6974 6572 616c 733a 3a20 2020 2020 literals:: \n-00008d80: 2020 2020 2020 2020 4a75 7374 2074 6865 Just the\n-00008d90: 206e 756d 6265 7273 2e0a 2a20 506f 696e numbers..* Poin\n-00008da0: 7420 616e 6420 7665 6374 6f72 206c 6974 t and vector lit\n-00008db0: 6572 616c 733a 3a20 2020 3364 2071 7561 erals:: 3d qua\n-00008dc0: 6e74 6974 6965 732e 0a2a 2054 7261 6e73 ntities..* Trans\n-00008dd0: 666f 726d 206c 6974 6572 616c 733a 3a20 form literals:: \n-00008de0: 2020 2020 2020 2020 204d 6174 7269 7820 Matrix \n-00008df0: 666f 726d 2e0a 0a1f 0a46 696c 653a 2073 form.....File: s\n-00008e00: 6b65 7463 682e 696e 666f 2c20 204e 6f64 ketch.info, Nod\n-00008e10: 653a 2053 6361 6c61 7220 6c69 7465 7261 e: Scalar litera\n-00008e20: 6c73 2c20 204e 6578 743a 2050 6f69 6e74 ls, Next: Point\n-00008e30: 2061 6e64 2076 6563 746f 7220 6c69 7465 and vector lite\n-00008e40: 7261 6c73 2c20 2050 7265 763a 204c 6974 rals, Prev: Lit\n-00008e50: 6572 616c 732c 2020 5570 3a20 4c69 7465 erals, Up: Lite\n-00008e60: 7261 6c73 0a0a 332e 312e 332e 3120 5363 rals..3.1.3.1 Sc\n-00008e70: 616c 6172 206c 6974 6572 616c 730a 2e2e alar literals...\n-00008e80: 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e ................\n-00008e90: 2e2e 2e2e 2e0a 0a53 6361 6c61 7220 6c69 .......Scalar li\n-00008ea0: 7465 7261 6c73 2061 7265 2070 6f73 6974 terals are posit\n-00008eb0: 6976 6520 666c 6f61 7469 6e67 2070 6f69 ive floating poi\n-00008ec0: 6e74 206e 756d 6265 7273 2077 6974 6820 nt numbers with \n-00008ed0: 7379 6e74 6178 0a61 6363 6f72 6469 6e67 syntax.according\n-00008ee0: 2074 6f20 4320 636f 6e76 656e 7469 6f6e to C convention\n-00008ef0: 732e 2020 5468 6520 666f 6c6c 6f77 696e s. The followin\n-00008f00: 6720 6172 6520 736f 6d65 2065 7861 6d70 g are some examp\n-00008f10: 6c65 732e 0a20 2020 2020 3020 3130 3034 les.. 0 1004\n-00008f20: 202e 3030 3120 382e 3331 3433 2033 2e20 .001 8.3143 3. \n-00008f30: 312e 3630 452d 3139 2036 2e30 3265 2b32 1.60E-19 6.02e+2\n-00008f40: 330a 5363 616c 6172 206c 6974 6572 616c 3.Scalar literal\n-00008f50: 7320 6d61 7920 6e6f 7420 636f 6e74 6169 s may not contai\n-00008f60: 6e20 656d 6265 6464 6564 2073 7061 6365 n embedded space\n-00008f70: 732e 0a0a 1f0a 4669 6c65 3a20 736b 6574 s.....File: sket\n-00008f80: 6368 2e69 6e66 6f2c 2020 4e6f 6465 3a20 ch.info, Node: \n-00008f90: 506f 696e 7420 616e 6420 7665 6374 6f72 Point and vector\n-00008fa0: 206c 6974 6572 616c 732c 2020 4e65 7874 literals, Next\n-00008fb0: 3a20 5472 616e 7366 6f72 6d20 6c69 7465 : Transform lite\n-00008fc0: 7261 6c73 2c20 2050 7265 763a 2053 6361 rals, Prev: Sca\n-00008fd0: 6c61 7220 6c69 7465 7261 6c73 2c20 2055 lar literals, U\n-00008fe0: 703a 204c 6974 6572 616c 730a 0a33 2e31 p: Literals..3.1\n-00008ff0: 2e33 2e32 2050 6f69 6e74 2061 6e64 2076 .3.2 Point and v\n-00009000: 6563 746f 7220 6c69 7465 7261 6c73 0a2e ector literals..\n-00009010: 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e ................\n+00005220: 2020 2020 2520 6f72 6967 696e 616c 0a20 % original. \n+00005230: 2020 2070 6f6c 7967 6f6e 2870 3129 2870 polygon(p1)(p\n+00005240: 3429 2870 3229 2020 2020 2020 2020 2020 4)(p2) \n+00005250: 2020 2020 2020 2020 2020 2020 2520 626f % bo\n+00005260: 7474 6f6d 0a20 2020 2070 6f6c 7967 6f6e ttom. polygon\n+00005270: 2870 3129 2870 3329 2870 3429 2020 2020 (p1)(p3)(p4) \n+00005280: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00005290: 2020 2520 6c65 6674 0a20 2020 2070 6f6c % left. pol\n+000052a0: 7967 6f6e 5b66 696c 6c63 6f6c 6f72 3d6c ygon[fillcolor=l\n+000052b0: 6967 6874 6772 6179 5d28 7033 2928 7032 ightgray](p3)(p2\n+000052c0: 2928 7034 2920 2520 7265 6172 0a20 2020 )(p4) % rear. \n+000052d0: 206c 696e 655b 6c69 6e65 636f 6c6f 723d line[linecolor=\n+000052e0: 7265 645d 282d 312c 2d31 2c2d 3129 2832 red](-1,-1,-1)(2\n+000052f0: 2c32 2c32 290a 2020 7d0a 2020 7b70 6965 ,2,2). }. {pie\n+00005300: 7263 6564 5f74 6574 7261 6865 6472 6f6e rced_tetrahedron\n+00005310: 7d20 2025 2074 6574 7261 6865 6472 6f6e } % tetrahedron\n+00005320: 2069 6e20 6f72 6967 696e 616c 2070 6f73 in original pos\n+00005330: 6974 696f 6e0a 2020 7075 7420 7b20 726f ition. put { ro\n+00005340: 7461 7465 2839 302c 2028 302c 302c 3029 tate(90, (0,0,0)\n+00005350: 2c20 5b31 2c30 2c30 5d29 2025 2063 6f70 , [1,0,0]) % cop\n+00005360: 7920 696e 206e 6577 2070 6f73 6974 696f y in new positio\n+00005370: 6e0a 2020 2020 2020 2020 7468 656e 2074 n. then t\n+00005380: 7261 6e73 6c61 7465 285b 322e 352c 302c ranslate([2.5,0,\n+00005390: 305d 2920 7d20 7b70 6965 7263 6564 5f74 0]) } {pierced_t\n+000053a0: 6574 7261 6865 6472 6f6e 7d0a 4865 7265 etrahedron}.Here\n+000053b0: 2074 6865 2065 6e74 6972 6520 636f 6465 the entire code\n+000053c0: 206f 6620 7468 6520 7072 6576 696f 7573 of the previous\n+000053d0: 2065 7861 6d70 6c65 2068 6173 2062 6565 example has bee\n+000053e0: 6e20 7772 6170 7065 6420 696e 2061 0a64 n wrapped in a.d\n+000053f0: 6566 696e 6974 696f 6e20 6279 2066 6f72 efinition by for\n+00005400: 6d69 6e67 2061 20e2 809c 626c 6f63 6be2 ming a ...block.\n+00005410: 809d 2077 6974 6820 6272 6163 6573 2028 .. with braces (\n+00005420: 6120 7369 6e67 6c65 2069 7465 6d20 776f a single item wo\n+00005430: 756c 6420 6e6f 740a 6e65 6564 2074 6865 uld not.need the\n+00005440: 6d29 2e20 2054 6865 2070 6f69 6e74 2064 m). The point d\n+00005450: 6566 696e 6974 696f 6e73 206e 6573 7465 efinitions neste\n+00005460: 6420 696e 7369 6465 2074 6865 2062 7261 d inside the bra\n+00005470: 6365 7320 6172 650a e280 9c6c 6578 6963 ces are....lexic\n+00005480: 616c 6c79 2073 636f 7065 64e2 809d 2e20 ally scoped.... \n+00005490: 2054 6865 6972 206d 6561 6e69 6e67 2065 Their meaning e\n+000054a0: 7874 656e 6473 206f 6e6c 7920 746f 2074 xtends only to t\n+000054b0: 6865 2065 6e64 206f 6620 7468 6520 626c he end of the bl\n+000054c0: 6f63 6b2e 0a54 6865 206f 7574 6572 2064 ock..The outer d\n+000054d0: 6566 2069 7320 6361 6c6c 6564 2061 20e2 ef is called a .\n+000054e0: 809c 6472 6177 6162 6c65 e280 9d20 6465 ..drawable... de\n+000054f0: 6669 6e69 7469 6f6e 2062 6563 6175 7365 finition because\n+00005500: 2069 7420 6465 7363 7269 6265 730a 736f it describes.so\n+00005510: 6d65 7468 696e 6720 7468 6174 2063 616e mething that can\n+00005520: 2062 6520 6472 6177 6e2e 0a0a 2020 2041 be drawn... A\n+00005530: 2064 7261 7761 626c 6520 6465 6669 6e69 drawable defini\n+00005540: 7469 6f6e 2062 7920 6974 7365 6c66 2063 tion by itself c\n+00005550: 6175 7365 7320 6e6f 7468 696e 6720 746f auses nothing to\n+00005560: 2068 6170 7065 6e20 756e 7469 6c20 6974 happen until it\n+00005570: 730a 6e61 6d65 2069 7320 7265 6665 7265 s.name is refere\n+00005580: 6e63 6564 2e20 2044 7261 7761 626c 6520 nced. Drawable \n+00005590: 7265 6665 7265 6e63 6573 206d 7573 7420 references must \n+000055a0: 6265 2065 6e63 6c6f 7365 6420 696e 2063 be enclosed in c\n+000055b0: 7572 6c79 0a62 7261 6365 732c 2065 2e67 urly.braces, e.g\n+000055c0: 2e20 7b66 6f6f 7d2c 2077 6974 6820 6e6f . {foo}, with no\n+000055d0: 2069 6e74 6572 7665 6e69 6e67 2077 6869 intervening whi\n+000055e0: 7465 2073 7061 6365 2e20 2049 6e20 7468 te space. In th\n+000055f0: 6520 636f 6465 2061 626f 7665 2c0a 7468 e code above,.th\n+00005600: 6520 6669 7273 7420 7265 6665 7265 6e63 e first referenc\n+00005610: 6520 7b70 6965 7263 6564 5f74 6574 7261 e {pierced_tetra\n+00005620: 6865 6472 6f6e 7d20 6973 2061 2070 6c61 hedron} is a pla\n+00005630: 696e 206f 6e65 2e20 2049 7473 2065 6666 in one. Its eff\n+00005640: 6563 7420 6973 0a6d 6572 656c 7920 746f ect is.merely to\n+00005650: 2064 7570 6c69 6361 7465 2074 6865 2065 duplicate the e\n+00005660: 6172 6c69 6572 2064 7261 7769 6e67 2e20 arlier drawing. \n+00005670: 2041 6c6d 6f73 7420 616e 7920 7365 7269 Almost any seri\n+00005680: 6573 206f 6620 e280 9873 6b65 7463 68e2 es of ...sketch.\n+00005690: 8099 0a63 6f6d 6d61 6e64 7320 7374 7566 ...commands stuf\n+000056a0: 6620 6d61 7920 6265 2072 6570 6c61 6365 f may be replace\n+000056b0: 6420 7769 7468 2064 6566 2066 6f6f 207b d with def foo {\n+000056c0: 2073 7475 6666 207d 207b 666f 6f7d 2077 stuff } {foo} w\n+000056d0: 6974 686f 7574 0a63 6861 6e67 696e 6720 ithout.changing \n+000056e0: 6974 7320 6d65 616e 696e 672e 0a0a 2020 its meaning... \n+000056f0: 2054 6865 2070 7574 2063 6f6d 6d61 6e64 The put command\n+00005700: 2073 7570 706c 6965 7320 6120 7365 636f supplies a seco\n+00005710: 6e64 2072 6566 6572 656e 6365 2c20 7468 nd reference, th\n+00005720: 6973 2074 696d 6520 7769 7468 2061 0ae2 is time with a..\n+00005730: 809c 7472 616e 7366 6f72 6de2 809d 2061 ..transform... a\n+00005740: 7070 6c69 6564 2066 6972 7374 2e20 2054 pplied first. T\n+00005750: 6865 2072 6f74 6174 6520 7472 616e 7366 he rotate transf\n+00005760: 6f72 6d20 7475 726e 7320 7468 6520 7465 orm turns the te\n+00005770: 7472 6168 6564 726f 6e0a 3930 2064 6567 trahedron.90 deg\n+00005780: 7265 6573 2061 626f 7574 2074 6865 206f rees about the o\n+00005790: 7269 6769 6e2e 2020 5468 6520 6178 6973 rigin. The axis\n+000057a0: 206f 6620 726f 7461 7469 6f6e 2069 7320 of rotation is \n+000057b0: 7468 6520 7665 6374 6f72 0a5b 312c 302c the vector.[1,0,\n+000057c0: 305d 2e20 2042 7920 7468 6520 e280 9c72 0]. By the ...r\n+000057d0: 6967 6874 2068 616e 6420 7275 6c65 e280 ight hand rule..\n+000057e0: 9d2c 2074 6869 7320 6361 7573 6573 2074 ., this causes t\n+000057f0: 6865 2074 6f70 206f 6620 7468 650a 7465 he top of the.te\n+00005800: 7472 6168 6564 726f 6e20 746f 2072 6f74 trahedron to rot\n+00005810: 6174 6520 746f 7761 7264 2074 6865 2076 ate toward the v\n+00005820: 6965 7765 7220 616e 6420 7468 6520 626f iewer and the bo\n+00005830: 7474 6f6d 2061 7761 792e 2020 5468 6520 ttom away. The \n+00005840: 7275 6c65 0a72 6563 6569 7665 7320 6974 rule.receives it\n+00005850: 7320 6e61 6d65 2066 726f 6d20 7468 6520 s name from the \n+00005860: 666f 6c6c 6f77 696e 6720 6465 6669 6e69 following defini\n+00005870: 7469 6f6e 3a0a 2020 2020 202a 5269 6768 tion:. *Righ\n+00005880: 7420 6861 6e64 2072 756c 652e 2a20 2049 t hand rule.* I\n+00005890: 6620 7468 6520 7269 6768 7420 6861 6e64 f the right hand\n+000058a0: 2069 7320 7772 6170 7065 6420 6172 6f75 is wrapped arou\n+000058b0: 6e64 2061 6e79 2061 7869 730a 2020 2020 nd any axis. \n+000058c0: 2077 6974 6820 7468 6520 7468 756d 6220 with the thumb \n+000058d0: 706f 696e 7469 6e67 2069 6e20 7468 6520 pointing in the \n+000058e0: 6178 6973 2064 6972 6563 7469 6f6e 2c20 axis direction, \n+000058f0: 7468 656e 2074 6865 2066 696e 6765 7273 then the fingers\n+00005900: 0a20 2020 2020 6375 726c 2069 6e20 7468 . curl in th\n+00005910: 6520 6469 7265 6374 696f 6e20 6f66 2070 e direction of p\n+00005920: 6f73 6974 6976 6520 726f 7461 7469 6f6e ositive rotation\n+00005930: 2061 626f 7574 2074 6861 7420 6178 6973 about that axis\n+00005940: 2e0a 2020 2054 6865 2074 7261 6e73 6c61 .. The transla\n+00005950: 7465 2074 7261 6e73 666f 726d 206d 6f76 te transform mov\n+00005960: 6573 2074 6865 2070 7972 616d 6964 206c es the pyramid l\n+00005970: 6174 6572 616c 6c79 2074 6f20 7468 6520 aterally to the \n+00005980: 7269 6768 7420 6279 0a61 6464 696e 6720 right by.adding \n+00005990: 7468 6520 7665 6374 6f72 205b 322e 352c the vector [2.5,\n+000059a0: 302c 305d 2074 6f20 6561 6368 2076 6572 0,0] to each ver\n+000059b0: 7465 7820 636f 6f72 6469 6e61 7465 2e20 tex coordinate. \n+000059c0: 2054 6865 2072 6573 756c 7420 6973 0a73 The result is.s\n+000059d0: 686f 776e 2068 6572 652e 0a0a 0008 5b69 hown here.....[i\n+000059e0: 6d61 6765 2073 7263 3d22 6578 3035 302e mage src=\"ex050.\n+000059f0: 706e 6722 2074 6578 743d 2249 6d61 6765 png\" text=\"Image\n+00005a00: 2065 7830 3530 206f 6d69 7474 6564 2069 ex050 omitted i\n+00005a10: 6e20 7465 7874 2076 6572 7369 6f6e 206f n text version o\n+00005a20: 6620 7468 6973 2064 6f63 756d 656e 742e f this document.\n+00005a30: 2200 085d 0a0a 1f0a 4669 6c65 3a20 736b \"..]....File: sk\n+00005a40: 6574 6368 2e69 6e66 6f2c 2020 4e6f 6465 etch.info, Node\n+00005a50: 3a20 5265 7065 6174 6564 206f 626a 6563 : Repeated objec\n+00005a60: 7473 2c20 204e 6578 743a 2053 7765 7074 ts, Next: Swept\n+00005a70: 206f 626a 6563 7473 2c20 2050 7265 763a objects, Prev:\n+00005a80: 204f 626a 6563 7420 7472 616e 7366 6f72 Object transfor\n+00005a90: 6d73 2c20 2055 703a 2049 6e74 726f 6475 ms, Up: Introdu\n+00005aa0: 6374 696f 6e20 6279 2065 7861 6d70 6c65 ction by example\n+00005ab0: 0a0a 322e 3620 5265 7065 6174 6564 206f ..2.6 Repeated o\n+00005ac0: 626a 6563 7473 0a3d 3d3d 3d3d 3d3d 3d3d bjects.=========\n+00005ad0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6f20 ===========..To \n+00005ae0: 6472 6177 2073 6576 656e 2069 6e73 7461 draw seven insta\n+00005af0: 6e63 6573 206f 6620 7468 6520 7465 7472 nces of the tetr\n+00005b00: 6168 6564 726f 6e2c 2065 6163 6820 6469 ahedron, each di\n+00005b10: 6666 6572 696e 6720 6672 6f6d 2074 6865 ffering from the\n+00005b20: 206c 6173 740a 6279 2074 6865 2073 616d last.by the sam\n+00005b30: 6520 7472 616e 7366 6f72 6d2c 2072 6570 e transform, rep\n+00005b40: 6c61 6365 2074 6865 206c 6173 7420 7477 lace the last tw\n+00005b50: 6f20 636f 6d6d 616e 6473 206f 6620 7468 o commands of th\n+00005b60: 6520 7072 6576 696f 7573 0a65 7861 6d70 e previous.examp\n+00005b70: 6c65 2077 6974 680a 2020 7265 7065 6174 le with. repeat\n+00005b80: 207b 2037 2c20 726f 7461 7465 2831 352c { 7, rotate(15,\n+00005b90: 2028 302c 302c 3029 2c20 5b31 2c30 2c30 (0,0,0), [1,0,0\n+00005ba0: 5d29 2025 2063 6f70 7920 696e 206e 6577 ]) % copy in new\n+00005bb0: 2070 6f73 6974 696f 6e0a 2020 2020 2020 position. \n+00005bc0: 2020 2020 2020 2020 7468 656e 2074 7261 then tra\n+00005bd0: 6e73 6c61 7465 285b 322c 302c 305d 2920 nslate([2,0,0]) \n+00005be0: 7d20 7b70 6965 7263 6564 5f74 6574 7261 } {pierced_tetra\n+00005bf0: 6865 6472 6f6e 7d0a 416e 6420 7468 6520 hedron}.And the \n+00005c00: 7265 7375 6c74 2e2e 2e0a 0a00 085b 696d result.......[im\n+00005c10: 6167 6520 7372 633d 2265 7830 3630 2e70 age src=\"ex060.p\n+00005c20: 6e67 2220 7465 7874 3d22 496d 6167 6520 ng\" text=\"Image \n+00005c30: 6578 3036 3020 6f6d 6974 7465 6420 696e ex060 omitted in\n+00005c40: 2074 6578 7420 7665 7273 696f 6e20 6f66 text version of\n+00005c50: 2074 6869 7320 646f 6375 6d65 6e74 2e22 this document.\"\n+00005c60: 0008 5d0a 0a1f 0a46 696c 653a 2073 6b65 ..]....File: ske\n+00005c70: 7463 682e 696e 666f 2c20 204e 6f64 653a tch.info, Node:\n+00005c80: 2053 7765 7074 206f 626a 6563 7473 2c20 Swept objects, \n+00005c90: 2050 7265 763a 2052 6570 6561 7465 6420 Prev: Repeated \n+00005ca0: 6f62 6a65 6374 732c 2020 5570 3a20 496e objects, Up: In\n+00005cb0: 7472 6f64 7563 7469 6f6e 2062 7920 6578 troduction by ex\n+00005cc0: 616d 706c 650a 0a32 2e37 2053 7765 7074 ample..2.7 Swept\n+00005cd0: 206f 626a 6563 7473 0a3d 3d3d 3d3d 3d3d objects.=======\n+00005ce0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4d61 6e79 ==========..Many\n+00005cf0: 2066 616d 696c 6961 7220 7368 6170 6573 familiar shapes\n+00005d00: 2063 616e 2062 6520 6765 6e65 7261 7465 can be generate\n+00005d10: 6420 6279 2073 7765 6570 696e 6720 7369 d by sweeping si\n+00005d20: 6d70 6c65 7220 6f6e 6573 2074 6872 6f75 mpler ones throu\n+00005d30: 6768 0a73 7061 6365 2061 6e64 2063 6f6e gh.space and con\n+00005d40: 7369 6465 7269 6e67 2074 6865 2072 6573 sidering the res\n+00005d50: 756c 7469 6e67 2070 6174 682c 2073 7572 ulting path, sur\n+00005d60: 6661 6365 2c20 6f72 2076 6f6c 756d 652e face, or volume.\n+00005d70: 2020 e280 9853 6b65 7463 68e2 8099 0a69 ...Sketch....i\n+00005d80: 6d70 6c65 6d65 6e74 7320 7468 6973 2069 mplements this i\n+00005d90: 6465 6120 696e 2074 6865 2073 7765 6570 dea in the sweep\n+00005da0: 2063 6f6d 6d61 6e64 2e0a 2020 6465 6620 command.. def \n+00005db0: 6e5f 7365 6773 2038 0a20 2073 7765 6570 n_segs 8. sweep\n+00005dc0: 207b 206e 5f73 6567 732c 2072 6f74 6174 { n_segs, rotat\n+00005dd0: 6528 3138 3020 2f20 6e5f 7365 6773 2c20 e(180 / n_segs, \n+00005de0: 2830 2c30 2c30 292c 205b 302c 302c 315d (0,0,0), [0,0,1]\n+00005df0: 2920 7d20 2831 2c30 2c30 290a 5468 6973 ) } (1,0,0).This\n+00005e00: 2063 6f64 6520 7377 6565 7073 2074 6865 code sweeps the\n+00005e10: 2070 6f69 6e74 2028 312c 302c 3029 2065 point (1,0,0) e\n+00005e20: 6967 6874 2074 696d 6573 2062 7920 726f ight times by ro\n+00005e30: 7461 7469 6e67 2069 7420 3138 302f 3820 tating it 180/8 \n+00005e40: 3d0a 3232 2e35 2064 6567 7265 6573 2065 =.22.5 degrees e\n+00005e50: 6163 6820 7469 6d65 2061 6e64 2063 6f6e ach time and con\n+00005e60: 6e65 6374 696e 6720 7468 6520 7265 7375 necting the resu\n+00005e70: 6c74 696e 6720 706f 696e 7473 2077 6974 lting points wit\n+00005e80: 6820 6c69 6e65 0a73 6567 6d65 6e74 732e h line.segments.\n+00005e90: 2020 5468 6520 6465 6620 7573 6564 2068 The def used h\n+00005ea0: 6572 6520 6973 2061 20e2 809c 7363 616c ere is a ...scal\n+00005eb0: 6172 e280 9d20 6465 6669 6e69 7469 6f6e ar... definition\n+00005ec0: 2e20 2052 6566 6572 656e 6365 7320 746f . References to\n+00005ed0: 0a73 6361 6c61 7273 2068 6176 6520 6e6f .scalars have no\n+00005ee0: 2065 6e63 6c6f 7369 6e67 2062 7261 636b enclosing brack\n+00005ef0: 6574 7320 6174 2061 6c6c 2e0a 0a2a 204d ets at all...* M\n+00005f00: 656e 753a 0a0a 2a20 506f 696e 7420 7377 enu:..* Point sw\n+00005f10: 6565 7073 3a3a 2020 2020 2020 2020 2020 eeps:: \n+00005f20: 2020 2020 2020 5377 6570 7420 706f 696e Swept poin\n+00005f30: 7473 206d 616b 6520 6c69 6e65 7320 616e ts make lines an\n+00005f40: 6420 706f 6c79 676f 6e73 2e0a 2a20 506f d polygons..* Po\n+00005f50: 6c79 6c69 6e65 2073 7765 6570 733a 3a20 lyline sweeps:: \n+00005f60: 2020 2020 2020 2020 2020 2020 5377 6570 Swep\n+00005f70: 7420 6c69 6e65 7320 6d61 6b65 2073 7572 t lines make sur\n+00005f80: 6661 6365 732e 0a2a 204e 6573 7465 6420 faces..* Nested \n+00005f90: 7377 6565 7073 3a3a 2020 2020 2020 2020 sweeps:: \n+00005fa0: 2020 2020 2020 2053 7765 7074 2073 7765 Swept swe\n+00005fb0: 6570 7320 6172 6520 7573 6566 756c 210a eps are useful!.\n+00005fc0: 2a20 506f 6c79 676f 6e20 7377 6565 7073 * Polygon sweeps\n+00005fd0: 3a3a 2020 2020 2020 2020 2020 2020 2020 :: \n+00005fe0: 5377 6570 7420 706f 6c79 676f 6e73 206d Swept polygons m\n+00005ff0: 616b 6520 736f 6c69 6473 2e2e 2e0a 2a20 ake solids....* \n+00006000: 506f 6c79 6c69 6e65 2073 7765 6570 7320 Polyline sweeps \n+00006010: 7769 7468 2063 6c6f 7375 7265 3a3a 2020 with closure:: \n+00006020: 616e 6420 736f 2064 6f20 636c 6f73 6564 and so do closed\n+00006030: 2070 6f6c 796c 696e 6520 7377 6565 7073 polyline sweeps\n+00006040: 2e0a 2a20 4166 6669 6e65 2061 7269 7468 ..* Affine arith\n+00006050: 6d65 7469 633a 3a20 2020 2020 2020 2020 metic:: \n+00006060: 2020 536b 6574 6368 2075 7365 6675 6c20 Sketch useful \n+00006070: 6d61 7468 2065 7870 7265 7373 696f 6e2e math expression.\n+00006080: 0a2a 204d 6f72 6520 746f 206c 6561 726e .* More to learn\n+00006090: 3a3a 2020 2020 2020 2020 2020 2020 2020 :: \n+000060a0: 2043 6865 636b 206f 7574 2074 6865 204d Check out the M\n+000060b0: 6f62 6975 7320 7374 7269 7021 0a0a 1f0a obius strip!....\n+000060c0: 4669 6c65 3a20 736b 6574 6368 2e69 6e66 File: sketch.inf\n+000060d0: 6f2c 2020 4e6f 6465 3a20 506f 696e 7420 o, Node: Point \n+000060e0: 7377 6565 7073 2c20 204e 6578 743a 2050 sweeps, Next: P\n+000060f0: 6f6c 796c 696e 6520 7377 6565 7073 2c20 olyline sweeps, \n+00006100: 2050 7265 763a 2053 7765 7074 206f 626a Prev: Swept obj\n+00006110: 6563 7473 2c20 2055 703a 2053 7765 7074 ects, Up: Swept\n+00006120: 206f 626a 6563 7473 0a0a 322e 372e 3120 objects..2.7.1 \n+00006130: 506f 696e 7420 7377 6565 7073 0a2d 2d2d Point sweeps.---\n+00006140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------.\n+00006150: 0a53 7765 6570 696e 6720 6120 706f 696e .Sweeping a poin\n+00006160: 7420 6d61 6b65 7320 6120 6f6e 652d 6469 t makes a one-di\n+00006170: 6d65 6e73 696f 6e61 6c20 7061 7468 2c20 mensional path, \n+00006180: 7768 6963 6820 6973 2061 2070 6f6c 796c which is a polyl\n+00006190: 696e 652e 0a53 696e 6365 2077 6520 6861 ine..Since we ha\n+000061a0: 7665 2073 7765 7074 2077 6974 6820 6120 ve swept with a \n+000061b0: 726f 7461 7469 6f6e 2c20 7468 6520 7265 rotation, the re\n+000061c0: 7375 6c74 2069 7320 6120 6369 7263 756c sult is a circul\n+000061d0: 6172 2061 7263 2e20 2048 6572 650a 6973 ar arc. Here.is\n+000061e0: 2077 6861 7420 6974 206c 6f6f 6b73 206c what it looks l\n+000061f0: 696b 652e 0a0a 0008 5b69 6d61 6765 2073 ike.....[image s\n+00006200: 7263 3d22 6578 3037 302e 706e 6722 2074 rc=\"ex070.png\" t\n+00006210: 6578 743d 2249 6d61 6765 2065 7830 3730 ext=\"Image ex070\n+00006220: 206f 6d69 7474 6564 2069 6e20 7465 7874 omitted in text\n+00006230: 2076 6572 7369 6f6e 206f 6620 7468 6973 version of this\n+00006240: 2064 6f63 756d 656e 742e 2200 085d 0a0a document.\"..]..\n+00006250: 2020 2054 6869 7320 6973 2074 6865 2066 This is the f\n+00006260: 6972 7374 2065 7861 6d70 6c65 2077 6520 irst example we \n+00006270: 6861 7665 2073 6565 6e20 6f66 20e2 8098 have seen of ...\n+00006280: 736b 6574 6368 e280 9920 6172 6974 686d sketch... arithm\n+00006290: 6574 6963 2e20 2054 6865 0a65 7870 7265 etic. The.expre\n+000062a0: 7373 696f 6e20 3138 3020 2f20 6e5f 7365 ssion 180 / n_se\n+000062b0: 6773 2063 6175 7365 7320 7468 6520 6569 gs causes the ei\n+000062c0: 6768 7420 726f 7461 7469 6f6e 7320 746f ght rotations to\n+000062d0: 2061 6464 2074 6f20 3138 302e 2020 4966 add to 180. If\n+000062e0: 0a79 6f75 2772 6520 7061 7969 6e67 2061 .you're paying a\n+000062f0: 7474 656e 7469 6f6e 2c20 796f 7527 6c6c ttention, you'll\n+00006300: 2068 6176 6520 616c 7265 6164 7920 6e6f have already no\n+00006310: 7465 6420 7468 6174 2074 6865 7265 2061 ted that there a\n+00006320: 7265 205f 6e69 6e65 5f0a 706f 696e 7473 re _nine_.points\n+00006330: 2c20 7072 6f64 7563 696e 6720 6569 6768 , producing eigh\n+00006340: 7420 6c69 6e65 2073 6567 6d65 6e74 732e t line segments.\n+00006350: 0a0a 2020 2059 6f75 2063 616e 2063 6175 .. You can cau\n+00006360: 7365 2074 6865 2073 7765 7074 2070 6f69 se the swept poi\n+00006370: 6e74 2074 6f20 6765 6e65 7261 7465 2061 nt to generate a\n+00006380: 2073 696e 676c 6520 706f 6c79 676f 6e20 single polygon \n+00006390: 7261 7468 6572 0a74 6861 6e20 6120 706f rather.than a po\n+000063a0: 6c79 6c69 6e65 2062 7920 7573 696e 6720 lyline by using \n+000063b0: 7468 6520 e280 9c63 6c6f 7375 7265 2074 the ...closure t\n+000063c0: 6167 e280 9d20 3c3e 2061 6674 6572 2074 ag... <> after t\n+000063d0: 6865 206e 756d 6265 7220 6f66 2073 7765 he number of swe\n+000063e0: 7074 0a6f 626a 6563 7473 2e20 2043 6f64 pt.objects. Cod\n+000063f0: 6520 616e 6420 7265 7375 6c74 2066 6f6c e and result fol\n+00006400: 6c6f 770a 2020 6465 6620 6e5f 7365 6773 low. def n_segs\n+00006410: 2038 0a20 2073 7765 6570 207b 206e 5f73 8. sweep { n_s\n+00006420: 6567 733c 3e2c 2072 6f74 6174 6528 3138 egs<>, rotate(18\n+00006430: 3020 2f20 6e5f 7365 6773 2c20 2830 2c30 0 / n_segs, (0,0\n+00006440: 2c30 292c 205b 302c 302c 315d 2920 7d20 ,0), [0,0,1]) } \n+00006450: 2831 2c30 2c30 290a 0008 5b69 6d61 6765 (1,0,0)...[image\n+00006460: 2073 7263 3d22 6578 3038 302e 706e 6722 src=\"ex080.png\"\n+00006470: 2074 6578 743d 2249 6d61 6765 2065 7830 text=\"Image ex0\n+00006480: 3830 206f 6d69 7474 6564 2069 6e20 7465 80 omitted in te\n+00006490: 7874 2076 6572 7369 6f6e 206f 6620 7468 xt version of th\n+000064a0: 6973 2064 6f63 756d 656e 742e 2200 085d is document.\"..]\n+000064b0: 0a0a 1f0a 4669 6c65 3a20 736b 6574 6368 ....File: sketch\n+000064c0: 2e69 6e66 6f2c 2020 4e6f 6465 3a20 506f .info, Node: Po\n+000064d0: 6c79 6c69 6e65 2073 7765 6570 732c 2020 lyline sweeps, \n+000064e0: 4e65 7874 3a20 4e65 7374 6564 2073 7765 Next: Nested swe\n+000064f0: 6570 732c 2020 5072 6576 3a20 506f 696e eps, Prev: Poin\n+00006500: 7420 7377 6565 7073 2c20 2055 703a 2053 t sweeps, Up: S\n+00006510: 7765 7074 206f 626a 6563 7473 0a0a 322e wept objects..2.\n+00006520: 372e 3220 506f 6c79 6c69 6e65 2073 7765 7.2 Polyline swe\n+00006530: 6570 730a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d eps.------------\n+00006540: 2d2d 2d2d 2d2d 2d2d 2d0a 0a53 7765 6570 ---------..Sweep\n+00006550: 696e 6720 6120 706f 6c79 6c69 6e65 2070 ing a polyline p\n+00006560: 726f 6475 6365 7320 6120 7375 7266 6163 roduces a surfac\n+00006570: 6520 636f 6d70 6f73 6564 206f 6620 6d61 e composed of ma\n+00006580: 6e79 2066 6163 6573 2e20 2054 6865 0a75 ny faces. The.u\n+00006590: 6e62 726f 6b65 6e20 6865 6c69 7820 696e nbroken helix in\n+000065a0: 2074 6865 2065 7861 6d70 6c65 202a 6e6f the example *no\n+000065b0: 7465 2048 656c 6978 2077 6974 6820 6375 te Helix with cu\n+000065c0: 6c6c 2073 6574 2066 616c 7365 2074 6865 ll set false the\n+000065d0: 6e0a 7472 7565 3a3a 2069 7320 7072 6f64 n.true:: is prod\n+000065e0: 7563 6564 2062 7920 7468 6973 2063 6f64 uced by this cod\n+000065f0: 6520 2870 6c75 7320 6120 7375 7272 6f75 e (plus a surrou\n+00006600: 6e64 696e 6720 7075 7420 726f 7461 7469 nding put rotati\n+00006610: 6f6e 2074 6f20 6d61 6b65 0a61 6e20 696e on to make.an in\n+00006620: 7465 7265 7374 696e 6720 7669 6577 3b20 teresting view; \n+00006630: 7468 6973 2068 6173 2062 6565 6e20 6f6d this has been om\n+00006640: 6974 7465 6429 2e0a 2020 6465 6620 4b20 itted).. def K \n+00006650: 5b30 2c30 2c31 5d0a 2020 7377 6565 705b [0,0,1]. sweep[\n+00006660: 6375 6c6c 3d66 616c 7365 5d20 7b0a 2020 cull=false] {. \n+00006670: 2020 3630 2c20 0a20 2020 2072 6f74 6174 60, . rotat\n+00006680: 6528 3130 2c20 2830 2c30 2c30 292c 205b e(10, (0,0,0), [\n+00006690: 4b5d 2920 7468 656e 2074 7261 6e73 6c61 K]) then transla\n+000066a0: 7465 2831 2f36 202a 205b 4b5d 2920 0a20 te(1/6 * [K]) . \n+000066b0: 207d 206c 696e 655b 6c69 6e65 7769 6474 } line[linewidt\n+000066c0: 683d 3270 745d 282d 312c 3029 2831 2c30 h=2pt](-1,0)(1,0\n+000066d0: 290a 4167 6169 6e2c 2036 3020 7365 676d ).Again, 60 segm\n+000066e0: 656e 7473 206f 6620 7468 6520 6865 6c69 ents of the heli\n+000066f0: 7820 6172 6520 7072 6f64 7563 6564 2062 x are produced b\n+00006700: 7920 636f 6e6e 6563 7469 6e67 2036 3120 y connecting 61 \n+00006710: 696e 7374 616e 6365 730a 6f66 2074 6865 instances.of the\n+00006720: 2073 7765 7074 206c 696e 652e 2020 4f70 swept line. Op\n+00006730: 7469 6f6e 7320 6170 706c 6965 6420 746f tions applied to\n+00006740: 2074 6865 2073 7765 6570 2c20 6865 7265 the sweep, here\n+00006750: 2063 756c 6c3d 6661 6c73 652c 2061 7265 cull=false, are\n+00006760: 0a74 7265 6174 6564 2061 7320 6f70 7469 .treated as opti\n+00006770: 6f6e 7320 666f 7220 7468 6520 6765 6e65 ons for the gene\n+00006780: 7261 7465 6420 706f 6c79 676f 6e20 6f72 rated polygon or\n+00006790: 2070 6f6c 796c 696e 652e 2020 4f70 7469 polyline. Opti\n+000067a0: 6f6e 7320 6f66 0a74 6865 2073 7765 7074 ons of.the swept\n+000067b0: 206c 696e 6520 6974 7365 6c66 2c20 6865 line itself, he\n+000067c0: 7265 206c 696e 6577 6964 7468 3d32 7074 re linewidth=2pt\n+000067d0: 2c20 6172 6520 6967 6e6f 7265 642c 2074 , are ignored, t\n+000067e0: 686f 7567 6820 7769 7468 2061 0a77 6172 hough with a.war\n+000067f0: 6e69 6e67 2e20 2054 6869 7320 6465 6620 ning. This def \n+00006800: 6973 2061 20e2 809c 7665 6374 6f72 e280 is a ...vector..\n+00006810: 9d20 6465 6669 6e69 7469 6f6e 2c20 7768 . definition, wh\n+00006820: 6963 6820 6d75 7374 2062 6520 7265 6665 ich must be refe\n+00006830: 7265 6e63 6564 0a77 6974 6820 7371 7561 renced.with squa\n+00006840: 7265 2062 7261 636b 6574 732c 2065 2e67 re brackets, e.g\n+00006850: 2e20 5b66 6f6f 5d2e 0a0a 1f0a 4669 6c65 . [foo].....File\n+00006860: 3a20 736b 6574 6368 2e69 6e66 6f2c 2020 : sketch.info, \n+00006870: 4e6f 6465 3a20 4e65 7374 6564 2073 7765 Node: Nested swe\n+00006880: 6570 732c 2020 4e65 7874 3a20 506f 6c79 eps, Next: Poly\n+00006890: 676f 6e20 7377 6565 7073 2c20 2050 7265 gon sweeps, Pre\n+000068a0: 763a 2050 6f6c 796c 696e 6520 7377 6565 v: Polyline swee\n+000068b0: 7073 2c20 2055 703a 2053 7765 7074 206f ps, Up: Swept o\n+000068c0: 626a 6563 7473 0a0a 322e 372e 3320 4e65 bjects..2.7.3 Ne\n+000068d0: 7374 6564 2073 7765 6570 730a 2d2d 2d2d sted sweeps.----\n+000068e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------.\n+000068f0: 0a57 6865 6e20 7468 6520 6365 6e74 6572 .When the center\n+00006900: 2070 6f69 6e74 206f 6620 726f 7461 7469 point of rotati\n+00006910: 6f6e 2069 7320 6f6d 6974 7465 642c 2074 on is omitted, t\n+00006920: 6865 206f 7269 6769 6e20 6973 2061 7373 he origin is ass\n+00006930: 756d 6564 2e0a 5768 656e 2061 2070 6f69 umed..When a poi\n+00006940: 6e74 2068 6173 206f 6e6c 7920 7477 6f20 nt has only two \n+00006950: 636f 6f72 6469 6e61 7465 732c 2074 6865 coordinates, the\n+00006960: 7920 6172 6520 7461 6b65 6e20 6173 2078 y are taken as x\n+00006970: 2061 6e64 2079 2c20 7769 7468 0a7a 3d30 and y, with.z=0\n+00006980: 2061 7373 756d 6564 2e20 2041 2074 6f72 assumed. A tor\n+00006990: 6f69 6420 6973 2074 6865 7265 666f 7265 oid is therefore\n+000069a0: 206f 6274 6169 6e65 6420 7769 7468 2074 obtained with t\n+000069b0: 6869 7320 636f 6465 2e0a 2020 6465 6620 his code.. def \n+000069c0: 6e5f 746f 726f 6964 5f73 6567 7320 3230 n_toroid_segs 20\n+000069d0: 2020 2064 6566 206e 5f63 6972 636c 655f def n_circle_\n+000069e0: 7365 6773 2031 360a 2020 6465 6620 725f segs 16. def r_\n+000069f0: 6d69 6e6f 7220 3120 2020 2020 2020 2020 minor 1 \n+00006a00: 2064 6566 2072 5f6d 616a 6f72 2031 2e35 def r_major 1.5\n+00006a10: 0a20 2073 7765 6570 207b 206e 5f74 6f72 . sweep { n_tor\n+00006a20: 6f69 645f 7365 6773 2c20 726f 7461 7465 oid_segs, rotate\n+00006a30: 2833 3630 202f 206e 5f74 6f72 6f69 645f (360 / n_toroid_\n+00006a40: 7365 6773 2c20 5b30 2c31 2c30 5d29 207d segs, [0,1,0]) }\n+00006a50: 0a20 2020 2073 7765 6570 207b 206e 5f63 . sweep { n_c\n+00006a60: 6972 636c 655f 7365 6773 2c20 726f 7461 ircle_segs, rota\n+00006a70: 7465 2833 3630 202f 206e 5f63 6972 636c te(360 / n_circl\n+00006a80: 655f 7365 6773 2c20 2872 5f6d 616a 6f72 e_segs, (r_major\n+00006a90: 2c30 2c30 2929 207d 200a 2020 2020 2020 ,0,0)) } . \n+00006aa0: 2872 5f6d 616a 6f72 202b 2072 5f6d 696e (r_major + r_min\n+00006ab0: 6f72 2c20 3029 0a20 2020 466f 7220 696e or, 0). For in\n+00006ac0: 7475 6974 696f 6e2c 2074 6865 2069 6465 tuition, the ide\n+00006ad0: 6120 6f66 2074 6865 2063 6f64 6520 6973 a of the code is\n+00006ae0: 2074 6f20 736b 6574 6368 2061 2063 6972 to sketch a cir\n+00006af0: 636c 6520 746f 2074 6865 0a72 6967 6874 cle to the.right\n+00006b00: 206f 6620 7468 6520 6f72 6967 696e 2069 of the origin i\n+00006b10: 6e20 7468 6520 7879 2d70 6c61 6e65 2c20 n the xy-plane, \n+00006b20: 7468 656e 2072 6f74 6174 6520 7468 6174 then rotate that\n+00006b30: 2063 6972 636c 6520 226f 7574 206f 6620 circle \"out of \n+00006b40: 7468 650a 706c 616e 6522 2061 626f 7574 the.plane\" about\n+00006b50: 2074 6865 2079 2d61 7869 7320 746f 206d the y-axis to m\n+00006b60: 616b 6520 7468 6520 6669 6e61 6c20 6669 ake the final fi\n+00006b70: 6775 7265 2e20 2054 6869 7320 7072 6f64 gure. This prod\n+00006b80: 7563 6573 2074 6865 0a66 6f6c 6c6f 7769 uces the.followi\n+00006b90: 6e67 2e20 2028 4120 7669 6577 2072 6f74 ng. (A view rot\n+00006ba0: 6174 696f 6e20 616e 6420 736f 6d65 2061 ation and some a\n+00006bb0: 7865 7320 6861 7665 2062 6565 6e20 6164 xes have been ad\n+00006bc0: 6465 642e 290a 0a00 085b 696d 6167 6520 ded.)....[image \n+00006bd0: 7372 633d 2265 7830 3930 2e70 6e67 2220 src=\"ex090.png\" \n+00006be0: 7465 7874 3d22 496d 6167 6520 6578 3039 text=\"Image ex09\n+00006bf0: 3020 6f6d 6974 7465 6420 696e 2074 6578 0 omitted in tex\n+00006c00: 7420 7665 7273 696f 6e20 6f66 2074 6869 t version of thi\n+00006c10: 7320 646f 6375 6d65 6e74 2e22 0008 5d0a s document.\"..].\n+00006c20: 0a20 2020 5468 6973 2065 7861 6d70 6c65 . This example\n+00006c30: 2061 6c73 6f20 7368 6f77 7320 7468 6174 also shows that\n+00006c40: 2074 6865 2073 7765 7074 206f 626a 6563 the swept objec\n+00006c50: 7420 6d61 7920 6974 7365 6c66 2062 6520 t may itself be \n+00006c60: 616e 6f74 6865 720a e280 9873 7765 6570 another....sweep\n+00006c70: e280 992e 2020 496e 2066 6163 742c 2069 .... In fact, i\n+00006c80: 7420 6d61 7920 6265 205f 616e 795f 20e2 t may be _any_ .\n+00006c90: 8098 736b 6574 6368 e280 9920 6578 7072 ..sketch... expr\n+00006ca0: 6573 7369 6f6e 2074 6861 7420 7265 7375 ession that resu\n+00006cb0: 6c74 7320 696e 2061 0a6c 6973 7420 6f66 lts in a.list of\n+00006cc0: 206f 6e65 206f 7220 6d6f 7265 2070 6f69 one or more poi\n+00006cd0: 6e74 7320 6f72 2c20 616c 7465 726e 6174 nts or, alternat\n+00006ce0: 656c 792c 2061 206c 6973 7420 6f66 206f ely, a list of o\n+00006cf0: 6e65 206f 7220 6d6f 7265 0a70 6f6c 796c ne or more.polyl\n+00006d00: 696e 6573 2061 6e64 2070 6f6c 7967 6f6e ines and polygon\n+00006d10: 732e 2020 5468 6520 6c61 7474 6572 206b s. The latter k\n+00006d20: 696e 6420 6f66 206c 6973 7420 6361 6e20 ind of list can \n+00006d30: 6265 2063 7265 6174 6564 2077 6974 6820 be created with \n+00006d40: 610a 7b20 7d2d 656e 636c 6f73 6564 2062 a.{ }-enclosed b\n+00006d50: 6c6f 636b 2c20 7065 7268 6170 7320 666f lock, perhaps fo\n+00006d60: 6c6c 6f77 696e 6720 6120 7075 7420 6f72 llowing a put or\n+00006d70: 2072 6570 6561 742e 0a0a 1f0a 4669 6c65 repeat.....File\n+00006d80: 3a20 736b 6574 6368 2e69 6e66 6f2c 2020 : sketch.info, \n+00006d90: 4e6f 6465 3a20 506f 6c79 676f 6e20 7377 Node: Polygon sw\n+00006da0: 6565 7073 2c20 204e 6578 743a 2050 6f6c eeps, Next: Pol\n+00006db0: 796c 696e 6520 7377 6565 7073 2077 6974 yline sweeps wit\n+00006dc0: 6820 636c 6f73 7572 652c 2020 5072 6576 h closure, Prev\n+00006dd0: 3a20 4e65 7374 6564 2073 7765 6570 732c : Nested sweeps,\n+00006de0: 2020 5570 3a20 5377 6570 7420 6f62 6a65 Up: Swept obje\n+00006df0: 6374 730a 0a32 2e37 2e34 2050 6f6c 7967 cts..2.7.4 Polyg\n+00006e00: 6f6e 2073 7765 6570 730a 2d2d 2d2d 2d2d on sweeps.------\n+00006e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------..\n+00006e20: 5377 6565 7069 6e67 2061 2070 6f6c 7967 Sweeping a polyg\n+00006e30: 6f6e 2063 7265 6174 6573 2061 2063 6c6f on creates a clo\n+00006e40: 7365 6420 7375 7266 6163 6520 7769 7468 sed surface with\n+00006e50: 2070 6f6c 7967 6f6e 7320 6174 2074 6865 polygons at the\n+00006e60: 2065 6e64 732c 0a77 6869 6368 2061 7265 ends,.which are\n+00006e70: 206a 7573 7420 636f 7069 6573 206f 6620 just copies of \n+00006e80: 7468 6520 6f72 6967 696e 616c 2c20 6170 the original, ap\n+00006e90: 7072 6f70 7269 6174 656c 7920 706f 7369 propriately posi\n+00006ea0: 7469 6f6e 6564 2e20 202a 4e6f 7465 0a53 tioned. *Note.S\n+00006eb0: 6f6c 6964 2063 6f69 6c20 6578 616d 706c olid coil exampl\n+00006ec0: 653a 3a2e 2020 4f70 7469 6f6e 7320 6f6e e::. Options on\n+00006ed0: 2074 6865 2073 7765 7074 2070 6f6c 7967 the swept polyg\n+00006ee0: 6f6e 2c20 6966 2074 6865 7920 6578 6973 on, if they exis\n+00006ef0: 742c 2061 7265 0a61 7070 6c69 6564 2074 t, are.applied t\n+00006f00: 6f20 7468 6520 656e 6473 2e20 204f 7468 o the ends. Oth\n+00006f10: 6572 7769 7365 2074 6865 2073 7765 6570 erwise the sweep\n+00006f20: 206f 7074 696f 6e73 2061 7265 2075 7365 options are use\n+00006f30: 6420 7468 726f 7567 686f 7574 2e0a 0a1f d throughout....\n+00006f40: 0a46 696c 653a 2073 6b65 7463 682e 696e .File: sketch.in\n+00006f50: 666f 2c20 204e 6f64 653a 2050 6f6c 796c fo, Node: Polyl\n+00006f60: 696e 6520 7377 6565 7073 2077 6974 6820 ine sweeps with \n+00006f70: 636c 6f73 7572 652c 2020 4e65 7874 3a20 closure, Next: \n+00006f80: 4166 6669 6e65 2061 7269 7468 6d65 7469 Affine arithmeti\n+00006f90: 632c 2020 5072 6576 3a20 506f 6c79 676f c, Prev: Polygo\n+00006fa0: 6e20 7377 6565 7073 2c20 2055 703a 2053 n sweeps, Up: S\n+00006fb0: 7765 7074 206f 626a 6563 7473 0a0a 322e wept objects..2.\n+00006fc0: 372e 3520 506f 6c79 6c69 6e65 2073 7765 7.5 Polyline swe\n+00006fd0: 6570 7320 7769 7468 2063 6c6f 7375 7265 eps with closure\n+00006fe0: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .---------------\n+00006ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ----------------\n+00007000: 2d2d 2d0a 0a41 2070 6f6c 796c 696e 6520 ---..A polyline \n+00007010: 7377 6565 7020 7769 7468 2061 2063 6c6f sweep with a clo\n+00007020: 7375 7265 2074 6167 2063 7265 6174 6573 sure tag creates\n+00007030: 2061 6e6f 7468 6572 206b 696e 6420 6f66 another kind of\n+00007040: 2063 6c6f 7365 640a 7375 7266 6163 652e closed.surface.\n+00007050: 2020 4669 7273 742c 2074 6865 2070 6f6c First, the pol\n+00007060: 796c 696e 6520 7365 676d 656e 7473 2061 yline segments a\n+00007070: 7265 2063 6f6e 6e65 6374 6564 2062 7920 re connected by \n+00007080: 6661 6365 732c 206a 7573 7420 6173 0a77 faces, just as.w\n+00007090: 6974 686f 7574 2074 6865 2063 6c6f 7375 ithout the closu\n+000070a0: 7265 2074 6167 2e20 2054 6865 6e2c 2065 re tag. Then, e\n+000070b0: 6163 6820 7365 7420 6f66 2065 6e64 2070 ach set of end p\n+000070c0: 6f69 6e74 7320 6973 206a 6f69 6e65 6420 oints is joined \n+000070d0: 746f 206d 616b 650a 6120 706f 6c79 676f to make.a polygo\n+000070e0: 6e2c 206f 6e65 2066 6f72 2065 6163 6820 n, one for each \n+000070f0: 656e 642e 2020 4120 636f 6465 2066 6f72 end. A code for\n+00007100: 2073 6576 6572 616c 2076 6965 7773 206f several views o\n+00007110: 6620 6120 6379 6c69 6e64 7269 6361 6c0a f a cylindrical.\n+00007120: 7072 6973 6d20 666f 6c6c 6f77 732e 0a20 prism follows.. \n+00007130: 2064 6566 206e 5f63 796c 5f73 6567 7320 def n_cyl_segs \n+00007140: 3230 2020 6465 6620 6e5f 7669 6577 7320 20 def n_views \n+00007150: 3520 2064 6566 2049 205b 312c 302c 305d 5 def I [1,0,0]\n+00007160: 0a20 2064 6566 2065 6e64 6f70 7473 205b . def endopts [\n+00007170: 6669 6c6c 636f 6c6f 723d 6c69 6768 7467 fillcolor=lightg\n+00007180: 7261 795d 0a20 2072 6570 6561 7420 7b20 ray]. repeat { \n+00007190: 6e5f 7669 6577 732c 2072 6f74 6174 6528 n_views, rotate(\n+000071a0: 3138 302f 6e5f 7669 6577 732c 205b 495d 180/n_views, [I]\n+000071b0: 2920 7468 656e 2074 7261 6e73 6c61 7465 ) then translate\n+000071c0: 285b 495d 202a 2032 2e31 2920 7d20 0a20 ([I] * 2.1) } . \n+000071d0: 2020 2073 7765 6570 5b65 6e64 6f70 7473 sweep[endopts\n+000071e0: 5d7b 206e 5f63 796c 5f73 6567 733c 3e2c ]{ n_cyl_segs<>,\n+000071f0: 2072 6f74 6174 6528 3336 302f 6e5f 6379 rotate(360/n_cy\n+00007200: 6c5f 7365 6773 2c20 5b30 2c31 2c30 5d29 l_segs, [0,1,0])\n+00007210: 207d 200a 2020 2020 2020 6c69 6e65 5b66 } . line[f\n+00007220: 696c 6c63 6f6c 6f72 3d77 6869 7465 5d28 illcolor=white](\n+00007230: 312c 2d31 2928 312c 3129 0a49 7420 7072 1,-1)(1,1).It pr\n+00007240: 6f64 7563 6573 2074 6869 7320 6472 6177 oduces this draw\n+00007250: 696e 672e 0a0a 0008 5b69 6d61 6765 2073 ing.....[image s\n+00007260: 7263 3d22 6578 3131 302e 706e 6722 2074 rc=\"ex110.png\" t\n+00007270: 6578 743d 2249 6d61 6765 2065 7831 3130 ext=\"Image ex110\n+00007280: 206f 6d69 7474 6564 2069 6e20 7465 7874 omitted in text\n+00007290: 2076 6572 7369 6f6e 206f 6620 7468 6973 version of this\n+000072a0: 2064 6f63 756d 656e 742e 2200 085d 0a0a document.\"..]..\n+000072b0: 5468 6520 6f70 7469 6f6e 7320 6f66 2074 The options of t\n+000072c0: 6865 2073 7765 7074 206c 696e 652c 2069 he swept line, i\n+000072d0: 6620 616e 792c 2061 7265 2061 7070 6c69 f any, are appli\n+000072e0: 6564 2074 6f20 7468 6520 6661 6365 7320 ed to the faces \n+000072f0: 7072 6f64 7563 6564 0a62 7920 7377 6565 produced.by swee\n+00007300: 7069 6e67 2074 6865 206c 696e 652c 2062 ping the line, b\n+00007310: 7574 206e 6f74 2074 6865 2065 6e64 2070 ut not the end p\n+00007320: 6f6c 7967 6f6e 732e 2020 4f74 6865 7277 olygons. Otherw\n+00007330: 6973 652c 2074 6865 2073 7765 6570 0a6f ise, the sweep.o\n+00007340: 7074 696f 6e73 2061 7265 2061 7070 6c69 ptions are appli\n+00007350: 6564 2074 6872 6f75 6768 6f75 742e 2020 ed throughout. \n+00007360: 5468 6520 6465 6620 696e 2074 6869 7320 The def in this \n+00007370: 6578 616d 706c 6520 6973 2061 6e20 e280 example is an ..\n+00007380: 9c6f 7074 696f 6ee2 809d 0a64 6566 696e .option....defin\n+00007390: 6974 696f 6e2e 2020 5265 6665 7265 6e63 ition. Referenc\n+000073a0: 6573 2074 6f20 6f70 7469 6f6e 7320 6d75 es to options mu\n+000073b0: 7374 2062 6520 656e 636c 6f73 6564 2069 st be enclosed i\n+000073c0: 6e20 7371 7561 7265 2062 7261 636b 6574 n square bracket\n+000073d0: 732c 0a65 2e67 2e20 2020 5b66 6f6f 5d2e s,.e.g. [foo].\n+000073e0: 2020 596f 7520 6361 6e20 636f 6e63 6174 You can concat\n+000073f0: 656e 6174 6520 7365 7665 7261 6c20 7365 enate several se\n+00007400: 7473 206f 6620 6f70 7469 6f6e 7320 7769 ts of options wi\n+00007410: 7468 2061 2073 696e 676c 650a 7265 6665 th a single.refe\n+00007420: 7265 6e63 652c 2065 2e67 2e20 2020 5b63 rence, e.g. [c\n+00007430: 6f6c 6f72 2c20 7369 7a65 2c20 7374 796c olor, size, styl\n+00007440: 655d 2020 776f 756c 6420 6361 7573 6520 e] would cause \n+00007450: 7468 6520 6f70 7469 6f6e 0a64 6566 696e the option.defin\n+00007460: 6974 696f 6e73 2066 6f72 20e2 8098 636f itions for ...co\n+00007470: 6c6f 72e2 8099 2c20 e280 9873 697a 65e2 lor..., ...size.\n+00007480: 8099 2c20 616e 6420 e280 9873 7479 6c65 .., and ...style\n+00007490: e280 9920 746f 2061 7070 6561 7220 696e ... to appear in\n+000074a0: 2073 6571 7565 6e63 6520 696e 0a74 6865 sequence in.the\n+000074b0: 206f 7574 7075 7420 6372 6561 7465 6420 output created \n+000074c0: 6279 2074 6865 20e2 8098 736b 6574 6368 by the ...sketch\n+000074d0: e280 9920 636f 6d6d 616e 6420 636f 6e74 ... command cont\n+000074e0: 6169 6e69 6e67 2074 6865 2072 6566 6572 aining the refer\n+000074f0: 656e 6365 2e0a 4861 7070 696c 792c 2074 ence..Happily, t\n+00007500: 6865 2073 796e 7461 7820 6f66 20e2 8098 he syntax of ...\n+00007510: 736b 6574 6368 e280 9920 6973 2073 7563 sketch... is suc\n+00007520: 6820 7468 6174 206f 7074 696f 6e73 2072 h that options r\n+00007530: 6566 6572 656e 6365 7320 6361 6e0a 6e65 eferences can.ne\n+00007540: 7665 7220 6265 2063 6f6e 6675 7365 6420 ver be confused \n+00007550: 7769 7468 2076 6563 746f 7220 7265 6665 with vector refe\n+00007560: 7265 6e63 6573 2e20 2057 6869 6c65 206e rences. While n\n+00007570: 6f74 2061 7070 6172 656e 7420 696e 2074 ot apparent in t\n+00007580: 6869 730a 6578 616d 706c 652c 206f 7074 his.example, opt\n+00007590: 696f 6e73 2072 6566 6572 656e 6365 7320 ions references \n+000075a0: 6172 6520 7573 6566 756c 2077 6865 6e20 are useful when \n+000075b0: 6465 6669 6e69 6e67 206d 616e 7920 6f62 defining many ob\n+000075c0: 6a65 6374 7320 7769 7468 2061 0a73 696d jects with a.sim\n+000075d0: 696c 6172 2061 7070 6561 7261 6e63 652e ilar appearance.\n+000075e0: 0a0a 1f0a 4669 6c65 3a20 736b 6574 6368 ....File: sketch\n+000075f0: 2e69 6e66 6f2c 2020 4e6f 6465 3a20 4166 .info, Node: Af\n+00007600: 6669 6e65 2061 7269 7468 6d65 7469 632c fine arithmetic,\n+00007610: 2020 4e65 7874 3a20 4d6f 7265 2074 6f20 Next: More to \n+00007620: 6c65 6172 6e2c 2020 5072 6576 3a20 506f learn, Prev: Po\n+00007630: 6c79 6c69 6e65 2073 7765 6570 7320 7769 lyline sweeps wi\n+00007640: 7468 2063 6c6f 7375 7265 2c20 2055 703a th closure, Up:\n+00007650: 2053 7765 7074 206f 626a 6563 7473 0a0a Swept objects..\n+00007660: 322e 372e 3620 4166 6669 6e65 2061 7269 2.7.6 Affine ari\n+00007670: 7468 6d65 7469 630a 2d2d 2d2d 2d2d 2d2d thmetic.--------\n+00007680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------.\n+00007690: 0a54 6865 2061 7269 7468 6d65 7469 6320 .The arithmetic \n+000076a0: 5b49 5d20 2a20 322e 3120 6162 6f76 6520 [I] * 2.1 above \n+000076b0: 6869 6e74 7320 6174 2061 206c 6172 6765 hints at a large\n+000076c0: 7220 7472 7574 682e 2020 e280 9853 6b65 r truth. ...Ske\n+000076d0: 7463 68e2 8099 0a6f 7065 7261 746f 7273 tch....operators\n+000076e0: 2077 6f72 6b20 6f6e 2073 6361 6c61 7273 work on scalars\n+000076f0: 2c20 7665 6374 6f72 732c 2070 6f69 6e74 , vectors, point\n+00007700: 732c 2061 6e64 2074 7261 6e73 666f 726d s, and transform\n+00007710: 7320 6163 636f 7264 696e 6720 746f 0a74 s according to.t\n+00007720: 6865 2067 656e 6572 616c 2072 756c 6573 he general rules\n+00007730: 206f 6620 e280 9c61 6666 696e 6520 616c of ...affine al\n+00007740: 6765 6272 61e2 809d 2e20 2054 6869 7320 gebra.... This \n+00007750: 6361 6e20 6265 2068 656c 7066 756c 2066 can be helpful f\n+00007760: 6f72 2073 6574 7469 6e67 0a75 7020 6469 or setting.up di\n+00007770: 6167 7261 6d73 2077 6974 6820 636f 6d70 agrams with comp\n+00007780: 7574 6564 2067 656f 6d65 7472 792e 2020 uted geometry. \n+00007790: 466f 7220 6578 616d 706c 652c 2069 6620 For example, if \n+000077a0: 796f 7520 6861 7665 2074 7269 616e 676c you have triangl\n+000077b0: 650a 7665 7274 6963 6573 2028 7031 2920 e.vertices (p1) \n+000077c0: 7468 726f 7567 6820 2870 3329 2061 6e64 through (p3) and\n+000077d0: 206e 6565 6420 746f 2064 7261 7720 6120 need to draw a \n+000077e0: 756e 6974 206e 6f72 6d61 6c20 7665 6374 unit normal vect\n+000077f0: 6f72 0a70 6f69 6e74 696e 6720 6f75 7420 or.pointing out \n+00007800: 6f66 2074 6865 2063 656e 7465 7220 6f66 of the center of\n+00007810: 2074 6865 2074 7269 616e 676c 652c 2074 the triangle, t\n+00007820: 6869 7320 636f 6465 2064 6f65 7320 7468 his code does th\n+00007830: 6520 7472 6963 6b2e 0a20 2064 6566 2070 e trick.. def p\n+00007840: 3120 2831 2c30 2c30 2920 2064 6566 2070 1 (1,0,0) def p\n+00007850: 3220 2830 2c30 2e35 2c30 2920 2064 6566 2 (0,0.5,0) def\n+00007860: 2070 3320 282d 302e 352c 2d31 2c32 290a p3 (-0.5,-1,2).\n+00007870: 2020 6465 6620 4f20 2830 2c30 2c30 290a def O (0,0,0).\n+00007880: 2020 6465 6620 4e20 756e 6974 2820 2828 def N unit( ((\n+00007890: 7033 2920 2d20 2870 3229 2920 2a20 2828 p3) - (p2)) * ((\n+000078a0: 7031 2920 2d20 2870 3229 2920 290a 2020 p1) - (p2)) ). \n+000078b0: 6465 6620 6e31 2028 2870 3129 2d28 4f29 def n1 ((p1)-(O)\n+000078c0: 202b 2028 7032 292d 284f 2920 2b20 2870 + (p2)-(O) + (p\n+000078d0: 3329 2d28 4f29 2920 2f20 3320 2b20 284f 3)-(O)) / 3 + (O\n+000078e0: 290a 2020 6465 6620 6e32 2028 6e31 292b ). def n2 (n1)+\n+000078f0: 5b4e 5d0a 2020 706f 6c79 676f 6e28 7031 [N]. polygon(p1\n+00007900: 2928 7032 2928 7033 290a 2020 6c69 6e65 )(p2)(p3). line\n+00007910: 5b61 7272 6f77 733d 2a2d 3e5d 286e 3129 [arrows=*->](n1)\n+00007920: 286e 3229 0a54 6865 2066 6972 7374 206c (n2).The first l\n+00007930: 696e 6520 636f 6d70 7574 6573 2074 6865 ine computes the\n+00007940: 2063 726f 7373 2070 726f 6475 6374 206f cross product o\n+00007950: 6620 7477 6f20 6564 6765 2076 6563 746f f two edge vecto\n+00007960: 7273 206f 6620 7468 650a 7472 6961 6e67 rs of the.triang\n+00007970: 6c65 2061 6e64 2073 6361 6c65 7320 6974 le and scales it\n+00007980: 2074 6f20 756e 6974 206c 656e 6774 682e to unit length.\n+00007990: 2020 5468 6520 7365 636f 6e64 2063 6f6d The second com\n+000079a0: 7075 7465 7320 7468 6520 6176 6572 6167 putes the averag\n+000079b0: 650a 6f66 2074 6865 2076 6572 7469 6365 e.of the vertice\n+000079c0: 732e 2020 4e6f 7465 2074 6861 7420 7375 s. Note that su\n+000079d0: 6274 7261 6374 696f 6e20 616e 6420 6164 btraction and ad\n+000079e0: 6469 7469 6f6e 206f 6620 7468 6520 6f72 dition of the or\n+000079f0: 6967 696e 0a65 6666 6563 7469 7665 6c79 igin.effectively\n+00007a00: 2063 6f6e 7665 7274 2076 6563 746f 7273 convert vectors\n+00007a10: 2074 6f20 706f 696e 7473 2061 6e64 205f to points and _\n+00007a20: 7669 6365 2076 6572 7361 5f2e 2020 5468 vice versa_. Th\n+00007a30: 6520 6c69 6e65 0a63 6f6d 6d61 6e64 2064 e line.command d\n+00007a40: 7261 7773 2074 6865 206e 6f72 6d61 6c20 raws the normal \n+00007a50: 6174 2074 6865 2063 6f72 7265 6374 2073 at the correct s\n+00007a60: 706f 742e 0a0a 0008 5b69 6d61 6765 2073 pot.....[image s\n+00007a70: 7263 3d22 6578 3130 302e 706e 6722 2074 rc=\"ex100.png\" t\n+00007a80: 6578 743d 2249 6d61 6765 2065 7831 3030 ext=\"Image ex100\n+00007a90: 206f 6d69 7474 6564 2069 6e20 7465 7874 omitted in text\n+00007aa0: 2076 6572 7369 6f6e 206f 6620 7468 6973 version of this\n+00007ab0: 2064 6f63 756d 656e 742e 2200 085d 0a0a document.\"..]..\n+00007ac0: 2020 2054 776f 2063 6176 6561 7473 2072 Two caveats r\n+00007ad0: 6567 6172 6469 6e67 2074 6869 7320 6578 egarding this ex\n+00007ae0: 616d 706c 6520 7265 6d61 696e 2e20 2046 ample remain. 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The\n+000090c0: 207a 2d63 6f6d 706f 6e65 6e74 730a 6172 z-components.ar\n+000090d0: 6520 6f70 7469 6f6e 616c 2061 6e64 2064 e optional and d\n+000090e0: 6566 6175 6c74 2074 6f20 7a65 726f 2e0a efault to zero..\n+000090f0: 0a1f 0a46 696c 653a 2073 6b65 7463 682e ...File: sketch.\n+00009100: 696e 666f 2c20 204e 6f64 653a 2054 7261 info, Node: Tra\n+00009110: 6e73 666f 726d 206c 6974 6572 616c 732c nsform literals,\n+00009120: 2020 5072 6576 3a20 506f 696e 7420 616e Prev: Point an\n+00009130: 6420 7665 6374 6f72 206c 6974 6572 616c d vector literal\n+00009140: 732c 2020 5570 3a20 4c69 7465 7261 6c73 s, Up: Literals\n+00009150: 0a0a 332e 312e 332e 3320 5472 616e 7366 ..3.1.3.3 Transf\n+00009160: 6f72 6d20 6c69 7465 7261 6c73 0a2e 2e2e orm literals....\n 00009170: 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e ................\n-00009180: 2e2e 2e0a 0a4d 6f73 7420 7472 616e 7366 .....Most transf\n-00009190: 6f72 6d20 6c69 7465 7261 6c73 2061 7265 orm literals are\n-000091a0: 2066 6f72 6d65 6420 6279 20e2 809c 636f formed by ...co\n-000091b0: 6e73 7472 7563 746f 7273 e280 9d2e 2020 nstructors.... \n-000091c0: 5468 6573 6520 6172 650a 7375 6d6d 6172 These are.summar\n-000091d0: 697a 6564 2069 6e20 7468 6520 666f 6c6c ized in the foll\n-000091e0: 6f77 696e 6720 7461 626c 652e 0a43 6f6e owing table..Con\n-000091f0: 7374 7275 6374 6f72 2020 2020 2020 2020 structor \n+00009180: 2e2e 2e2e 2e2e 2e0a 0a4d 6f73 7420 7472 .........Most tr\n+00009190: 616e 7366 6f72 6d20 6c69 7465 7261 6c73 ansform literals\n+000091a0: 2061 7265 2066 6f72 6d65 6420 6279 20e2 are formed by .\n+000091b0: 809c 636f 6e73 7472 7563 746f 7273 e280 ..constructors..\n+000091c0: 9d2e 2020 5468 6573 6520 6172 650a 7375 .. These are.su\n+000091d0: 6d6d 6172 697a 6564 2069 6e20 7468 6520 mmarized in the \n+000091e0: 666f 6c6c 6f77 696e 6720 7461 626c 652e following table.\n+000091f0: 0a43 6f6e 7374 7275 6374 6f72 2020 2020 .Constructor \n 00009200: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00009210: 2050 6172 616d 2074 7970 6573 2020 2020 Param types \n-00009220: 2020 2020 2020 2044 6573 6372 6970 7469 Descripti\n-00009230: 6f6e 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d on.-------------\n+00009210: 2020 2020 2050 6172 616d 2074 7970 6573 Param types\n+00009220: 2020 2020 2020 2020 2020 2044 6573 6372 Descr\n+00009230: 6970 7469 6f6e 0a2d 2d2d 2d2d 2d2d 2d2d iption.---------\n 00009240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ----------------\n 00009250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ----------------\n 00009260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ----------------\n 00009270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ----------------\n 00009280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ----------------\n-00009290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0ae2 --------------..\n-000092a0: 8098 726f 7461 7465 2841 2c50 2c58 29e2 ..rotate(A,P,X).\n-000092b0: 8099 2020 2020 2020 2020 2020 2020 2020 .. \n-000092c0: 2020 2020 2020 2073 6361 6c61 722c 706f scalar,po\n-000092d0: 696e 742c 7665 6374 6f72 2020 2052 6f74 int,vector Rot\n-000092e0: 6174 6520 e280 9841 e280 9920 6465 6772 ate ...A... degr\n-000092f0: 6565 7320 6162 6f75 7420 706f 696e 7420 ees about point \n-00009300: e280 9850 e280 9920 7769 7468 2061 7869 ...P... with axi\n-00009310: 730a 2020 2020 2020 2020 2020 2020 2020 s. \n+00009290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ----------------\n+000092a0: 2d2d 0ae2 8098 726f 7461 7465 2841 2c50 --....rotate(A,P\n+000092b0: 2c58 29e2 8099 2020 2020 2020 2020 2020 ,X)... \n+000092c0: 2020 2020 2020 2020 2020 2073 6361 6c61 scala\n+000092d0: 722c 706f 696e 742c 7665 6374 6f72 2020 r,point,vector \n+000092e0: 2052 6f74 6174 6520 e280 9841 e280 9920 Rotate ...A... \n+000092f0: 6465 6772 6565 7320 6162 6f75 7420 706f degrees about po\n+00009300: 696e 7420 e280 9850 e280 9920 7769 7468 int ...P... with\n+00009310: 2061 7869 730a 2020 2020 2020 2020 2020 axis. \n 00009320: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00009330: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00009340: 2020 2020 2020 2020 2020 2020 e280 9858 ...X\n-00009350: e280 9920 6163 636f 7264 696e 6720 746f ... according to\n-00009360: 2074 6865 2072 6967 6874 2068 616e 6420 the right hand \n-00009370: 7275 6c65 2e0a 2020 2020 2020 2020 2020 rule.. \n+00009340: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00009350: e280 9858 e280 9920 6163 636f 7264 696e ...X... accordin\n+00009360: 6720 746f 2074 6865 2072 6967 6874 2068 g to the right h\n+00009370: 616e 6420 7275 6c65 2e0a 2020 2020 2020 and rule.. \n 00009380: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00009390: 2020 2020 2020 2020 2020 2020 2020 2020 \n 000093a0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000093b0: 2a4e 6f74 6520 5269 6768 7420 6861 6e64 *Note Right hand\n-000093c0: 2072 756c 653a 3a2e 2020 e280 9850 e280 rule::. ...P..\n-000093d0: 9920 616e 6420 e280 9858 e280 9920 6172 . and ...X... ar\n-000093e0: 6520 626f 7468 0a20 2020 2020 2020 2020 e both. \n+000093b0: 2020 2020 2a4e 6f74 6520 5269 6768 7420 *Note Right \n+000093c0: 6861 6e64 2072 756c 653a 3a2e 2020 e280 hand rule::. ..\n+000093d0: 9850 e280 9920 616e 6420 e280 9858 e280 .P... and ...X..\n+000093e0: 9920 6172 6520 626f 7468 0a20 2020 2020 . are both. \n 000093f0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00009400: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00009410: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00009420: 206f 7074 696f 6e61 6c20 616e 6420 6465 optional and de\n-00009430: 6661 756c 7420 746f 2074 6865 206f 7269 fault to the ori\n-00009440: 6769 6e20 616e 6420 7468 650a 2020 2020 gin and the. \n+00009420: 2020 2020 206f 7074 696f 6e61 6c20 616e optional an\n+00009430: 6420 6465 6661 756c 7420 746f 2074 6865 d default to 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7728 452c 4c2c 5529 e280 9920 2020 iew(E,L,U)... \n-00009ca0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00009cb0: 2020 2020 706f 696e 742c 706f 696e 742c point,point,\n-00009cc0: 7665 6374 6f72 2020 2020 416e 2061 6c74 vector An alt\n-00009cd0: 6572 6e61 7465 2066 6f72 6d20 6f66 20e2 ernate form of .\n-00009ce0: 8098 7669 6577 2845 2c44 2c55 29e2 8099 ..view(E,D,U)...\n-00009cf0: 2061 626f 7665 2077 6865 7265 0a20 2020 above where. \n-00009d00: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00009c70: 2020 2020 2020 2020 2020 2020 2074 6f77 tow\n+00009c80: 6172 6420 7468 6520 6f72 6967 696e 2e0a ard the origin..\n+00009c90: e280 9876 6965 7728 452c 4c2c 5529 e280 ...view(E,L,U)..\n+00009ca0: 9920 2020 2020 2020 2020 2020 2020 2020 . \n+00009cb0: 2020 2020 2020 2020 706f 696e 742c 706f point,po\n+00009cc0: 696e 742c 7665 6374 6f72 2020 2020 416e int,vector An\n+00009cd0: 2061 6c74 6572 6e61 7465 2066 6f72 6d20 alternate form \n+00009ce0: 6f66 20e2 8098 7669 6577 2845 2c44 2c55 of ...view(E,D,U\n+00009cf0: 29e2 8099 2061 626f 7665 2077 6865 7265 )... above where\n+00009d00: 0a20 2020 2020 2020 2020 2020 2020 2020 . \n 00009d10: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00009d20: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00009d30: 2020 2020 2020 2074 6865 2076 6965 7720 the view \n-00009d40: 6469 7265 6374 696f 6e20 7061 7261 6d65 direction parame\n-00009d50: 7465 7220 e280 9844 e280 9920 6973 2072 ter ...D... is r\n-00009d60: 6570 6c61 6365 640a 2020 2020 2020 2020 eplaced. \n+00009d30: 2020 2020 2020 2020 2020 2074 6865 2076 the v\n+00009d40: 6965 7720 6469 7265 6374 696f 6e20 7061 iew direction pa\n+00009d50: 7261 6d65 7465 7220 e280 9844 e280 9920 rameter ...D... \n+00009d60: 6973 2072 6570 6c61 6365 640a 2020 2020 is replaced. \n 00009d70: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00009d80: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00009d90: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00009da0: 2020 7769 7468 2061 2022 6c6f 6f6b 2061 with a \"look a\n-00009db0: 7422 2070 6f69 6e74 20e2 8098 4ce2 8099 t\" point ...L...\n-00009dc0: 2c20 692e 652e 2c20 6120 706f 696e 740a , i.e., a point.\n-00009dd0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00009da0: 2020 2020 2020 7769 7468 2061 2022 6c6f with a \"lo\n+00009db0: 6f6b 2061 7422 2070 6f69 6e74 20e2 8098 ok at\" point ...\n+00009dc0: 4ce2 8099 2c20 692e 652e 2c20 6120 706f L..., i.e., a po\n+00009dd0: 696e 740a 2020 2020 2020 2020 2020 2020 int. \n 00009de0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00009df0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00009e00: 2020 2020 2020 2020 2020 7768 6572 6520 where \n-00009e10: 7468 6520 7669 6577 6572 2069 7320 666f the viewer is fo\n-00009e20: 6375 7369 6e67 2068 6572 2061 7474 656e cusing her atten\n-00009e30: 7469 6f6e 2e0a 2020 2020 2020 2020 2020 tion.. \n+00009e00: 2020 2020 2020 2020 2020 2020 2020 7768 wh\n+00009e10: 6572 6520 7468 6520 7669 6577 6572 2069 ere the viewer i\n+00009e20: 7320 666f 6375 7369 6e67 2068 6572 2061 s focusing her a\n+00009e30: 7474 656e 7469 6f6e 2e0a 2020 2020 2020 ttention.. \n 00009e40: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00009e50: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00009e60: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00009e70: 5468 6973 2066 6f72 6d20 6f66 2076 6965 This form of vie\n-00009e80: 7720 6973 2065 7175 6976 616c 656e 7420 w is equivalent \n-00009e90: 746f 20e2 8098 7669 6577 2845 2c0a 2020 to ...view(E,. \n-00009ea0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00009e70: 2020 2020 5468 6973 2066 6f72 6d20 6f66 This form of\n+00009e80: 2076 6965 7720 6973 2065 7175 6976 616c view is equival\n+00009e90: 656e 7420 746f 20e2 8098 7669 6577 2845 ent to ...view(E\n+00009ea0: 2c0a 2020 2020 2020 2020 2020 2020 2020 ,. \n 00009eb0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00009ec0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00009ed0: 2020 2020 2020 2020 284c 292d 2845 292c (L)-(E),\n-00009ee0: 2055 29e2 8099 2c20 7768 6572 6520 e280 U)..., where ..\n-00009ef0: 9828 4c29 2d28 4529 e280 9920 6973 2061 .(L)-(E)... is a\n-00009f00: 2064 6972 6563 7469 6f6e 0a20 2020 2020 direction. \n+00009ed0: 2020 2020 2020 2020 2020 2020 284c 292d (L)-\n+00009ee0: 2845 292c 2055 29e2 8099 2c20 7768 6572 (E), U)..., wher\n+00009ef0: 6520 e280 9828 4c29 2d28 4529 e280 9920 e ...(L)-(E)... \n+00009f00: 6973 2061 2064 6972 6563 7469 6f6e 0a20 is a direction. \n 00009f10: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00009f20: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00009f30: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00009f40: 2020 2020 2076 6563 746f 722e 2020 e280 vector. ..\n-00009f50: 9855 e280 9920 6973 206f 7074 696f 6e61 .U... is optiona\n-00009f60: 6c20 616e 6420 6465 6661 756c 7473 2074 l and defaults t\n-00009f70: 6f0a 2020 2020 2020 2020 2020 2020 2020 o. \n+00009f40: 2020 2020 2020 2020 2076 6563 746f 722e vector.\n+00009f50: 2020 e280 9855 e280 9920 6973 206f 7074 ...U... is opt\n+00009f60: 696f 6e61 6c20 616e 6420 6465 6661 756c ional and defaul\n+00009f70: 7473 2074 6f0a 2020 2020 2020 2020 2020 ts to. \n 00009f80: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00009f90: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00009fa0: 2020 2020 2020 2020 2020 2020 5b30 2c31 [0,1\n-00009fb0: 2c30 5d2e 0ae2 8098 5b5b 615f 7b31 317d ,0].....[[a_{11}\n-00009fc0: 2c61 5f7b 3132 7d2c 615f 7b31 337d 2c61 ,a_{12},a_{13},a\n-00009fd0: 5f7b 3134 7d5d e280 9920 2020 2031 3620 _{14}]... 16 \n-00009fe0: 7363 616c 6172 7320 2020 2020 2020 2020 scalars \n-00009ff0: 2020 2044 6972 6563 7420 7472 616e 7366 Direct transf\n-0000a000: 6f72 6d20 6d61 7472 6978 2064 6566 696e orm matrix defin\n-0000a010: 6974 696f 6e2e 2020 4561 6368 206f 660a ition. Each of.\n-0000a020: e280 9820 5b61 5f7b 3231 7d2c 615f 7b32 ... [a_{21},a_{2\n-0000a030: 327d 2c61 5f7b 3233 7d2c 615f 7b32 347d 2},a_{23},a_{24}\n-0000a040: 5de2 8099 2020 2020 2020 2020 2020 2020 ]... \n-0000a050: 2020 2020 2020 2020 2020 2020 2020 7468 th\n-0000a060: 6520 615f 7b69 6a7d 2069 7320 6120 7363 e a_{ij} is a sc\n-0000a070: 616c 6172 2065 7870 7265 7373 696f 6e2e alar expression.\n-0000a080: 2020 5365 6520 7465 7874 0ae2 8098 205b See text.... [\n-0000a090: 615f 7b33 317d 2c61 5f7b 3332 7d2c 615f a_{31},a_{32},a_\n-0000a0a0: 7b33 337d 2c61 5f7b 3334 7d5d e280 9920 {33},a_{34}]... \n-0000a0b0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0000a0c0: 2020 2020 2020 2020 2066 6f72 2064 6574 for det\n-0000a0d0: 6169 6c73 2e0a e280 9820 5b61 5f7b 3431 ails..... [a_{41\n-0000a0e0: 7d2c 615f 7b34 327d 2c61 5f7b 3433 7d2c },a_{42},a_{43},\n-0000a0f0: 615f 7b34 347d 5d5d e280 990a 5468 6520 a_{44}]]....The \n-0000a100: e280 9870 726f 6a65 6374 e280 9920 636f ...project... co\n-0000a110: 6e73 7472 7563 746f 7220 6973 206e 6f74 nstructor is not\n-0000a120: 2067 656e 6572 616c 6c79 2075 7365 6675 generally usefu\n-0000a130: 6c20 6265 6361 7573 6520 6974 2064 6566 l because it def\n-0000a140: 6561 7473 0a68 6964 6465 6e20 7375 7266 eats.hidden surf\n-0000a150: 6163 6520 7265 6d6f 7661 6c20 6279 2063 ace removal by c\n-0000a160: 6f6c 6c61 7073 696e 6720 7468 6520 7363 ollapsing the sc\n-0000a170: 656e 6520 6f6e 746f 2061 2073 696e 676c ene onto a singl\n-0000a180: 6520 706c 616e 652e 2020 4974 0a69 7320 e plane. It.is \n-0000a190: 6120 7370 6563 6961 6c20 7075 7270 6f73 a special purpos\n-0000a1a0: 6520 7472 616e 7366 6f72 6d20 666f 7220 e transform for \n-0000a1b0: 6472 6177 696e 6720 7069 6374 7572 6573 drawing pictures\n-0000a1c0: 206f 6620 7363 656e 6573 2077 6865 7265 of scenes where\n-0000a1d0: 0a74 6872 6565 2d64 696d 656e 7369 6f6e .three-dimension\n-0000a1e0: 616c 206f 626a 6563 7473 2061 7265 2062 al objects are b\n-0000a1f0: 6569 6e67 2070 726f 6a65 6374 6564 206f eing projected o\n-0000a200: 6e74 6f20 706c 616e 6573 2e20 2053 6565 nto planes. See\n-0000a210: 2c20 666f 720a 6578 616d 706c 652c 202a , for.example, *\n-0000a220: 6e6f 7465 204f 7665 7276 6965 773a 3a2e note Overview::.\n-0000a230: 0a0a 2020 2054 6865 2064 6972 6563 7420 .. The direct \n-0000a240: 7472 616e 7366 6f72 6d20 636f 6e73 7472 transform constr\n-0000a250: 7563 746f 7220 616c 6c6f 7773 2064 6972 uctor allows dir\n-0000a260: 6563 7420 656e 7472 7920 6f66 2061 2034 ect entry of a 4\n-0000a270: 7834 0a74 7261 6e73 666f 726d 6174 696f x4.transformatio\n-0000a280: 6e20 6d61 7472 6978 2e20 204d 6f73 7420 n matrix. Most \n-0000a290: 3364 2067 7261 7068 6963 7320 7379 7374 3d graphics syst\n-0000a2a0: 656d 732c 2069 6e63 6c75 6469 6e67 20e2 ems, including .\n-0000a2b0: 8098 736b 6574 6368 e280 992c 0a75 7365 ..sketch...,.use\n-0000a2c0: 2034 2d64 696d 656e 7369 6f6e 616c 2068 4-dimensional h\n-0000a2d0: 6f6d 6f67 656e 656f 7573 2063 6f6f 7264 omogeneous coord\n-0000a2e0: 696e 6174 6573 2069 6e74 6572 6e61 6c6c inates internall\n-0000a2f0: 792e 2020 5468 6573 6520 6172 650a 7472 y. These are.tr\n-0000a300: 616e 7366 6f72 6d65 6420 6279 206d 756c ansformed by mul\n-0000a310: 7469 706c 6963 6174 696f 6e20 7769 7468 tiplication with\n-0000a320: 2034 7834 206d 6174 7269 6365 732e 2020 4x4 matrices. \n-0000a330: 5468 6520 6275 696c 742d 696e 0a63 6f6e The built-in.con\n-0000a340: 7374 7275 6374 6f72 7320 28e2 8098 726f structors (...ro\n-0000a350: 7461 7465 e280 992c 20e2 8098 7363 616c tate..., ...scal\n-0000a360: 65e2 8099 2c20 6574 632e 2920 2061 7265 e..., etc.) are\n-0000a370: 2061 6c6c 2074 7261 6e73 6c61 7465 6420 all translated \n-0000a380: 696e 7465 726e 616c 6c79 2074 6f0a 7375 internally to.su\n-0000a390: 6368 2034 7834 206d 6174 7269 6365 732e ch 4x4 matrices.\n-0000a3a0: 2020 466f 7220 6d6f 7265 2069 6e66 6f72 For more infor\n-0000a3b0: 6d61 7469 6f6e 206f 6e20 686f 6d6f 6765 mation on homoge\n-0000a3c0: 6e65 6f75 7320 636f 6f72 6469 6e61 7465 neous coordinate\n-0000a3d0: 0a74 7261 6e73 666f 726d 6174 696f 6e73 .transformations\n-0000a3e0: 2c20 7365 6520 616e 7920 676f 6f64 2063 , see any good c\n-0000a3f0: 6f6d 7075 7465 7220 6772 6170 6869 6373 omputer graphics\n-0000a400: 2074 6578 7462 6f6f 6b2e 2020 5468 6520 textbook. The \n-0000a410: 6469 7265 6374 0a74 7261 6e73 666f 726d direct.transform\n-0000a420: 2066 6561 7475 7265 206f 6620 e280 9873 feature of ...s\n-0000a430: 6b65 7463 68e2 8099 2061 6c6c 6f77 7320 ketch... allows \n-0000a440: 796f 7520 746f 2065 6e74 6572 2061 206d you to enter a m\n-0000a450: 6174 7269 7820 6f66 2079 6f75 7220 6f77 atrix of your ow\n-0000a460: 6e0a 6368 6f69 6365 2e20 2054 6865 7265 n.choice. There\n-0000a470: 2061 7265 206e 6f74 206d 616e 7920 6361 are not many ca\n-0000a480: 7365 7320 7768 6572 6520 7468 6973 2069 ses where this i\n-0000a490: 7320 7573 6566 756c 2c20 6275 7420 6966 s useful, but if\n-0000a4a0: 2079 6f75 2064 6f0a 6e65 6564 2069 742c you do.need it,\n-0000a4b0: 2079 6f75 206e 6565 6420 6974 2062 6164 you need it bad\n-0000a4c0: 6c79 210a 0a1f 0a46 696c 653a 2073 6b65 ly!....File: ske\n-0000a4d0: 7463 682e 696e 666f 2c20 204e 6f64 653a tch.info, Node:\n-0000a4e0: 2041 7269 7468 6d65 7469 632c 2020 4e65 Arithmetic, Ne\n-0000a4f0: 7874 3a20 4f70 7469 6f6e 732c 2020 5072 xt: Options, Pr\n-0000a500: 6576 3a20 4c69 7465 7261 6c73 2c20 2055 ev: Literals, U\n-0000a510: 703a 204c 616e 6775 6167 6520 6261 7369 p: Language basi\n-0000a520: 6373 0a0a 332e 312e 3420 4172 6974 686d cs..3.1.4 Arithm\n-0000a530: 6574 6963 2065 7870 7265 7373 696f 6e73 etic expressions\n-0000a540: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .---------------\n-0000a550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a41 -------------..A\n-0000a560: 7269 7468 6d65 7469 6320 6578 7072 6573 rithmetic expres\n-0000a570: 7369 6f6e 7320 6f76 6572 20e2 8098 736b sions over ...sk\n-0000a580: 6574 6368 e280 9920 6c69 7465 7261 6c73 etch... literals\n-0000a590: 2061 6e64 2064 6566 696e 6564 2069 6465 and defined ide\n-0000a5a0: 6e74 6966 6965 7273 0a61 7265 2073 756d ntifiers.are sum\n-0000a5b0: 6d61 7269 7a65 6420 696e 2074 6865 2066 marized in the f\n-0000a5c0: 6f6c 6c6f 7769 6e67 2074 6162 6c65 732e ollowing tables.\n-0000a5d0: 0a0a 2a20 4d65 6e75 3a0a 0a2a 2054 776f ..* Menu:..* Two\n-0000a5e0: 2d6f 7065 7261 6e64 2028 6269 6e61 7279 -operand (binary\n-0000a5f0: 2920 666f 726d 733a 3a20 2041 206f 7020 ) forms:: A op \n-0000a600: 420a 2a20 556e 6172 7920 666f 726d 733a B.* Unary forms:\n-0000a610: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n-0000a620: 2020 6f70 2041 2028 616e 6420 6f74 6865 op A (and othe\n-0000a630: 7273 290a 0a1f 0a46 696c 653a 2073 6b65 rs)....File: ske\n-0000a640: 7463 682e 696e 666f 2c20 204e 6f64 653a tch.info, Node:\n-0000a650: 2054 776f 2d6f 7065 7261 6e64 2028 6269 Two-operand (bi\n-0000a660: 6e61 7279 2920 666f 726d 732c 2020 4e65 nary) forms, Ne\n-0000a670: 7874 3a20 556e 6172 7920 666f 726d 732c xt: Unary forms,\n-0000a680: 2020 5072 6576 3a20 4172 6974 686d 6574 Prev: Arithmet\n-0000a690: 6963 2c20 2055 703a 2041 7269 7468 6d65 ic, Up: Arithme\n-0000a6a0: 7469 630a 0a33 2e31 2e34 2e31 2054 776f tic..3.1.4.1 Two\n-0000a6b0: 2d6f 7065 7261 6e64 2028 6269 6e61 7279 -operand (binary\n-0000a6c0: 2920 666f 726d 7320 616e 6420 7072 6563 ) forms and prec\n-0000a6d0: 6564 656e 6365 0a2e 2e2e 2e2e 2e2e 2e2e edence..........\n+00009fa0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00009fb0: 5b30 2c31 2c30 5d2e 0ae2 8098 5b5b 615f [0,1,0].....[[a_\n+00009fc0: 7b31 317d 2c61 5f7b 3132 7d2c 615f 7b31 {11},a_{12},a_{1\n+00009fd0: 337d 2c61 5f7b 3134 7d5d e280 9920 2020 3},a_{14}]... \n+00009fe0: 2031 3620 7363 616c 6172 7320 2020 2020 16 scalars \n+00009ff0: 2020 2020 2020 2044 6972 6563 7420 7472 Direct tr\n+0000a000: 616e 7366 6f72 6d20 6d61 7472 6978 2064 ansform matrix d\n+0000a010: 6566 696e 6974 696f 6e2e 2020 4561 6368 efinition. Each\n+0000a020: 206f 660a e280 9820 5b61 5f7b 3231 7d2c of.... [a_{21},\n+0000a030: 615f 7b32 327d 2c61 5f7b 3233 7d2c 615f a_{22},a_{23},a_\n+0000a040: 7b32 347d 5de2 8099 2020 2020 2020 2020 {24}]... \n+0000a050: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0000a060: 2020 7468 6520 615f 7b69 6a7d 2069 7320 the a_{ij} is \n+0000a070: 6120 7363 616c 6172 2065 7870 7265 7373 a scalar express\n+0000a080: 696f 6e2e 2020 5365 6520 7465 7874 0ae2 ion. See text..\n+0000a090: 8098 205b 615f 7b33 317d 2c61 5f7b 3332 .. [a_{31},a_{32\n+0000a0a0: 7d2c 615f 7b33 337d 2c61 5f7b 3334 7d5d },a_{33},a_{34}]\n+0000a0b0: e280 9920 2020 2020 2020 2020 2020 2020 ... \n+0000a0c0: 2020 2020 2020 2020 2020 2020 2066 6f72 for\n+0000a0d0: 2064 6574 6169 6c73 2e0a e280 9820 5b61 details..... [a\n+0000a0e0: 5f7b 3431 7d2c 615f 7b34 327d 2c61 5f7b _{41},a_{42},a_{\n+0000a0f0: 3433 7d2c 615f 7b34 347d 5d5d e280 990a 43},a_{44}]]....\n+0000a100: 5468 6520 e280 9870 726f 6a65 6374 e280 The ...project..\n+0000a110: 9920 636f 6e73 7472 7563 746f 7220 6973 . constructor is\n+0000a120: 206e 6f74 2067 656e 6572 616c 6c79 2075 not generally u\n+0000a130: 7365 6675 6c20 6265 6361 7573 6520 6974 seful because it\n+0000a140: 2064 6566 6561 7473 0a68 6964 6465 6e20 defeats.hidden \n+0000a150: 7375 7266 6163 6520 7265 6d6f 7661 6c20 surface removal \n+0000a160: 6279 2063 6f6c 6c61 7073 696e 6720 7468 by collapsing th\n+0000a170: 6520 7363 656e 6520 6f6e 746f 2061 2073 e scene onto a s\n+0000a180: 696e 676c 6520 706c 616e 652e 2020 4974 ingle plane. It\n+0000a190: 0a69 7320 6120 7370 6563 6961 6c20 7075 .is a special pu\n+0000a1a0: 7270 6f73 6520 7472 616e 7366 6f72 6d20 rpose transform \n+0000a1b0: 666f 7220 6472 6177 696e 6720 7069 6374 for drawing pict\n+0000a1c0: 7572 6573 206f 6620 7363 656e 6573 2077 ures of scenes w\n+0000a1d0: 6865 7265 0a74 6872 6565 2d64 696d 656e here.three-dimen\n+0000a1e0: 7369 6f6e 616c 206f 626a 6563 7473 2061 sional objects a\n+0000a1f0: 7265 2062 6569 6e67 2070 726f 6a65 6374 re being project\n+0000a200: 6564 206f 6e74 6f20 706c 616e 6573 2e20 ed onto planes. \n+0000a210: 2053 6565 2c20 666f 720a 6578 616d 706c See, for.exampl\n+0000a220: 652c 202a 6e6f 7465 204f 7665 7276 6965 e, *note Overvie\n+0000a230: 773a 3a2e 0a0a 2020 2054 6865 2064 6972 w::... 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T\n+0000a470: 6865 7265 2061 7265 206e 6f74 206d 616e here are not man\n+0000a480: 7920 6361 7365 7320 7768 6572 6520 7468 y cases where th\n+0000a490: 6973 2069 7320 7573 6566 756c 2c20 6275 is is useful, bu\n+0000a4a0: 7420 6966 2079 6f75 2064 6f0a 6e65 6564 t if you do.need\n+0000a4b0: 2069 742c 2079 6f75 206e 6565 6420 6974 it, you need it\n+0000a4c0: 2062 6164 6c79 210a 0a1f 0a46 696c 653a badly!....File:\n+0000a4d0: 2073 6b65 7463 682e 696e 666f 2c20 204e sketch.info, N\n+0000a4e0: 6f64 653a 2041 7269 7468 6d65 7469 632c ode: Arithmetic,\n+0000a4f0: 2020 4e65 7874 3a20 4f70 7469 6f6e 732c Next: Options,\n+0000a500: 2020 5072 6576 3a20 4c69 7465 7261 6c73 Prev: Literals\n+0000a510: 2c20 2055 703a 204c 616e 6775 6167 6520 , Up: Language \n+0000a520: 6261 7369 6373 0a0a 332e 312e 3420 4172 basics..3.1.4 Ar\n+0000a530: 6974 686d 6574 6963 2065 7870 7265 7373 ithmetic express\n+0000a540: 696f 6e73 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d ions.-----------\n+0000a550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ----------------\n+0000a560: 2d0a 0a41 7269 7468 6d65 7469 6320 6578 -..Arithmetic ex\n+0000a570: 7072 6573 7369 6f6e 7320 6f76 6572 20e2 pressions over .\n+0000a580: 8098 736b 6574 6368 e280 9920 6c69 7465 ..sketch... lite\n+0000a590: 7261 6c73 2061 6e64 2064 6566 696e 6564 rals and defined\n+0000a5a0: 2069 6465 6e74 6966 6965 7273 0a61 7265 identifiers.are\n+0000a5b0: 2073 756d 6d61 7269 7a65 6420 696e 2074 summarized in t\n+0000a5c0: 6865 2066 6f6c 6c6f 7769 6e67 2074 6162 he following tab\n+0000a5d0: 6c65 732e 0a0a 2a20 4d65 6e75 3a0a 0a2a les...* Menu:..*\n+0000a5e0: 2054 776f 2d6f 7065 7261 6e64 2028 6269 Two-operand (bi\n+0000a5f0: 6e61 7279 2920 666f 726d 733a 3a20 2041 nary) forms:: A\n+0000a600: 206f 7020 420a 2a20 556e 6172 7920 666f op B.* Unary fo\n+0000a610: 726d 733a 3a20 2020 2020 2020 2020 2020 rms:: \n+0000a620: 2020 2020 2020 6f70 2041 2028 616e 6420 op A (and \n+0000a630: 6f74 6865 7273 290a 0a1f 0a46 696c 653a others)....File:\n+0000a640: 2073 6b65 7463 682e 696e 666f 2c20 204e sketch.info, N\n+0000a650: 6f64 653a 2054 776f 2d6f 7065 7261 6e64 ode: Two-operand\n+0000a660: 2028 6269 6e61 7279 2920 666f 726d 732c (binary) forms,\n+0000a670: 2020 4e65 7874 3a20 556e 6172 7920 666f Next: Unary fo\n+0000a680: 726d 732c 2020 5072 6576 3a20 4172 6974 rms, Prev: Arit\n+0000a690: 686d 6574 6963 2c20 2055 703a 2041 7269 hmetic, Up: Ari\n+0000a6a0: 7468 6d65 7469 630a 0a33 2e31 2e34 2e31 thmetic..3.1.4.1\n+0000a6b0: 2054 776f 2d6f 7065 7261 6e64 2028 6269 Two-operand (bi\n+0000a6c0: 6e61 7279 2920 666f 726d 7320 616e 6420 nary) forms and \n+0000a6d0: 7072 6563 6564 656e 6365 0a2e 2e2e 2e2e precedence......\n 0000a6e0: 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e ................\n 0000a6f0: 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e ................\n-0000a700: 2e2e 2e2e 2e2e 2e2e 0a0a 4d6f 7374 2074 ..........Most t\n-0000a710: 776f 2d6f 7065 7261 6e64 2062 696e 6172 wo-operand binar\n-0000a720: 7920 666f 726d 7320 6861 7665 206d 6561 y forms have mea\n-0000a730: 6e69 6e67 7320 6465 7065 6e64 656e 7420 nings dependent \n-0000a740: 6f6e 2074 6865 2074 7970 6573 206f 660a on the types of.\n-0000a750: 7468 6569 7220 6172 6775 6d65 6e74 732e their arguments.\n-0000a760: 2020 416e 2065 7868 6175 7374 6976 6520 An exhaustive \n-0000a770: 7375 6d6d 6172 7920 6f66 2074 6865 2070 summary of the p\n-0000a780: 6f73 7369 6269 6c69 7469 6573 2069 7320 ossibilities is \n-0000a790: 6769 7665 6e20 696e 0a74 6865 2066 6f6c given in.the fol\n-0000a7a0: 6c6f 7769 6e67 2074 6162 6c65 2e0a 4c65 lowing table..Le\n-0000a7b0: 6674 2020 2020 2020 2020 4f70 2020 2020 ft Op \n-0000a7c0: 2020 2052 6967 6874 2020 2020 2020 2052 Right R\n-0000a7d0: 6573 756c 7420 2020 2020 2044 6573 6372 esult Descr\n-0000a7e0: 6970 7469 6f6e 0a2d 2d2d 2d2d 2d2d 2d2d iption.---------\n+0000a700: 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 0a0a 4d6f ..............Mo\n+0000a710: 7374 2074 776f 2d6f 7065 7261 6e64 2062 st two-operand b\n+0000a720: 696e 6172 7920 666f 726d 7320 6861 7665 inary forms have\n+0000a730: 206d 6561 6e69 6e67 7320 6465 7065 6e64 meanings depend\n+0000a740: 656e 7420 6f6e 2074 6865 2074 7970 6573 ent on the types\n+0000a750: 206f 660a 7468 6569 7220 6172 6775 6d65 of.their argume\n+0000a760: 6e74 732e 2020 416e 2065 7868 6175 7374 nts. 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7363 -------------.sc\n-0000a860: 616c 6172 2020 2020 2020 e280 982b e280 alar ...+..\n-0000a870: 9920 2020 2020 2073 6361 6c61 7220 2020 . scalar \n-0000a880: 2020 2073 6361 6c61 7220 2020 2020 2053 scalar S\n-0000a890: 6361 6c61 7220 7375 6d2e 0a76 6563 746f calar sum..vecto\n-0000a8a0: 7220 2020 2020 20e2 8098 2be2 8099 2020 r ...+... \n-0000a8b0: 2020 2020 7665 6374 6f72 2020 2020 2020 vector \n-0000a8c0: 7665 6374 6f72 2020 2020 2020 5665 6374 vector Vect\n-0000a8d0: 6f72 2073 756d 2e0a 706f 696e 7420 2020 or sum..point \n-0000a8e0: 2020 2020 e280 982b e280 9920 2020 2020 ...+... \n-0000a8f0: 2076 6563 746f 7220 2020 2020 2070 6f69 vector poi\n-0000a900: 6e74 2020 2020 2020 2050 6f69 6e74 2d76 nt Point-v\n-0000a910: 6563 746f 7220 6166 6669 6e65 2073 756d ector affine sum\n-0000a920: 2e0a 7665 6374 6f72 2020 2020 2020 e280 ..vector ..\n-0000a930: 982b e280 9920 2020 2020 2070 6f69 6e74 .+... point\n-0000a940: 2020 2020 2020 2022 2020 2020 2020 2020 \" \n-0000a950: 2020 2022 0a73 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\n-0000aa50: 2020 706f 696e 7420 2020 2020 2020 506f point Po\n-0000aa60: 696e 742d 7665 6374 6f72 2061 6666 696e int-vector affin\n-0000aa70: 6520 6469 6666 6572 656e 6365 2e0a 7363 e difference..sc\n-0000aa80: 616c 6172 2020 2020 2020 e280 982a e280 alar ...*..\n-0000aa90: 9920 6f72 2020 2073 6361 6c61 7220 2020 . or scalar \n-0000aaa0: 2020 2073 6361 6c61 7220 2020 2020 2053 scalar S\n-0000aab0: 6361 6c61 7220 7072 6f64 7563 742e 0a20 calar product.. \n-0000aac0: 2020 2020 2020 2020 2020 20e2 8098 2ee2 .....\n-0000aad0: 8099 0a73 6361 6c61 7220 2020 2020 20e2 ...scalar .\n-0000aae0: 8098 2ae2 8099 206f 7220 2020 7665 6374 ..*... or vect\n-0000aaf0: 6f72 2020 2020 2020 7665 6374 6f72 2020 or vector \n-0000ab00: 2020 2020 5363 616c 6172 2d76 6563 746f Scalar-vecto\n-0000ab10: 7220 7072 6f64 7563 742e 0a20 2020 2020 r product.. \n-0000ab20: 2020 2020 2020 20e2 8098 2ee2 8099 0a76 ........v\n-0000ab30: 6563 746f 7220 2020 2020 20e2 8098 2ae2 ector ...*.\n-0000ab40: 8099 206f 7220 2020 7363 616c 6172 2020 .. or scalar \n-0000ab50: 2020 2020 2220 2020 2020 2020 2020 2020 \" \n-0000ab60: 220a 2020 2020 2020 2020 2020 2020 e280 \". ..\n-0000ab70: 982e e280 990a 7665 6374 6f72 2020 2020 ......vector \n-0000ab80: 2020 e280 982a e280 9920 2020 2020 2076 ...*... v\n-0000ab90: 6563 746f 7220 2020 2020 2076 6563 746f ector vecto\n-0000aba0: 7220 2020 2020 2056 6563 746f 7220 6372 r Vector cr\n-0000abb0: 6f73 732d 7072 6f64 7563 742e 0a76 6563 oss-product..vec\n-0000abc0: 746f 7220 2020 2020 20e2 8098 2ee2 8099 tor .......\n-0000abd0: 2020 2020 2020 7665 6374 6f72 2020 2020 vector \n-0000abe0: 2020 7363 616c 6172 2020 2020 2020 5665 scalar Ve\n-0000abf0: 6374 6f72 2064 6f74 2070 726f 6475 6374 ctor dot product\n-0000ac00: 2e0a 7363 616c 6172 2020 2020 2020 e280 ..scalar ..\n-0000ac10: 985e e280 9920 2020 2020 2073 6361 6c61 .^... scala\n-0000ac20: 7220 2020 2020 2073 6361 6c61 7220 2020 r scalar \n-0000ac30: 2020 2052 6169 7365 2073 6361 6c61 7220 Raise scalar \n-0000ac40: 746f 2073 6361 6c61 7220 706f 7765 722e to scalar power.\n-0000ac50: 0a74 7261 6e73 666f 726d 2020 20e2 8098 .transform ...\n-0000ac60: 5ee2 8099 2020 2020 2020 696e 7465 6765 ^... intege\n-0000ac70: 7220 2020 2020 7472 616e 7366 6f72 6d20 r transform \n-0000ac80: 2020 5261 6973 6520 7472 616e 7366 6f72 Raise transfor\n-0000ac90: 6d20 746f 2069 6e74 6567 6572 2070 6f77 m to integer pow\n-0000aca0: 6572 2e0a 7472 616e 7366 6f72 6d20 2020 er..transform \n-0000acb0: e280 982a e280 9920 6f72 2020 2070 6f69 ...*... or poi\n-0000acc0: 6e74 2020 2020 2020 2070 6f69 6e74 2020 nt point \n-0000acd0: 2020 2020 2041 6666 696e 6520 706f 696e Affine poin\n-0000ace0: 7420 7472 616e 7366 6f72 6d20 2872 6967 t transform (rig\n-0000acf0: 6874 2d74 6f2d 6c65 6674 292e 0a20 2020 ht-to-left).. \n-0000ad00: 2020 2020 2020 2020 20e2 8098 2ee2 8099 .......\n-0000ad10: 0a74 7261 6e73 666f 726d 2020 20e2 8098 .transform ...\n-0000ad20: 2ae2 8099 206f 7220 2020 7665 6374 6f72 *... or vector\n-0000ad30: 2020 2020 2020 7665 6374 6f72 2020 2020 vector \n-0000ad40: 2020 4166 6669 6e65 2076 6563 746f 7220 Affine vector \n-0000ad50: 7472 616e 7366 6f72 6d20 2872 6967 6874 transform (right\n-0000ad60: 2d74 6f2d 6c65 6674 292e 0a20 2020 2020 -to-left).. \n-0000ad70: 2020 2020 2020 20e2 8098 2ee2 8099 0a74 ........t\n-0000ad80: 7261 6e73 666f 726d 2020 20e2 8098 2ae2 ransform ...*.\n-0000ad90: 8099 206f 7220 2020 7472 616e 7366 6f72 .. or transfor\n-0000ada0: 6d20 2020 7472 616e 7366 6f72 6d20 2020 m transform \n-0000adb0: 5472 616e 7366 6f72 6d20 636f 6d70 6f73 Transform compos\n-0000adc0: 6974 696f 6e20 2872 6967 6874 2d74 6f2d ition (right-to-\n-0000add0: 6c65 6674 292e 0a20 2020 2020 2020 2020 left).. \n-0000ade0: 2020 20e2 8098 2ee2 8099 0a70 6f69 6e74 ........point\n-0000adf0: 2020 2020 2020 20e2 8098 7468 656e e280 ...then..\n-0000ae00: 9920 2020 7472 616e 7366 6f72 6d20 2020 . transform \n-0000ae10: 706f 696e 7420 2020 2020 2020 4166 6669 point Affi\n-0000ae20: 6e65 2070 6f69 6e74 2074 7261 6e73 666f ne point transfo\n-0000ae30: 726d 2028 6c65 6674 2d74 6f2d 7269 6768 rm (left-to-righ\n-0000ae40: 7429 2e0a 7665 6374 6f72 2020 2020 2020 t)..vector \n-0000ae50: e280 9874 6865 6ee2 8099 2020 2074 7261 ...then... tra\n-0000ae60: 6e73 666f 726d 2020 2076 6563 746f 7220 nsform vector \n-0000ae70: 2020 2020 2041 6666 696e 6520 7665 6374 Affine vect\n-0000ae80: 6f72 2074 7261 6e73 666f 726d 2028 6c65 or transform (le\n-0000ae90: 6674 2d74 6f2d 7269 6768 7429 2e0a 7472 ft-to-right)..tr\n-0000aea0: 616e 7366 6f72 6d20 2020 e280 9874 6865 ansform ...the\n-0000aeb0: 6ee2 8099 2020 2074 7261 6e73 666f 726d n... transform\n-0000aec0: 2020 2074 7261 6e73 666f 726d 2020 2054 transform T\n-0000aed0: 7261 6e73 666f 726d 2063 6f6d 706f 7369 ransform composi\n-0000aee0: 7469 6f6e 2028 6c65 6674 2d74 6f2d 7269 tion (left-to-ri\n-0000aef0: 6768 7429 2e0a 7363 616c 6172 2020 2020 ght)..scalar \n-0000af00: 2020 e280 982f e280 9920 2020 2020 2073 .../... s\n-0000af10: 6361 6c61 7220 2020 2020 2073 6361 6c61 calar scala\n-0000af20: 7220 2020 2020 2053 6361 6c61 7220 6469 r Scalar di\n-0000af30: 7669 7369 6f6e 2e0a 7665 6374 6f72 2020 vision..vector \n-0000af40: 2020 2020 e280 982f e280 9920 2020 2020 .../... \n-0000af50: 2073 6361 6c61 7220 2020 2020 2076 6563 scalar vec\n-0000af60: 746f 7220 2020 2020 2056 6563 746f 7220 tor Vector \n-0000af70: 636f 6d70 6f6e 656e 742d 7769 7365 2064 component-wise d\n-0000af80: 6976 6973 696f 6e20 6279 2073 6361 6c61 ivision by scala\n-0000af90: 722e 0a70 6f69 6e74 2020 2020 2020 20e2 r..point .\n-0000afa0: 8098 27e2 8099 2020 2020 2020 e280 9878 ..'... ...x\n-0000afb0: e280 992c 20e2 8098 79e2 8099 2c20 2020 ..., ...y..., \n-0000afc0: 7363 616c 6172 2020 2020 2020 506f 696e scalar Poin\n-0000afd0: 7420 636f 6d70 6f6e 656e 7420 6578 7472 t component extr\n-0000afe0: 6163 7469 6f6e 2e0a 2020 2020 2020 2020 action.. \n-0000aff0: 2020 2020 2020 2020 2020 2020 206f 7220 or \n-0000b000: e280 987a e280 990a 7665 6374 6f72 2020 ...z....vector \n-0000b010: 2020 2020 e280 9827 e280 9920 2020 2020 ...'... \n-0000b020: 20e2 8098 78e2 8099 2c20 e280 9879 e280 ...x..., ...y..\n-0000b030: 992c 2020 2073 6361 6c61 7220 2020 2020 ., scalar \n-0000b040: 2056 6563 746f 7220 636f 6d70 6f6e 656e Vector componen\n-0000b050: 7420 6578 7472 6163 7469 6f6e 2e0a 2020 t extraction.. \n-0000b060: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0000b070: 2020 206f 7220 e280 987a e280 990a 0a4f or ...z.....O\n-0000b080: 7065 7261 746f 7220 7072 6563 6564 656e perator preceden\n-0000b090: 6365 2069 7320 7368 6f77 6e20 696e 2074 ce is shown in t\n-0000b0a0: 6869 7320 7461 626c 652e 0a4f 7020 2020 his table..Op \n-0000b0b0: 2020 2020 5072 6563 6564 656e 6365 0a2d Precedence.-\n-0000b0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ----------------\n+0000a850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ----------------\n+0000a860: 2d0a 7363 616c 6172 2020 2020 2020 e280 -.scalar ..\n+0000a870: 982b e280 9920 2020 2020 2073 6361 6c61 .+... scala\n+0000a880: 7220 2020 2020 2073 6361 6c61 7220 2020 r scalar \n+0000a890: 2020 2053 6361 6c61 7220 7375 6d2e 0a76 Scalar sum..v\n+0000a8a0: 6563 746f 7220 2020 2020 20e2 8098 2be2 ector ...+.\n+0000a8b0: 8099 2020 2020 2020 7665 6374 6f72 2020 .. vector \n+0000a8c0: 2020 2020 7665 6374 6f72 2020 2020 2020 vector \n+0000a8d0: 5665 6374 6f72 2073 756d 2e0a 706f 696e Vector sum..poin\n+0000a8e0: 7420 2020 2020 2020 e280 982b e280 9920 t ...+... \n+0000a8f0: 2020 2020 2076 6563 746f 7220 2020 2020 vector \n+0000a900: 2070 6f69 6e74 2020 2020 2020 2050 6f69 point Poi\n+0000a910: 6e74 2d76 6563 746f 7220 6166 6669 6e65 nt-vector affine\n+0000a920: 2073 756d 2e0a 7665 6374 6f72 2020 2020 sum..vector \n+0000a930: 2020 e280 982b e280 9920 2020 2020 2070 ...+... p\n+0000a940: 6f69 6e74 2020 2020 2020 2022 2020 2020 oint \" \n+0000a950: 2020 2020 2020 2022 0a73 6361 6c61 7220 \".scalar \n+0000a960: 2020 2020 20e2 8098 2de2 8099 2020 2020 ...-... \n+0000a970: 2020 7363 616c 6172 2020 2020 2020 7363 scalar sc\n+0000a980: 616c 6172 2020 2020 2020 5363 616c 6172 alar Scalar\n+0000a990: 2064 6966 6665 7265 6e63 652e 0a76 6563 difference..vec\n+0000a9a0: 746f 7220 2020 2020 20e2 8098 2de2 8099 tor ...-...\n+0000a9b0: 2020 2020 2020 7665 6374 6f72 2020 2020 vector \n+0000a9c0: 2020 7665 6374 6f72 2020 2020 2020 5665 vector Ve\n+0000a9d0: 6374 6f72 2064 6966 6665 7265 6e63 652e ctor difference.\n+0000a9e0: 0a70 6f69 6e74 2020 2020 2020 20e2 8098 .point ...\n+0000a9f0: 2de2 8099 2020 2020 2020 706f 696e 7420 -... point \n+0000aa00: 2020 2020 2020 7665 6374 6f72 2020 2020 vector \n+0000aa10: 2020 506f 696e 742d 706f 696e 7420 6166 Point-point af\n+0000aa20: 6669 6e65 2064 6966 6665 7265 6e63 652e fine difference.\n+0000aa30: 0a70 6f69 6e74 2020 2020 2020 20e2 8098 .point ...\n+0000aa40: 2de2 8099 2020 2020 2020 7665 6374 6f72 -... vector\n+0000aa50: 2020 2020 2020 706f 696e 7420 2020 2020 point \n+0000aa60: 2020 506f 696e 742d 7665 6374 6f72 2061 Point-vector a\n+0000aa70: 6666 696e 6520 6469 6666 6572 656e 6365 ffine difference\n+0000aa80: 2e0a 7363 616c 6172 2020 2020 2020 e280 ..scalar ..\n+0000aa90: 982a e280 9920 6f72 2020 2073 6361 6c61 .*... or scala\n+0000aaa0: 7220 2020 2020 2073 6361 6c61 7220 2020 r scalar \n+0000aab0: 2020 2053 6361 6c61 7220 7072 6f64 7563 Scalar produc\n+0000aac0: 742e 0a20 2020 2020 2020 2020 2020 20e2 t.. .\n+0000aad0: 8098 2ee2 8099 0a73 6361 6c61 7220 2020 .......scalar \n+0000aae0: 2020 20e2 8098 2ae2 8099 206f 7220 2020 ...*... or \n+0000aaf0: 7665 6374 6f72 2020 2020 2020 7665 6374 vector vect\n+0000ab00: 6f72 2020 2020 2020 5363 616c 6172 2d76 or Scalar-v\n+0000ab10: 6563 746f 7220 7072 6f64 7563 742e 0a20 ector product.. \n+0000ab20: 2020 2020 2020 2020 2020 20e2 8098 2ee2 .....\n+0000ab30: 8099 0a76 6563 746f 7220 2020 2020 20e2 ...vector .\n+0000ab40: 8098 2ae2 8099 206f 7220 2020 7363 616c ..*... or scal\n+0000ab50: 6172 2020 2020 2020 2220 2020 2020 2020 ar \" \n+0000ab60: 2020 2020 220a 2020 2020 2020 2020 2020 \". \n+0000ab70: 2020 e280 982e e280 990a 7665 6374 6f72 ........vector\n+0000ab80: 2020 2020 2020 e280 982a e280 9920 2020 ...*... \n+0000ab90: 2020 2076 6563 746f 7220 2020 2020 2076 vector v\n+0000aba0: 6563 746f 7220 2020 2020 2056 6563 746f ector Vecto\n+0000abb0: 7220 6372 6f73 732d 7072 6f64 7563 742e r cross-product.\n+0000abc0: 0a76 6563 746f 7220 2020 2020 20e2 8098 .vector ...\n+0000abd0: 2ee2 8099 2020 2020 2020 7665 6374 6f72 .... vector\n+0000abe0: 2020 2020 2020 7363 616c 6172 2020 2020 scalar \n+0000abf0: 2020 5665 6374 6f72 2064 6f74 2070 726f Vector dot pro\n+0000ac00: 6475 6374 2e0a 7363 616c 6172 2020 2020 duct..scalar \n+0000ac10: 2020 e280 985e e280 9920 2020 2020 2073 ...^... s\n+0000ac20: 6361 6c61 7220 2020 2020 2073 6361 6c61 calar scala\n+0000ac30: 7220 2020 2020 2052 6169 7365 2073 6361 r Raise sca\n+0000ac40: 6c61 7220 746f 2073 6361 6c61 7220 706f lar to scalar po\n+0000ac50: 7765 722e 0a74 7261 6e73 666f 726d 2020 wer..transform \n+0000ac60: 20e2 8098 5ee2 8099 2020 2020 2020 696e ...^... in\n+0000ac70: 7465 6765 7220 2020 2020 7472 616e 7366 teger transf\n+0000ac80: 6f72 6d20 2020 5261 6973 6520 7472 616e orm Raise tran\n+0000ac90: 7366 6f72 6d20 746f 2069 6e74 6567 6572 sform to integer\n+0000aca0: 2070 6f77 6572 2e0a 7472 616e 7366 6f72 power..transfor\n+0000acb0: 6d20 2020 e280 982a e280 9920 6f72 2020 m ...*... or \n+0000acc0: 2070 6f69 6e74 2020 2020 2020 2070 6f69 point poi\n+0000acd0: 6e74 2020 2020 2020 2041 6666 696e 6520 nt Affine \n+0000ace0: 706f 696e 7420 7472 616e 7366 6f72 6d20 point transform \n+0000acf0: 2872 6967 6874 2d74 6f2d 6c65 6674 292e (right-to-left).\n+0000ad00: 0a20 2020 2020 2020 2020 2020 20e2 8098 . ...\n+0000ad10: 2ee2 8099 0a74 7261 6e73 666f 726d 2020 .....transform \n+0000ad20: 20e2 8098 2ae2 8099 206f 7220 2020 7665 ...*... or ve\n+0000ad30: 6374 6f72 2020 2020 2020 7665 6374 6f72 ctor vector\n+0000ad40: 2020 2020 2020 4166 6669 6e65 2076 6563 Affine vec\n+0000ad50: 746f 7220 7472 616e 7366 6f72 6d20 2872 tor transform (r\n+0000ad60: 6967 6874 2d74 6f2d 6c65 6674 292e 0a20 ight-to-left).. \n+0000ad70: 2020 2020 2020 2020 2020 20e2 8098 2ee2 .....\n+0000ad80: 8099 0a74 7261 6e73 666f 726d 2020 20e2 ...transform .\n+0000ad90: 8098 2ae2 8099 206f 7220 2020 7472 616e ..*... or tran\n+0000ada0: 7366 6f72 6d20 2020 7472 616e 7366 6f72 sform transfor\n+0000adb0: 6d20 2020 5472 616e 7366 6f72 6d20 636f m Transform co\n+0000adc0: 6d70 6f73 6974 696f 6e20 2872 6967 6874 mposition (right\n+0000add0: 2d74 6f2d 6c65 6674 292e 0a20 2020 2020 -to-left).. \n+0000ade0: 2020 2020 2020 20e2 8098 2ee2 8099 0a70 ........p\n+0000adf0: 6f69 6e74 2020 2020 2020 20e2 8098 7468 oint ...th\n+0000ae00: 656e e280 9920 2020 7472 616e 7366 6f72 en... transfor\n+0000ae10: 6d20 2020 706f 696e 7420 2020 2020 2020 m point \n+0000ae20: 4166 6669 6e65 2070 6f69 6e74 2074 7261 Affine point tra\n+0000ae30: 6e73 666f 726d 2028 6c65 6674 2d74 6f2d nsform (left-to-\n+0000ae40: 7269 6768 7429 2e0a 7665 6374 6f72 2020 right)..vector \n+0000ae50: 2020 2020 e280 9874 6865 6ee2 8099 2020 ...then... \n+0000ae60: 2074 7261 6e73 666f 726d 2020 2076 6563 transform vec\n+0000ae70: 746f 7220 2020 2020 2041 6666 696e 6520 tor Affine \n+0000ae80: 7665 6374 6f72 2074 7261 6e73 666f 726d vector transform\n+0000ae90: 2028 6c65 6674 2d74 6f2d 7269 6768 7429 (left-to-right)\n+0000aea0: 2e0a 7472 616e 7366 6f72 6d20 2020 e280 ..transform ..\n+0000aeb0: 9874 6865 6ee2 8099 2020 2074 7261 6e73 .then... trans\n+0000aec0: 666f 726d 2020 2074 7261 6e73 666f 726d form transform\n+0000aed0: 2020 2054 7261 6e73 666f 726d 2063 6f6d Transform com\n+0000aee0: 706f 7369 7469 6f6e 2028 6c65 6674 2d74 position (left-t\n+0000aef0: 6f2d 7269 6768 7429 2e0a 7363 616c 6172 o-right)..scalar\n+0000af00: 2020 2020 2020 e280 982f e280 9920 2020 .../... \n+0000af10: 2020 2073 6361 6c61 7220 2020 2020 2073 scalar s\n+0000af20: 6361 6c61 7220 2020 2020 2053 6361 6c61 calar Scala\n+0000af30: 7220 6469 7669 7369 6f6e 2e0a 7665 6374 r division..vect\n+0000af40: 6f72 2020 2020 2020 e280 982f e280 9920 or .../... \n+0000af50: 2020 2020 2073 6361 6c61 7220 2020 2020 scalar \n+0000af60: 2076 6563 746f 7220 2020 2020 2056 6563 vector Vec\n+0000af70: 746f 7220 636f 6d70 6f6e 656e 742d 7769 tor component-wi\n+0000af80: 7365 2064 6976 6973 696f 6e20 6279 2073 se division by s\n+0000af90: 6361 6c61 722e 0a70 6f69 6e74 2020 2020 calar..point \n+0000afa0: 2020 20e2 8098 27e2 8099 2020 2020 2020 ...'... \n+0000afb0: e280 9878 e280 992c 20e2 8098 79e2 8099 ...x..., ...y...\n+0000afc0: 2c20 2020 7363 616c 6172 2020 2020 2020 , scalar \n+0000afd0: 506f 696e 7420 636f 6d70 6f6e 656e 7420 Point component \n+0000afe0: 6578 7472 6163 7469 6f6e 2e0a 2020 2020 extraction.. \n+0000aff0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0000b000: 206f 7220 e280 987a e280 990a 7665 6374 or ...z....vect\n+0000b010: 6f72 2020 2020 2020 e280 9827 e280 9920 or ...'... \n+0000b020: 2020 2020 20e2 8098 78e2 8099 2c20 e280 ...x..., ..\n+0000b030: 9879 e280 992c 2020 2073 6361 6c61 7220 .y..., scalar \n+0000b040: 2020 2020 2056 6563 746f 7220 636f 6d70 Vector comp\n+0000b050: 6f6e 656e 7420 6578 7472 6163 7469 6f6e onent extraction\n+0000b060: 2e0a 2020 2020 2020 2020 2020 2020 2020 .. \n+0000b070: 2020 2020 2020 206f 7220 e280 987a e280 or ...z..\n+0000b080: 990a 0a4f 7065 7261 746f 7220 7072 6563 ...Operator prec\n+0000b090: 6564 656e 6365 2069 7320 7368 6f77 6e20 edence is shown \n+0000b0a0: 696e 2074 6869 7320 7461 626c 652e 0a4f in this table..O\n+0000b0b0: 7020 2020 2020 2020 5072 6563 6564 656e p Preceden\n+0000b0c0: 6365 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ce.-------------\n 0000b0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ----------------\n-0000b0e0: 2d2d 2d2d 2d2d 2d2d 2d0a e280 9827 e280 ---------....'..\n-0000b0f0: 9920 2020 2020 2068 6967 6865 7374 2028 . highest (\n-0000b100: 6d6f 7374 2074 6967 6874 6c79 2062 696e most tightly bin\n-0000b110: 6469 6e67 290a e280 985e e280 990a e280 ding)....^......\n-0000b120: 982d e280 9920 2020 2020 2028 756e 6172 .-... (unar\n-0000b130: 7920 6e65 6761 7469 6f6e 290a e280 982a y negation)....*\n-0000b140: e280 990a e280 982e e280 990a e280 982f .............../\n-0000b150: e280 990a e280 982b e280 990a e280 982d .......+.......-\n-0000b160: e280 990a e280 9874 6865 6ee2 8099 2020 .......then... \n-0000b170: 206c 6f77 6573 7420 286c 6561 7374 2074 lowest (least t\n-0000b180: 6967 6874 6c79 2062 696e 6469 6e67 290a ightly binding).\n-0000b190: 0a41 6c6c 206f 7065 7261 7469 6f6e 7320 .All operations \n-0000b1a0: 6172 6520 6c65 6674 2d61 7373 6f63 6961 are left-associa\n-0000b1b0: 7469 7665 2065 7863 6570 7420 666f 7220 tive except for \n-0000b1c0: e280 985e e280 992e 2020 5061 7265 6e74 ...^.... Parent\n-0000b1d0: 6865 7365 7320 e280 9828 2029 e280 990a heses ...( )....\n-0000b1e0: 6172 6520 7573 6564 2066 6f72 2067 726f are used for gro\n-0000b1f0: 7570 696e 6720 746f 206f 7665 7272 6964 uping to overrid\n-0000b200: 6520 7072 6563 6564 656e 6365 2069 6e20 e precedence in \n-0000b210: 7468 6520 7573 7561 6c20 7761 792e 0a0a the usual way...\n-0000b220: 2020 2041 7320 796f 7520 6361 6e20 7365 As you can se\n-0000b230: 652c 2074 6865 2064 6f74 206f 7065 7261 e, the dot opera\n-0000b240: 746f 7220 e280 982e e280 9920 6973 2075 tor ....... is u\n-0000b250: 7375 616c 6c79 2061 2073 796e 6f6e 796d sually a synonym\n-0000b260: 2066 6f72 0a72 756e 2d6f 662d 7468 652d for.run-of-the-\n-0000b270: 6d69 6c6c 206d 756c 7469 706c 6963 6174 mill multiplicat\n-0000b280: 696f 6e2c 20e2 8098 2ae2 8099 2e20 2054 ion, ...*.... T\n-0000b290: 6865 206d 6561 6e69 6e67 7320 6469 6666 he meanings diff\n-0000b2a0: 6572 206f 6e6c 7920 666f 720a 7665 6374 er only for.vect\n-0000b2b0: 6f72 206f 7065 7261 6e64 732e 2020 5468 or operands. Th\n-0000b2c0: 6520 e280 9874 6865 6ee2 8099 206f 7065 e ...then... ope\n-0000b2d0: 7261 746f 7220 6d65 7265 6c79 2072 6576 rator merely rev\n-0000b2e0: 6572 7365 7320 7468 6520 6f70 6572 616e erses the operan\n-0000b2f0: 6420 6f72 6465 720a 7769 7468 2072 6573 d order.with res\n-0000b300: 7065 6374 2074 6f20 6e6f 726d 616c 206d pect to normal m\n-0000b310: 756c 7469 706c 6963 6174 696f 6e20 e280 ultiplication ..\n-0000b320: 982a e280 992e 2020 5468 6520 696e 7465 .*.... The inte\n-0000b330: 6e74 2068 6572 6520 6973 2074 6f20 6d61 nt here is to ma\n-0000b340: 6b65 0a63 6f6d 706f 7369 7469 6f6e 7320 ke.compositions \n-0000b350: 7265 6164 206d 6f72 6520 6e61 7475 7261 read more natura\n-0000b360: 6c6c 792e 2020 5468 6520 636f 6465 0a20 lly. The code. \n-0000b370: 2020 2020 2831 2c32 2c33 2920 7468 656e (1,2,3) then\n-0000b380: 2073 6361 6c65 2832 2920 7468 656e 2072 scale(2) then r\n-0000b390: 6f74 6174 6528 3330 2920 7468 656e 2074 otate(30) then t\n-0000b3a0: 7261 6e73 6c61 7465 285b 312c 332c 305d ranslate([1,3,0]\n-0000b3b0: 290a 6578 7072 6573 7365 7320 6120 7365 ).expresses a se\n-0000b3c0: 7269 6573 206f 6620 7375 6363 6573 7369 ries of successi\n-0000b3d0: 7665 206d 6f64 6966 6963 6174 696f 6e73 ve modifications\n-0000b3e0: 2074 6f20 7468 6520 706f 696e 742c 2077 to the point, w\n-0000b3f0: 6865 7265 6173 2074 6865 0a65 7175 6976 hereas the.equiv\n-0000b400: 616c 656e 7420 666f 726d 0a20 2020 2020 alent form. \n-0000b410: 7472 616e 736c 6174 6528 5b31 2c33 2c30 translate([1,3,0\n-0000b420: 5d29 202a 2072 6f74 6174 6528 3330 2920 ]) * rotate(30) \n-0000b430: 2a20 7363 616c 6528 3229 202a 2028 312c * scale(2) * (1,\n-0000b440: 322c 3329 0a77 696c 6c20 6265 2069 6e74 2,3).will be int\n-0000b450: 7569 7469 7665 206f 6e6c 7920 746f 206d uitive only to m\n-0000b460: 6174 6865 6d61 7469 6369 616e 7320 2861 athematicians (a\n-0000b470: 6e64 2070 6572 6861 7073 2041 7261 6269 nd perhaps Arabi\n-0000b480: 6320 6c61 6e67 7561 6765 0a72 6561 6465 c language.reade\n-0000b490: 7273 292e 0a0a 1f0a 4669 6c65 3a20 736b rs).....File: sk\n-0000b4a0: 6574 6368 2e69 6e66 6f2c 2020 4e6f 6465 etch.info, Node\n-0000b4b0: 3a20 556e 6172 7920 666f 726d 732c 2020 : Unary forms, \n-0000b4c0: 5072 6576 3a20 5477 6f2d 6f70 6572 616e Prev: Two-operan\n-0000b4d0: 6420 2862 696e 6172 7929 2066 6f72 6d73 d (binary) forms\n-0000b4e0: 2c20 2055 703a 2041 7269 7468 6d65 7469 , Up: Arithmeti\n-0000b4f0: 630a 0a33 2e31 2e34 2e32 2055 6e61 7279 c..3.1.4.2 Unary\n-0000b500: 2066 6f72 6d73 0a2e 2e2e 2e2e 2e2e 2e2e forms..........\n-0000b510: 2e2e 2e2e 2e2e 2e2e 2e2e 0a0a 556e 6172 ............Unar\n-0000b520: 7920 6f72 206f 6e65 2d6f 7065 7261 6e64 y or one-operand\n-0000b530: 2066 6f72 6d73 2061 7265 2073 756d 6d61 forms are summa\n-0000b540: 7269 7a65 6420 696e 2074 6865 2066 6f6c rized in the fol\n-0000b550: 6c6f 7769 6e67 2074 6162 6c65 2c20 7768 lowing table, wh\n-0000b560: 6572 650a e280 9858 e280 9920 7374 616e ere....X... stan\n-0000b570: 6473 2066 6f72 2074 6865 206f 7065 7261 ds for the opera\n-0000b580: 6e64 2e0a 4f70 2020 2020 2020 2020 2020 nd..Op \n-0000b590: 2020 204f 7065 7261 6e64 2020 2020 2052 Operand R\n-0000b5a0: 6573 756c 7420 2020 2020 2044 6573 6372 esult Descr\n-0000b5b0: 6970 7469 6f6e 0a2d 2d2d 2d2d 2d2d 2d2d iption.---------\n+0000b0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a e280 -------------...\n+0000b0f0: 9827 e280 9920 2020 2020 2068 6967 6865 .'... highe\n+0000b100: 7374 2028 6d6f 7374 2074 6967 6874 6c79 st (most tightly\n+0000b110: 2062 696e 6469 6e67 290a e280 985e e280 binding)....^..\n+0000b120: 990a e280 982d e280 9920 2020 2020 2028 .....-... (\n+0000b130: 756e 6172 7920 6e65 6761 7469 6f6e 290a unary negation).\n+0000b140: e280 982a e280 990a e280 982e e280 990a ...*............\n+0000b150: e280 982f e280 990a e280 982b e280 990a .../.......+....\n+0000b160: e280 982d e280 990a e280 9874 6865 6ee2 ...-.......then.\n+0000b170: 8099 2020 206c 6f77 6573 7420 286c 6561 .. lowest (lea\n+0000b180: 7374 2074 6967 6874 6c79 2062 696e 6469 st tightly bindi\n+0000b190: 6e67 290a 0a41 6c6c 206f 7065 7261 7469 ng)..All operati\n+0000b1a0: 6f6e 7320 6172 6520 6c65 6674 2d61 7373 ons are left-ass\n+0000b1b0: 6f63 6961 7469 7665 2065 7863 6570 7420 ociative except \n+0000b1c0: 666f 7220 e280 985e e280 992e 2020 5061 for ...^.... Pa\n+0000b1d0: 7265 6e74 6865 7365 7320 e280 9828 2029 rentheses ...( )\n+0000b1e0: e280 990a 6172 6520 7573 6564 2066 6f72 ....are used for\n+0000b1f0: 2067 726f 7570 696e 6720 746f 206f 7665 grouping to ove\n+0000b200: 7272 6964 6520 7072 6563 6564 656e 6365 rride precedence\n+0000b210: 2069 6e20 7468 6520 7573 7561 6c20 7761 in the usual wa\n+0000b220: 792e 0a0a 2020 2041 7320 796f 7520 6361 y... As you ca\n+0000b230: 6e20 7365 652c 2074 6865 2064 6f74 206f n see, the dot o\n+0000b240: 7065 7261 746f 7220 e280 982e e280 9920 perator ....... \n+0000b250: 6973 2075 7375 616c 6c79 2061 2073 796e is usually a syn\n+0000b260: 6f6e 796d 2066 6f72 0a72 756e 2d6f 662d onym for.run-of-\n+0000b270: 7468 652d 6d69 6c6c 206d 756c 7469 706c the-mill multipl\n+0000b280: 6963 6174 696f 6e2c 20e2 8098 2ae2 8099 ication, ...*...\n+0000b290: 2e20 2054 6865 206d 6561 6e69 6e67 7320 . 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the\n+0000b550: 2066 6f6c 6c6f 7769 6e67 2074 6162 6c65 following table\n+0000b560: 2c20 7768 6572 650a e280 9858 e280 9920 , where....X... \n+0000b570: 7374 616e 6473 2066 6f72 2074 6865 206f stands for the o\n+0000b580: 7065 7261 6e64 2e0a 4f70 2020 2020 2020 perand..Op \n+0000b590: 2020 2020 2020 204f 7065 7261 6e64 2020 Operand \n+0000b5a0: 2020 2052 6573 756c 7420 2020 2020 2044 Result D\n+0000b5b0: 6573 6372 6970 7469 6f6e 0a2d 2d2d 2d2d escription.-----\n 0000b5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ----------------\n 0000b5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ----------------\n 0000b5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ----------------\n 0000b5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ----------------\n-0000b600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a e280 982d -----------....-\n-0000b610: 58e2 8099 2020 2020 2020 2020 2020 2073 X... s\n-0000b620: 6361 6c61 7220 2020 2020 2073 6361 6c61 calar scala\n-0000b630: 7220 2020 2020 2055 6e61 7279 2073 6361 r Unary 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Options k\n-0000bff0: 6e6f 776e 2074 6f20 e280 9873 6b65 7463 nown to ...sketc\n-0000c000: 68e2 8099 2066 6f72 2070 7572 706f 7365 h... for purpose\n-0000c010: 7320 6f66 2074 6869 7320 7370 6c69 7474 s of this splitt\n-0000c020: 696e 670a 6f70 6572 6174 696f 6e20 696e ing.operation in\n-0000c030: 636c 7564 6520 e280 9861 7272 6f77 73e2 clude ...arrows.\n-0000c040: 8099 2c20 e280 9864 6173 68e2 8099 2c20 .., ...dash..., \n-0000c050: e280 9864 6f74 7365 70e2 8099 2c20 e280 ...dotsep..., ..\n-0000c060: 9866 696c 6c63 6f6c 6f72 e280 992c 20e2 .fillcolor..., .\n-0000c070: 8098 6669 6c6c 7374 796c 65e2 8099 2c0a ..fillstyle...,.\n-0000c080: e280 986c 696e 6563 6f6c 6f72 e280 992c ...linecolor...,\n-0000c090: 20e2 8098 6c69 6e65 7374 796c 65e2 8099 ...linestyle...\n-0000c0a0: 2c20 e280 986c 696e 6577 6964 7468 e280 , ...linewidth..\n-0000c0b0: 992c 20e2 8098 6f70 6163 6974 79e2 8099 ., ...opacity...\n-0000c0c0: 2c20 e280 9873 686f 7770 6f69 6e74 73e2 , ...showpoints.\n-0000c0d0: 8099 2c0a e280 9873 7472 6f6b 656f 7061 ..,....strokeopa\n-0000c0e0: 6369 7479 e280 992c 2061 6e64 20e2 8098 city..., and ...\n-0000c0f0: 7472 616e 7370 616c 7068 61e2 8099 2e0a transpalpha.....\n-0000c100: 0a1f 0a46 696c 653a 2073 6b65 7463 682e ...File: sketch.\n-0000c110: 696e 666f 2c20 204e 6f64 653a 2054 696b info, Node: Tik\n-0000c120: 5a2f 5047 4620 6f70 7469 6f6e 732c 2020 Z/PGF options, \n-0000c130: 4e65 7874 3a20 446f 7473 2069 6e20 5469 Next: Dots in Ti\n-0000c140: 6b5a 2f50 4746 2c20 2050 7265 763a 2050 kZ/PGF, Prev: P\n-0000c150: 5354 7269 636b 7320 6f70 7469 6f6e 732c STricks options,\n-0000c160: 2020 5570 3a20 4f70 7469 6f6e 730a 0a33 Up: Options..3\n-0000c170: 2e31 2e35 2e32 20e2 8098 5469 6b5a 2f50 .1.5.2 ...TikZ/P\n-0000c180: 4746 e280 9920 6f70 7469 6f6e 730a 2e2e GF... options...\n-0000c190: 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e ................\n-0000c1a0: 2e2e 2e2e 2e2e 2e2e 0a0a e280 9854 696b .............Tik\n-0000c1b0: 5a2f 5047 46e2 8099 206f 7074 696f 6e73 Z/PGF... options\n-0000c1c0: 2061 7265 2068 616e 646c 6564 206d 7563 are handled muc\n-0000c1d0: 6820 6173 2066 6f72 20e2 8098 5053 5472 h as for ...PSTr\n-0000c1e0: 6963 6b73 e280 992e 2020 5468 6f75 6768 icks.... Though\n-0000c1f0: 0ae2 8098 5469 6b5a 2f50 4746 e280 9920 ....TikZ/PGF... \n-0000c200: 6f66 7465 6e20 616c 6c6f 7773 2063 6f6c often allows col\n-0000c210: 6f72 7320 616e 6420 7374 796c 6573 2074 ors and styles t\n-0000c220: 6f20 6265 2067 6976 656e 2077 6974 686f o be given witho\n-0000c230: 7574 0a63 6f72 7265 7370 6f6e 6469 6e67 ut.corresponding\n-0000c240: 206b 6579 732c 2066 6f72 2065 7861 6d70 keys, for examp\n-0000c250: 6c65 2c0a 2020 5c64 7261 775b 7265 642c le,. \\draw[red,\n-0000c260: 756c 7472 6120 7468 6963 6b5d 2830 2c30 ultra thick](0,0\n-0000c270: 292d 2d28 312c 3129 3b0a 7468 6973 2069 )--(1,1);.this i\n-0000c280: 7320 6e6f 7420 7065 726d 6974 7465 6420 s not permitted \n-0000c290: 696e 20e2 8098 736b 6574 6368 e280 992e in ...sketch....\n-0000c2a0: 2020 546f 2064 7261 7720 6120 7265 642c To draw a red,\n-0000c2b0: 2075 6c74 7261 2d74 6869 636b 206c 696e ultra-thick lin\n-0000c2c0: 6520 696e 0ae2 8098 736b 6574 6368 e280 e in....sketch..\n-0000c2d0: 992c 2074 6865 2066 6f72 6d20 6973 0a20 ., the form is. \n-0000c2e0: 206c 696e 655b 6472 6177 3d72 6564 2c73 line[draw=red,s\n-0000c2f0: 7479 6c65 3d75 6c74 7261 2074 6869 636b tyle=ultra thick\n-0000c300: 5d28 302c 3029 2831 2c31 290a 0a20 2020 ](0,0)(1,1).. \n-0000c310: 4a75 7374 2061 7320 666f 7220 e280 9850 Just as for ...P\n-0000c320: 5354 7269 636b 73e2 8099 2c20 7768 656e STricks..., when\n-0000c330: 2061 20e2 8098 706f 6c79 676f 6ee2 8099 a ...polygon...\n-0000c340: 2068 6173 206f 7074 696f 6e73 2066 6f72 has options for\n-0000c350: 2062 6f74 6820 6974 730a 6661 6365 2061 both its.face a\n-0000c360: 6e64 2069 7473 2065 6467 6573 2c20 616e nd its edges, an\n-0000c370: 6420 7468 6520 706f 6c79 676f 6e20 6973 d the polygon is\n-0000c380: 2073 706c 6974 2062 7920 7468 6520 6869 split by the hi\n-0000c390: 6464 656e 2073 7572 6661 6365 0a61 6c67 dden surface.alg\n-0000c3a0: 6f72 6974 686d 2c20 e280 9873 6b65 7463 orithm, ...sketc\n-0000c3b0: 68e2 8099 206d 7573 7420 636f 7079 2074 h... must copy t\n-0000c3c0: 6865 2065 6467 6520 6f70 7469 6f6e 7320 he edge options \n-0000c3d0: 746f 20e2 8098 7073 6c69 6e65 e280 9973 to ...psline...s\n-0000c3e0: 2066 6f72 2074 6865 2065 6467 650a 7365 for the edge.se\n-0000c3f0: 676d 656e 7473 2061 6e64 2074 6865 2066 gments and the f\n-0000c400: 6163 6520 6f70 7469 6f6e 7320 746f 20e2 ace options to .\n-0000c410: 8098 7073 706f 6c79 676f 6ee2 8099 732e ..pspolygon...s.\n-0000c420: 2020 e280 9854 696b 5a2f 5047 46e2 8099 ...TikZ/PGF...\n-0000c430: 206f 7074 696f 6e73 206b 6e6f 776e 0a74 options known.t\n-0000c440: 6f20 e280 9873 6b65 7463 68e2 8099 2066 o ...sketch... f\n-0000c450: 6f72 2070 7572 706f 7365 7320 6f66 2074 or purposes of t\n-0000c460: 6869 7320 7370 6c69 7474 696e 6720 6f70 his splitting op\n-0000c470: 6572 6174 696f 6e20 696e 636c 7564 6520 eration include \n-0000c480: e280 9861 7272 6f77 73e2 8099 2c0a e280 ...arrows...,...\n-0000c490: 9863 6170 e280 992c 20e2 8098 636f 6c6f .cap..., ...colo\n-0000c4a0: 72e2 8099 2c20 e280 9864 6173 6820 7061 r..., ...dash pa\n-0000c4b0: 7474 6572 6ee2 8099 2c20 e280 9864 6173 ttern..., ...das\n-0000c4c0: 6820 7068 6173 65e2 8099 2c20 e280 9864 h phase..., ...d\n-0000c4d0: 6f75 626c 65e2 8099 2c20 e280 9864 6f75 ouble..., ...dou\n-0000c4e0: 626c 650a 6469 7374 616e 6365 e280 992c ble.distance...,\n-0000c4f0: 20e2 8098 6472 6177 e280 992c 20e2 8098 ...draw..., ...\n-0000c500: 6472 6177 206f 7061 6369 7479 e280 992c draw opacity...,\n-0000c510: 20e2 8098 6669 6c6c e280 992c 20e2 8098 ...fill..., ...\n-0000c520: 6669 6c6c 206f 7061 6369 7479 e280 992c fill opacity...,\n-0000c530: 20e2 8098 6a6f 696e e280 992c 20e2 8098 ...join..., ...\n-0000c540: 6c69 6e65 0a77 6964 7468 e280 992c 20e2 line.width..., .\n-0000c550: 8098 6d69 7465 7220 6c69 6d69 74e2 8099 ..miter limit...\n-0000c560: 2c20 e280 9870 6174 7465 726e e280 992c , ...pattern...,\n-0000c570: 20e2 8098 7061 7474 6572 6e20 636f 6c6f ...pattern colo\n-0000c580: 72e2 8099 2c20 616e 6420 e280 9873 7479 r..., and ...sty\n-0000c590: 6c65 e280 992e 0a0a 2020 2054 6865 20e2 le...... 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sketch.inf\n-0000c7e0: 6f2c 2020 4e6f 6465 3a20 446f 7473 2069 o, Node: Dots i\n-0000c7f0: 6e20 5469 6b5a 2f50 4746 2c20 204e 6578 n TikZ/PGF, Nex\n-0000c800: 743a 2054 696b 5a2f 5047 4620 7573 6572 t: TikZ/PGF user\n-0000c810: 2d64 6566 696e 6564 2073 7479 6c65 732c -defined styles,\n-0000c820: 2020 5072 6576 3a20 5469 6b5a 2f50 4746 Prev: TikZ/PGF\n-0000c830: 206f 7074 696f 6e73 2c20 2055 703a 204f options, Up: O\n-0000c840: 7074 696f 6e73 0a0a 332e 312e 352e 3320 ptions..3.1.5.3 \n-0000c850: 446f 7473 2069 6e20 e280 9854 696b 5a2f Dots in ...TikZ/\n-0000c860: 5047 46e2 8099 0a2e 2e2e 2e2e 2e2e 2e2e PGF.............\n+0000b600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------.\n+0000b610: e280 982d 58e2 8099 2020 2020 2020 2020 ...-X... \n+0000b620: 2020 2073 6361 6c61 7220 2020 2020 2073 scalar s\n+0000b630: 6361 6c61 7220 2020 2020 2055 6e61 7279 calar Unary\n+0000b640: 2073 6361 6c61 7220 6e65 6761 7469 6f6e scalar negation\n+0000b650: 2e0a e280 982d 58e2 8099 2020 2020 2020 .....-X... \n+0000b660: 2020 2020 2076 6563 746f 7220 2020 2020 vector \n+0000b670: 2076 6563 746f 7220 2020 2020 2055 6e61 vector Una\n+0000b680: 7279 2076 6563 746f 7220 6e65 6761 7469 ry vector negati\n+0000b690: 6f6e 2e0a e280 987c 587c e280 9920 2020 on.....|X|... \n+0000b6a0: 2020 2020 2020 2076 6563 746f 7220 2020 vector \n+0000b6b0: 2020 2073 6361 6c61 7220 2020 2020 2056 scalar V\n+0000b6c0: 6563 746f 7220 6c65 6e67 7468 2e0a e280 ector length....\n+0000b6d0: 9875 6e69 7428 5829 e280 9920 2020 2020 .unit(X)... \n+0000b6e0: 2076 6563 746f 7220 2020 2020 2076 6563 vector vec\n+0000b6f0: 746f 7220 2020 2020 2055 6e69 7420 7665 tor Unit ve\n+0000b700: 6374 6f72 2077 6974 6820 7361 6d65 2064 ctor with same d\n+0000b710: 6972 6563 7469 6f6e 2e0a e280 9873 7172 irection.....sqr\n+0000b720: 7428 5829 e280 9920 2020 2020 2073 6361 t(X)... sca\n+0000b730: 6c61 7220 2020 2020 2073 6361 6c61 7220 lar scalar \n+0000b740: 2020 2020 2053 6361 6c61 7220 7371 7561 Scalar squa\n+0000b750: 7265 2072 6f6f 742e 0ae2 8098 7369 6e28 re root.....sin(\n+0000b760: 5829 e280 9920 2020 2020 2020 7363 616c X)... scal\n+0000b770: 6172 2020 2020 2020 7363 616c 6172 2020 ar scalar \n+0000b780: 2020 2020 5472 6967 6f6e 6f6d 6574 7269 Trigonometri\n+0000b790: 6320 7369 6e65 2028 e280 9858 e280 9920 c sine (...X... \n+0000b7a0: 696e 2064 6567 7265 6573 292e 0ae2 8098 in degrees).....\n+0000b7b0: 636f 7328 5829 e280 9920 2020 2020 2020 cos(X)... \n+0000b7c0: 7363 616c 6172 2020 2020 2020 7363 616c scalar scal\n+0000b7d0: 6172 2020 2020 2020 5472 6967 6f6e 6f6d ar Trigonom\n+0000b7e0: 6574 7269 6320 636f 7369 6e65 2028 e280 etric cosine (..\n+0000b7f0: 9858 e280 9920 696e 2064 6567 7265 6573 .X... in degrees\n+0000b800: 292e 0ae2 8098 6173 696e 2858 29e2 8099 ).....asin(X)...\n+0000b810: 2020 2020 2020 7363 616c 6172 2020 2020 scalar \n+0000b820: 2020 7363 616c 6172 2020 2020 2020 496e scalar In\n+0000b830: 7665 7273 6520 7369 6e65 2028 e280 9858 verse sine (...X\n+0000b840: e280 9920 696e 2064 6567 7265 6573 292e ... in degrees).\n+0000b850: 0ae2 8098 6163 6f73 2858 29e2 8099 2020 ....acos(X)... \n+0000b860: 2020 2020 7363 616c 6172 2020 2020 2020 scalar \n+0000b870: 7363 616c 6172 2020 2020 2020 496e 7665 scalar Inve\n+0000b880: 7273 6520 636f 7369 6e65 2028 e280 9858 rse cosine (...X\n+0000b890: e280 9920 696e 2064 6567 7265 6573 292e ... in degrees).\n+0000b8a0: 0ae2 8098 6174 616e 3228 582c 5929 e280 ....atan2(X,Y)..\n+0000b8b0: 9920 2020 7363 616c 6172 2020 2020 2020 . scalar \n+0000b8c0: 7363 616c 6172 2020 2020 2020 506f 6c61 scalar Pola\n+0000b8d0: 7220 616e 676c 6520 696e 2064 6567 7265 r angle in degre\n+0000b8e0: 6573 206f 6620 7665 6374 6f72 205b 582c es of vector [X,\n+0000b8f0: 595d 2e0a e280 9869 6e76 6572 7365 2858 Y].....inverse(X\n+0000b900: 29e2 8099 2020 2074 7261 6e73 666f 726d )... transform\n+0000b910: 2020 2074 7261 6e73 666f 726d 2020 2049 transform I\n+0000b920: 6e76 6572 7365 2074 7261 6e73 666f 726d nverse transform\n+0000b930: 2e0a 0a45 7272 6f72 7320 6172 6520 7265 ...Errors are re\n+0000b940: 706f 7274 6564 2077 6865 6e20 e280 987c ported when ...|\n+0000b950: 587c e280 992c 20e2 8098 756e 6974 e280 X|..., ...unit..\n+0000b960: 992c 20e2 8098 6173 696e e280 992c 20e2 ., ...asin..., .\n+0000b970: 8098 6163 6f73 e280 992c 20e2 8098 7371 ..acos..., ...sq\n+0000b980: 7274 e280 992c 20e2 8098 6174 616e 32e2 rt..., ...atan2.\n+0000b990: 8099 2c0a 616e 6420 e280 9869 6e76 6572 ..,.and ...inver\n+0000b9a0: 7365 e280 9920 6661 696c 2064 7565 2074 se... fail due t\n+0000b9b0: 6f20 6f70 6572 616e 6473 206f 7574 7369 o operands outsi\n+0000b9c0: 6465 2074 6865 6972 2072 6573 7065 6374 de their respect\n+0000b9d0: 6976 6520 646f 6d61 696e 732e 0a0a 1f0a ive domains.....\n+0000b9e0: 4669 6c65 3a20 736b 6574 6368 2e69 6e66 File: sketch.inf\n+0000b9f0: 6f2c 2020 4e6f 6465 3a20 4f70 7469 6f6e o, Node: Option\n+0000ba00: 732c 2020 5072 6576 3a20 4172 6974 686d s, Prev: Arithm\n+0000ba10: 6574 6963 2c20 2055 703a 204c 616e 6775 etic, 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O\n+0000be90: 7074 696f 6e20 6c69 7374 7320 6172 6520 ption lists are \n+0000bea0: 616c 7761 7973 206f 7074 696f 6e61 6c2e always optional.\n+0000beb0: 2020 4120 6d69 7373 696e 6720 6f70 7469 A missing opti\n+0000bec0: 6f6e 7320 6c69 7374 0a69 7320 6571 7569 ons list.is equi\n+0000bed0: 7661 6c65 6e74 2074 6f20 616e 2065 6d70 valent to an emp\n+0000bee0: 7479 206f 6e65 20e2 8098 5b5d e280 992e ty one ...[]....\n+0000bef0: 0a0a 2020 2057 6865 6e20 6120 e280 9870 .. 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Th\n+0000c1f0: 6f75 6768 0ae2 8098 5469 6b5a 2f50 4746 ough....TikZ/PGF\n+0000c200: e280 9920 6f66 7465 6e20 616c 6c6f 7773 ... often allows\n+0000c210: 2063 6f6c 6f72 7320 616e 6420 7374 796c colors and styl\n+0000c220: 6573 2074 6f20 6265 2067 6976 656e 2077 es to be given w\n+0000c230: 6974 686f 7574 0a63 6f72 7265 7370 6f6e ithout.correspon\n+0000c240: 6469 6e67 206b 6579 732c 2066 6f72 2065 ding keys, for e\n+0000c250: 7861 6d70 6c65 2c0a 2020 5c64 7261 775b xample,. \\draw[\n+0000c260: 7265 642c 756c 7472 6120 7468 6963 6b5d red,ultra thick]\n+0000c270: 2830 2c30 292d 2d28 312c 3129 3b0a 7468 (0,0)--(1,1);.th\n+0000c280: 6973 2069 7320 6e6f 7420 7065 726d 6974 is is not permit\n+0000c290: 7465 6420 696e 20e2 8098 736b 6574 6368 ted in ...sketch\n+0000c2a0: e280 992e 2020 546f 2064 7261 7720 6120 .... To draw a \n+0000c2b0: 7265 642c 2075 6c74 7261 2d74 6869 636b red, ultra-thick\n+0000c2c0: 206c 696e 6520 696e 0ae2 8098 736b 6574 line in....sket\n+0000c2d0: 6368 e280 992c 2074 6865 2066 6f72 6d20 ch..., the form \n+0000c2e0: 6973 0a20 206c 696e 655b 6472 6177 3d72 is. line[draw=r\n+0000c2f0: 6564 2c73 7479 6c65 3d75 6c74 7261 2074 ed,style=ultra t\n+0000c300: 6869 636b 5d28 302c 3029 2831 2c31 290a hick](0,0)(1,1).\n+0000c310: 0a20 2020 4a75 7374 2061 7320 666f 7220 . Just as for \n+0000c320: e280 9850 5354 7269 636b 73e2 8099 2c20 ...PSTricks..., \n+0000c330: 7768 656e 2061 20e2 8098 706f 6c79 676f when a ...polygo\n+0000c340: 6ee2 8099 2068 6173 206f 7074 696f 6e73 n... has options\n+0000c350: 2066 6f72 2062 6f74 6820 6974 730a 6661 for both its.fa\n+0000c360: 6365 2061 6e64 2069 7473 2065 6467 6573 ce and its edges\n+0000c370: 2c20 616e 6420 7468 6520 706f 6c79 676f , and the polygo\n+0000c380: 6e20 6973 2073 706c 6974 2062 7920 7468 n is split by th\n+0000c390: 6520 6869 6464 656e 2073 7572 6661 6365 e hidden surface\n+0000c3a0: 0a61 6c67 6f72 6974 686d 2c20 e280 9873 .algorithm, ...s\n+0000c3b0: 6b65 7463 68e2 8099 206d 7573 7420 636f ketch... must co\n+0000c3c0: 7079 2074 6865 2065 6467 6520 6f70 7469 py the edge opti\n+0000c3d0: 6f6e 7320 746f 20e2 8098 7073 6c69 6e65 ons to ...psline\n+0000c3e0: e280 9973 2066 6f72 2074 6865 2065 6467 ...s for the edg\n+0000c3f0: 650a 7365 676d 656e 7473 2061 6e64 2074 e.segments and t\n+0000c400: 6865 2066 6163 6520 6f70 7469 6f6e 7320 he face options \n+0000c410: 746f 20e2 8098 7073 706f 6c79 676f 6ee2 to ...pspolygon.\n+0000c420: 8099 732e 2020 e280 9854 696b 5a2f 5047 ..s. ...TikZ/PG\n+0000c430: 46e2 8099 206f 7074 696f 6e73 206b 6e6f F... options kno\n+0000c440: 776e 0a74 6f20 e280 9873 6b65 7463 68e2 wn.to ...sketch.\n+0000c450: 8099 2066 6f72 2070 7572 706f 7365 7320 .. for purposes \n+0000c460: 6f66 2074 6869 7320 7370 6c69 7474 696e of this splittin\n+0000c470: 6720 6f70 6572 6174 696f 6e20 696e 636c g operation incl\n+0000c480: 7564 6520 e280 9861 7272 6f77 73e2 8099 ude ...arrows...\n+0000c490: 2c0a e280 9863 6170 e280 992c 20e2 8098 ,....cap..., ...\n+0000c4a0: 636f 6c6f 72e2 8099 2c20 e280 9864 6173 color..., ...das\n+0000c4b0: 6820 7061 7474 6572 6ee2 8099 2c20 e280 h pattern..., ..\n+0000c4c0: 9864 6173 6820 7068 6173 65e2 8099 2c20 .dash phase..., \n+0000c4d0: e280 9864 6f75 626c 65e2 8099 2c20 e280 ...double..., ..\n+0000c4e0: 9864 6f75 626c 650a 6469 7374 616e 6365 .double.distance\n+0000c4f0: e280 992c 20e2 8098 6472 6177 e280 992c ..., ...draw...,\n+0000c500: 20e2 8098 6472 6177 206f 7061 6369 7479 ...draw opacity\n+0000c510: e280 992c 20e2 8098 6669 6c6c e280 992c ..., ...fill...,\n+0000c520: 20e2 8098 6669 6c6c 206f 7061 6369 7479 ...fill opacity\n+0000c530: e280 992c 20e2 8098 6a6f 696e e280 992c ..., ...join...,\n+0000c540: 20e2 8098 6c69 6e65 0a77 6964 7468 e280 ...line.width..\n+0000c550: 992c 20e2 8098 6d69 7465 7220 6c69 6d69 ., ...miter limi\n+0000c560: 74e2 8099 2c20 e280 9870 6174 7465 726e t..., ...pattern\n+0000c570: e280 992c 20e2 8098 7061 7474 6572 6e20 ..., ...pattern \n+0000c580: 636f 6c6f 72e2 8099 2c20 616e 6420 e280 color..., and ..\n+0000c590: 9873 7479 6c65 e280 992e 0a0a 2020 2054 .style...... 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V\n+0000c610: 616c 7565 7320 6b6e 6f77 6e20 746f 20e2 alues known to .\n+0000c620: 8098 736b 6574 6368 e280 9920 696e 636c ..sketch... incl\n+0000c630: 7564 650a e280 9864 6173 6865 64e2 8099 ude....dashed...\n+0000c640: 2c20 e280 9864 656e 7365 6c79 2064 6173 , ...densely das\n+0000c650: 6865 64e2 8099 2c20 e280 9864 656e 7365 hed..., ...dense\n+0000c660: 6c79 2064 6f74 7465 64e2 8099 2c20 e280 ly dotted..., ..\n+0000c670: 9864 6f74 7465 64e2 8099 2c20 e280 9864 .dotted..., ...d\n+0000c680: 6f75 626c 65e2 8099 2c0a e280 986c 6f6f ouble...,....loo\n+0000c690: 7365 6c79 2064 6173 6865 64e2 8099 2c20 sely dashed..., \n+0000c6a0: e280 986c 6f6f 7365 6c79 2064 6f74 7465 ...loosely dotte\n+0000c6b0: 64e2 8099 2c20 e280 986e 6561 726c 7920 d..., ...nearly \n+0000c6c0: 6f70 6171 7565 e280 992c 20e2 8098 6e65 opaque..., ...ne\n+0000c6d0: 6172 6c79 0a74 7261 6e73 7061 7265 6e74 arly.transparent\n+0000c6e0: e280 992c 20e2 8098 7365 6d69 7468 6963 ..., ...semithic\n+0000c6f0: 6be2 8099 2c20 e280 9873 656d 6974 7261 k..., ...semitra\n+0000c700: 6e73 7061 7265 6e74 e280 992c 20e2 8098 nsparent..., ...\n+0000c710: 736f 6c69 64e2 8099 2c20 e280 9874 6869 solid..., ...thi\n+0000c720: 636b e280 992c 20e2 8098 7468 696e e280 ck..., ...thin..\n+0000c730: 992c 0ae2 8098 7472 616e 7370 6172 656e .,....transparen\n+0000c740: 74e2 8099 2c20 e280 9875 6c74 7261 206e t..., ...ultra n\n+0000c750: 6561 726c 7920 7472 616e 7370 6172 656e early transparen\n+0000c760: 74e2 8099 2c20 e280 9875 6c74 7261 2074 t..., ...ultra t\n+0000c770: 6869 636b e280 992c 20e2 8098 756c 7472 hick..., ...ultr\n+0000c780: 6120 7468 696e e280 992c 0ae2 8098 7665 a thin...,....ve\n+0000c790: 7279 206e 6561 726c 7920 7472 616e 7370 ry nearly transp\n+0000c7a0: 6172 656e 74e2 8099 2c20 e280 9876 6572 arent..., ...ver\n+0000c7b0: 7920 7468 6963 6be2 8099 2c20 616e 6420 y thick..., and \n+0000c7c0: e280 9876 6572 7920 7468 696e e280 992e ...very thin....\n+0000c7d0: 0a0a 1f0a 4669 6c65 3a20 736b 6574 6368 ....File: sketch\n+0000c7e0: 2e69 6e66 6f2c 2020 4e6f 6465 3a20 446f .info, Node: Do\n+0000c7f0: 7473 2069 6e20 5469 6b5a 2f50 4746 2c20 ts in TikZ/PGF, \n+0000c800: 204e 6578 743a 2054 696b 5a2f 5047 4620 Next: TikZ/PGF \n+0000c810: 7573 6572 2d64 6566 696e 6564 2073 7479 user-defined sty\n+0000c820: 6c65 732c 2020 5072 6576 3a20 5469 6b5a les, Prev: TikZ\n+0000c830: 2f50 4746 206f 7074 696f 6e73 2c20 2055 /PGF options, U\n+0000c840: 703a 204f 7074 696f 6e73 0a0a 332e 312e p: Options..3.1.\n+0000c850: 352e 3320 446f 7473 2069 6e20 e280 9854 5.3 Dots in ...T\n+0000c860: 696b 5a2f 5047 46e2 8099 0a2e 2e2e 2e2e ikZ/PGF.........\n 0000c870: 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e ................\n-0000c880: 2e0a 0ae2 8098 5469 6b5a 2f50 4746 e280 ......TikZ/PGF..\n-0000c890: 9920 646f 6573 206e 6f74 2068 6176 6520 . does not have \n-0000c8a0: 6120 e280 9864 6f74 73e2 8099 2063 6f6d a ...dots... com\n-0000c8b0: 6d61 6e64 2061 7320 646f 6573 2050 5354 mand as does PST\n-0000c8c0: 7269 636b 732e 2020 496e 7374 6561 642c ricks. Instead,\n-0000c8d0: 0ae2 8098 536b 6574 6368 e280 9920 656d ....Sketch... em\n-0000c8e0: 6974 7320 646f 7473 2061 7320 e280 9866 its dots as ...f\n-0000c8f0: 696c 6c64 7261 77e2 8099 2063 6972 636c illdraw... circl\n-0000c900: 6573 2e20 2054 6865 2064 6961 6d65 7465 es. The diamete\n-0000c910: 7220 6d61 7920 6265 2073 6574 0a75 7369 r may be set.usi\n-0000c920: 6e67 2074 6865 206f 7074 696f 6e20 e280 ng the option ..\n-0000c930: 9864 6f74 7369 7a65 e280 9920 626f 7272 .dotsize... borr\n-0000c940: 6f77 6564 2066 726f 6d20 5053 5472 6963 owed from PSTric\n-0000c950: 6b73 2e20 2054 6865 20e2 8098 646f 7473 ks. The ...dots\n-0000c960: 697a 65e2 8099 206f 7074 696f 6e0a 7769 ize... option.wi\n-0000c970: 6c6c 2062 6520 7265 6d6f 7665 6420 6672 ll be removed fr\n-0000c980: 6f6d 2074 6865 206f 7074 696f 6e20 6c69 om the option li\n-0000c990: 7374 2069 6e20 7468 6520 6f75 7470 7574 st in the output\n-0000c9a0: 20e2 8098 6669 6c6c 6472 6177 e280 9920 ...filldraw... \n-0000c9b0: 636f 6d6d 616e 642e 0a4f 7468 6572 206f command..Other o\n-0000c9c0: 7074 696f 6e73 2077 6f72 6b20 696e 2074 ptions work in t\n-0000c9d0: 6865 2065 7870 6563 7465 6420 7761 792e he expected way.\n-0000c9e0: 2020 466f 7220 6578 616d 706c 652c 20e2 For example, .\n-0000c9f0: 8098 6669 6c6c e280 9920 7365 7473 2066 ..fill... sets f\n-0000ca00: 696c 6c0a 636f 6c6f 7220 616e 6420 e280 ill.color and ..\n-0000ca10: 9863 6f6c 6f72 e280 9920 7365 7473 206c .color... sets l\n-0000ca20: 696e 6520 636f 6c6f 7220 6f66 2074 6865 ine color of the\n-0000ca30: 2063 6972 636c 6573 2e0a 0a1f 0a46 696c circles.....Fil\n-0000ca40: 653a 2073 6b65 7463 682e 696e 666f 2c20 e: sketch.info, \n-0000ca50: 204e 6f64 653a 2054 696b 5a2f 5047 4620 Node: TikZ/PGF \n-0000ca60: 7573 6572 2d64 6566 696e 6564 2073 7479 user-defined sty\n-0000ca70: 6c65 732c 2020 4e65 7874 3a20 5472 616e les, Next: Tran\n-0000ca80: 7370 6172 656e 6379 2c20 2050 7265 763a sparency, Prev:\n-0000ca90: 2044 6f74 7320 696e 2054 696b 5a2f 5047 Dots in TikZ/PG\n-0000caa0: 462c 2020 5570 3a20 4f70 7469 6f6e 730a F, Up: Options.\n-0000cab0: 0a33 2e31 2e35 2e34 20e2 8098 5469 6b5a .3.1.5.4 ...TikZ\n-0000cac0: 2f50 4746 e280 9920 7573 6572 2d64 6566 /PGF... user-def\n-0000cad0: 696e 6564 2073 7479 6c65 730a 2e2e 2e2e ined styles.....\n+0000c880: 2e2e 2e2e 2e0a 0ae2 8098 5469 6b5a 2f50 ..........TikZ/P\n+0000c890: 4746 e280 9920 646f 6573 206e 6f74 2068 GF... does not h\n+0000c8a0: 6176 6520 6120 e280 9864 6f74 73e2 8099 ave a ...dots...\n+0000c8b0: 2063 6f6d 6d61 6e64 2061 7320 646f 6573 command as does\n+0000c8c0: 2050 5354 7269 636b 732e 2020 496e 7374 PSTricks. Inst\n+0000c8d0: 6561 642c 0ae2 8098 536b 6574 6368 e280 ead,....Sketch..\n+0000c8e0: 9920 656d 6974 7320 646f 7473 2061 7320 . emits dots as \n+0000c8f0: e280 9866 696c 6c64 7261 77e2 8099 2063 ...filldraw... c\n+0000c900: 6972 636c 6573 2e20 2054 6865 2064 6961 ircles. The dia\n+0000c910: 6d65 7465 7220 6d61 7920 6265 2073 6574 meter may be set\n+0000c920: 0a75 7369 6e67 2074 6865 206f 7074 696f .using the optio\n+0000c930: 6e20 e280 9864 6f74 7369 7a65 e280 9920 n ...dotsize... \n+0000c940: 626f 7272 6f77 6564 2066 726f 6d20 5053 borrowed from PS\n+0000c950: 5472 6963 6b73 2e20 2054 6865 20e2 8098 Tricks. The ...\n+0000c960: 646f 7473 697a 65e2 8099 206f 7074 696f dotsize... optio\n+0000c970: 6e0a 7769 6c6c 2062 6520 7265 6d6f 7665 n.will be remove\n+0000c980: 6420 6672 6f6d 2074 6865 206f 7074 696f d from the optio\n+0000c990: 6e20 6c69 7374 2069 6e20 7468 6520 6f75 n list in the ou\n+0000c9a0: 7470 7574 20e2 8098 6669 6c6c 6472 6177 tput ...filldraw\n+0000c9b0: e280 9920 636f 6d6d 616e 642e 0a4f 7468 ... command..Oth\n+0000c9c0: 6572 206f 7074 696f 6e73 2077 6f72 6b20 er options work \n+0000c9d0: 696e 2074 6865 2065 7870 6563 7465 6420 in the expected \n+0000c9e0: 7761 792e 2020 466f 7220 6578 616d 706c way. For exampl\n+0000c9f0: 652c 20e2 8098 6669 6c6c e280 9920 7365 e, ...fill... se\n+0000ca00: 7473 2066 696c 6c0a 636f 6c6f 7220 616e ts fill.color an\n+0000ca10: 6420 e280 9863 6f6c 6f72 e280 9920 7365 d ...color... se\n+0000ca20: 7473 206c 696e 6520 636f 6c6f 7220 6f66 ts line color of\n+0000ca30: 2074 6865 2063 6972 636c 6573 2e0a 0a1f the circles....\n+0000ca40: 0a46 696c 653a 2073 6b65 7463 682e 696e .File: sketch.in\n+0000ca50: 666f 2c20 204e 6f64 653a 2054 696b 5a2f fo, Node: TikZ/\n+0000ca60: 5047 4620 7573 6572 2d64 6566 696e 6564 PGF user-defined\n+0000ca70: 2073 7479 6c65 732c 2020 4e65 7874 3a20 styles, Next: \n+0000ca80: 5472 616e 7370 6172 656e 6379 2c20 2050 Transparency, P\n+0000ca90: 7265 763a 2044 6f74 7320 696e 2054 696b rev: Dots in Tik\n+0000caa0: 5a2f 5047 462c 2020 5570 3a20 4f70 7469 Z/PGF, Up: Opti\n+0000cab0: 6f6e 730a 0a33 2e31 2e35 2e34 20e2 8098 ons..3.1.5.4 ...\n+0000cac0: 5469 6b5a 2f50 4746 e280 9920 7573 6572 TikZ/PGF... user\n+0000cad0: 2d64 6566 696e 6564 2073 7479 6c65 730a -defined styles.\n 0000cae0: 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e ................\n 0000caf0: 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e ................\n-0000cb00: 2e2e 0a0a e280 9854 696b 5a2f 5047 46e2 .......TikZ/PGF.\n-0000cb10: 8099 2061 6c6c 6f77 7320 6e61 6d65 6420 .. allows named \n-0000cb20: 7374 796c 6573 2064 6566 696e 6564 2062 styles defined b\n-0000cb30: 7920 7468 6520 7573 6572 2c20 666f 7220 y the user, for \n-0000cb40: 6578 616d 706c 650a 2020 5c74 696b 7a73 example. \\tikzs\n-0000cb50: 7479 6c65 7b6d 7970 6f6c 7967 6f6e 7374 tyle{mypolygonst\n-0000cb60: 796c 657d 203d 205b 6669 6c6c 3d62 6c75 yle} = [fill=blu\n-0000cb70: 6521 3230 2c66 696c 6c20 6f70 6163 6974 e!20,fill opacit\n-0000cb80: 793d 302e 385d 2020 0a20 205c 7469 6b7a y=0.8] . \\tikz\n-0000cb90: 7374 796c 657b 6d79 6c69 6e65 7374 796c style{mylinestyl\n-0000cba0: 657d 203d 205b 7265 6421 3230 2c64 6173 e} = [red!20,das\n-0000cbb0: 6865 645d 0a53 696e 6365 20e2 8098 736b hed].Since ...sk\n-0000cbc0: 6574 6368 e280 9920 6861 7320 6e6f 2069 etch... has no i\n-0000cbd0: 6e66 6f72 6d61 7469 6f6e 206f 6e20 7468 nformation on th\n-0000cbe0: 6520 636f 6e74 656e 7473 206f 6620 7375 e contents of su\n-0000cbf0: 6368 2073 7479 6c65 732c 2069 740a 6f6d ch styles, it.om\n-0000cc00: 6974 7320 7468 656d 2065 6e74 6972 656c its them entirel\n-0000cc10: 7920 6672 6f6d 206c 696e 6573 2c20 706f y from lines, po\n-0000cc20: 6c79 676f 6e73 2c20 616e 6420 7468 6569 lygons, and thei\n-0000cc30: 7220 6564 6765 7320 6475 7269 6e67 206f r edges during o\n-0000cc40: 7074 696f 6e0a 7370 6c69 7474 696e 672e ption.splitting.\n-0000cc50: 2020 466f 7220 6578 616d 706c 652c 0a20 For example,. \n-0000cc60: 2070 6f6c 7967 6f6e 5b73 7479 6c65 3d6d polygon[style=m\n-0000cc70: 7970 6f6c 7967 6f6e 7374 796c 652c 7374 ypolygonstyle,st\n-0000cc80: 796c 653d 7468 6963 6b5d 2830 2c30 2c31 yle=thick](0,0,1\n-0000cc90: 2928 312c 302c 3029 2830 2c31 2c30 290a )(1,0,0)(0,1,0).\n-0000cca0: 2020 6c69 6e65 5b73 7479 6c65 3d6d 796c line[style=myl\n-0000ccb0: 696e 6573 7479 6c65 5d28 2d31 2c2d 312c inestyle](-1,-1,\n-0000ccc0: 2d31 2928 322c 322c 3229 0a70 726f 6475 -1)(2,2,2).produ\n-0000ccd0: 6365 7320 7468 6520 e280 9854 696b 5ae2 ces the ...TikZ.\n-0000cce0: 8099 206f 7574 7075 740a 2020 5c64 7261 .. output. \\dra\n-0000ccf0: 7728 2d31 2c2d 3129 2d2d 282e 3333 332c w(-1,-1)--(.333,\n-0000cd00: 2e33 3333 293b 0a20 205c 6669 6c6c 6472 .333);. \\filldr\n-0000cd10: 6177 5b74 6869 636b 2c66 696c 6c3d 7768 aw[thick,fill=wh\n-0000cd20: 6974 655d 2830 2c30 292d 2d28 312c 3029 ite](0,0)--(1,0)\n-0000cd30: 2d2d 2830 2c31 292d 2d63 7963 6c65 3b0a --(0,1)--cycle;.\n-0000cd40: 2020 5c64 7261 7728 2e33 3333 2c2e 3333 \\draw(.333,.33\n-0000cd50: 3329 2d2d 2832 2c32 293b 0a4e 6f74 6520 3)--(2,2);.Note \n-0000cd60: 7468 6174 2074 6865 2075 7365 722d 6465 that the user-de\n-0000cd70: 6669 6e65 6420 7374 796c 6573 2061 7265 fined styles are\n-0000cd80: 206e 6f74 2070 7265 7365 6e74 2e20 2053 not present. S\n-0000cd90: 6b65 7463 6820 616c 736f 2069 7373 7565 ketch also issue\n-0000cda0: 730a 7761 726e 696e 6773 3a0a 2020 7761 s.warnings:. wa\n-0000cdb0: 726e 696e 672c 2075 6e6b 6e6f 776e 2070 rning, unknown p\n-0000cdc0: 6f6c 7967 6f6e 206f 7074 696f 6e20 7374 olygon option st\n-0000cdd0: 796c 653d 6d79 706f 6c79 676f 6e73 7479 yle=mypolygonsty\n-0000cde0: 6c65 2077 696c 6c20 6265 2069 676e 6f72 le will be ignor\n-0000cdf0: 6564 0a20 2077 6172 6e69 6e67 2c20 756e ed. warning, un\n-0000ce00: 6b6e 6f77 6e20 6c69 6e65 206f 7074 696f known line optio\n-0000ce10: 6e20 7374 796c 653d 6d79 6c69 6e65 7374 n style=mylinest\n-0000ce20: 796c 6520 7769 6c6c 2062 6520 6967 6e6f yle will be igno\n-0000ce30: 7265 640a 0a20 2020 5468 6520 7265 6d65 red.. The reme\n-0000ce40: 6479 2069 7320 746f 2073 7461 7465 2065 dy is to state e\n-0000ce50: 7870 6c69 6369 746c 7920 7768 6574 6865 xplicitly whethe\n-0000ce60: 7220 6120 7573 6572 2d64 6566 696e 6564 r a user-defined\n-0000ce70: 2073 7479 6c65 2073 686f 756c 640a 6265 style should.be\n-0000ce80: 2061 7474 6368 6564 2074 6f20 706f 6c79 attched to poly\n-0000ce90: 676f 6e73 206f 7220 6c69 6e65 7320 696e gons or lines in\n-0000cea0: 2074 6865 20e2 8098 5469 6b5a e280 9920 the ...TikZ... \n-0000ceb0: 6f75 7470 7574 2075 7369 6e67 0a5f 7073 output using._ps\n-0000cec0: 6575 646f 2d6f 7074 696f 6e73 5f20 e280 eudo-options_ ..\n-0000ced0: 9866 696c 6c20 7374 796c 65e2 8099 2061 .fill style... a\n-0000cee0: 6e64 20e2 8098 6c69 6e65 2073 7479 6c65 nd ...line style\n-0000cef0: e280 992c 0a20 2070 6f6c 7967 6f6e 5b66 ...,. polygon[f\n-0000cf00: 696c 6c20 7374 796c 653d 6d79 706f 6c79 ill style=mypoly\n-0000cf10: 676f 6e73 7479 6c65 2c73 7479 6c65 3d74 gonstyle,style=t\n-0000cf20: 6869 636b 5d28 302c 302c 3129 2831 2c30 hick](0,0,1)(1,0\n-0000cf30: 2c30 2928 302c 312c 3029 0a20 206c 696e ,0)(0,1,0). lin\n-0000cf40: 655b 6c69 6e65 2073 7479 6c65 3d6d 796c e[line style=myl\n-0000cf50: 696e 6573 7479 6c65 5d28 2d31 2c2d 312c inestyle](-1,-1,\n-0000cf60: 2d31 2928 322c 322c 3229 0a4e 6f77 2c20 -1)(2,2,2).Now, \n-0000cf70: 7468 6520 6f75 7470 7574 2069 730a 2020 the output is. \n-0000cf80: 5c64 7261 775b 6d79 6c69 6e65 7374 796c \\draw[mylinestyl\n-0000cf90: 655d 282d 312c 2d31 292d 2d28 2e33 3333 e](-1,-1)--(.333\n-0000cfa0: 2c2e 3333 3329 3b0a 2020 5c66 696c 6c64 ,.333);. \\filld\n-0000cfb0: 7261 775b 6d79 706f 6c79 676f 6e73 7479 raw[mypolygonsty\n-0000cfc0: 6c65 2c74 6869 636b 5d28 302c 3029 2d2d le,thick](0,0)--\n-0000cfd0: 2831 2c30 292d 2d28 302c 3129 2d2d 6379 (1,0)--(0,1)--cy\n-0000cfe0: 636c 653b 0a20 205c 6472 6177 5b6d 796c cle;. \\draw[myl\n-0000cff0: 696e 6573 7479 6c65 5d28 2e33 3333 2c2e inestyle](.333,.\n-0000d000: 3333 3329 2d2d 2832 2c32 293b 0a0a 2020 333)--(2,2);.. \n-0000d010: 2041 2075 7365 6675 6c20 7465 6368 6e69 A useful techni\n-0000d020: 7175 6520 6973 2074 6f20 696e 636c 7564 que is to includ\n-0000d030: 6520 7573 6572 2d64 6566 696e 6564 2073 e user-defined s\n-0000d040: 7479 6c65 2064 6566 696e 6974 696f 6e73 tyle definitions\n-0000d050: 2069 6e0a e280 9873 6b65 7463 68e2 8099 in....sketch...\n-0000d060: 2063 6f64 6520 6173 20e2 8098 7370 6563 code as ...spec\n-0000d070: 6961 6ce2 8099 7320 7769 7468 206f 7074 ial...s with opt\n-0000d080: 696f 6e20 e280 985b 6c61 793d 756e 6465 ion ...[lay=unde\n-0000d090: 725d e280 9920 746f 2065 6e73 7572 6520 r]... to ensure \n-0000d0a0: 7468 6174 2074 6865 0a73 7479 6c65 7320 that the.styles \n-0000d0b0: 6172 6520 656d 6974 7465 6420 6669 7273 are emitted firs\n-0000d0c0: 7420 696e 2074 6865 206f 7574 7075 742c t in the output,\n-0000d0d0: 2062 6566 6f72 6520 616e 7920 7573 6573 before any uses\n-0000d0e0: 206f 6620 7468 6520 7374 796c 650a 6e61 of the style.na\n-0000d0f0: 6d65 732e 2020 2831 2920 2046 6f72 2065 mes. (1) For e\n-0000d100: 7861 6d70 6c65 2c0a 2020 7370 6563 6961 xample,. specia\n-0000d110: 6c7c 5c74 696b 7a73 7479 6c65 7b6d 7970 l|\\tikzstyle{myp\n-0000d120: 6f6c 7967 6f6e 7374 796c 657d 203d 205b olygonstyle} = [\n-0000d130: 6669 6c6c 3d62 6c75 6521 3230 2c66 696c fill=blue!20,fil\n-0000d140: 6c20 6f70 6163 6974 793d 302e 385d 7c5b l opacity=0.8]|[\n-0000d150: 6c61 793d 756e 6465 725d 0a20 2073 7065 lay=under]. spe\n-0000d160: 6369 616c 7c5c 7469 6b7a 7374 796c 657b cial|\\tikzstyle{\n-0000d170: 6d79 6c69 6e65 7374 796c 657d 203d 205b mylinestyle} = [\n-0000d180: 7265 6421 3230 2c64 6173 6865 645d 7c5b red!20,dashed]|[\n-0000d190: 6c61 793d 756e 6465 725d 0a54 6865 2061 lay=under].The a\n-0000d1a0: 7574 686f 7220 6973 2072 6573 706f 6e73 uthor is respons\n-0000d1b0: 6962 6c65 2066 6f72 2075 7369 6e67 2074 ible for using t\n-0000d1c0: 6865 206b 6579 2c20 e280 986c 696e 6520 he key, ...line \n-0000d1d0: 7374 796c 65e2 8099 206f 7220 e280 9866 style... or ...f\n-0000d1e0: 696c 6c0a 7374 796c 65e2 8099 2c20 7468 ill.style..., th\n-0000d1f0: 6174 206d 6174 6368 6573 2074 6865 2063 at matches the c\n-0000d200: 6f6e 7465 6e74 206f 6620 7468 6520 7374 ontent of the st\n-0000d210: 796c 6520 6465 6669 6e69 7469 6f6e 2e0a yle definition..\n-0000d220: 0a20 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2046 . ---------- F\n-0000d230: 6f6f 746e 6f74 6573 202d 2d2d 2d2d 2d2d ootnotes -------\n-0000d240: 2d2d 2d0a 0a20 2020 2831 2920 5468 6973 ---.. (1) This\n-0000d250: 2063 6c65 7665 7220 7472 6963 6b20 6973 clever trick is\n-0000d260: 2064 7565 2074 6f20 4b6a 656c 6c20 4d61 due to Kjell Ma\n-0000d270: 676e 6520 4661 7573 6b65 2e0a 0a1f 0a46 gne Fauske.....F\n-0000d280: 696c 653a 2073 6b65 7463 682e 696e 666f ile: sketch.info\n-0000d290: 2c20 204e 6f64 653a 2054 7261 6e73 7061 , Node: Transpa\n-0000d2a0: 7265 6e63 792c 2020 4e65 7874 3a20 496e rency, Next: In\n-0000d2b0: 7465 726e 616c 206f 7074 696f 6e73 2c20 ternal options, \n-0000d2c0: 2050 7265 763a 2054 696b 5a2f 5047 4620 Prev: TikZ/PGF \n-0000d2d0: 7573 6572 2d64 6566 696e 6564 2073 7479 user-defined sty\n-0000d2e0: 6c65 732c 2020 5570 3a20 4f70 7469 6f6e les, Up: Option\n-0000d2f0: 730a 0a33 2e31 2e35 2e35 2054 7261 6e73 s..3.1.5.5 Trans\n-0000d300: 7061 7265 6e63 790a 2e2e 2e2e 2e2e 2e2e parency.........\n-0000d310: 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 0a0a 426f ..............Bo\n-0000d320: 7468 20e2 8098 5053 5472 6963 6b73 e280 th ...PSTricks..\n-0000d330: 9920 616e 6420 e280 9854 696b 5a2f 5047 . and ...TikZ/PG\n-0000d340: 46e2 8099 2073 7570 706f 7274 2070 6f6c F... support pol\n-0000d350: 7967 6f6e 206f 7074 696f 6e73 2074 6861 ygon options tha\n-0000d360: 7420 6861 7665 2074 6865 0a65 6666 6563 t have the.effec\n-0000d370: 7420 6f66 206d 616b 696e 6720 7468 6520 t of making the \n-0000d380: 706f 6c79 676f 6e20 6170 7065 6172 2074 polygon appear t\n-0000d390: 7261 6e73 7061 7265 6e74 2e20 2046 6f72 ransparent. For\n-0000d3a0: 20e2 8098 5053 5472 6963 6b73 e280 992c ...PSTricks...,\n-0000d3b0: 0a6b 6579 776f 7264 20e2 8098 7472 616e .keyword ...tran\n-0000d3c0: 7370 616c 7068 61e2 8099 2077 6173 2075 spalpha... was u\n-0000d3d0: 7365 6420 6475 7269 6e67 2069 6e69 7469 sed during initi\n-0000d3e0: 616c 2064 6576 656c 6f70 6d65 6e74 206f al development o\n-0000d3f0: 660a 7472 616e 7370 6172 656e 6379 2066 f.transparency f\n-0000d400: 6561 7475 7265 732c 2061 6e64 20e2 8098 eatures, and ...\n-0000d410: 6f70 6163 6974 79e2 8099 2077 6173 2061 opacity... was a\n-0000d420: 646f 7074 6564 206c 6174 6572 2e20 20e2 dopted later. .\n-0000d430: 8098 536b 6574 6368 e280 9920 686f 6e6f ..Sketch... hono\n-0000d440: 7273 0a62 6f74 682e 2020 e280 9854 696b rs.both. ...Tik\n-0000d450: 5a2f 5047 46e2 8099 2075 7365 7320 e280 Z/PGF... uses ..\n-0000d460: 986f 7061 6369 7479 e280 9920 6f6e 6c79 .opacity... only\n-0000d470: 2e20 2057 6865 6e20 7472 616e 7370 6172 . When transpar\n-0000d480: 656e 7420 706f 6c79 676f 6e73 2061 7265 ent polygons are\n-0000d490: 2069 6e0a 7468 6520 666f 7265 6772 6f75 in.the foregrou\n-0000d4a0: 6e64 2c20 6f62 6a65 6374 7320 6265 6869 nd, objects behi\n-0000d4b0: 6e64 2074 6865 6d20 2864 7261 776e 2065 nd them (drawn e\n-0000d4c0: 6172 6c69 6572 2920 6172 6520 7669 7369 arlier) are visi\n-0000d4d0: 626c 6520 7769 7468 0a63 6f6c 6f72 2073 ble with.color s\n-0000d4e0: 7562 6475 6564 2061 6e64 2074 696e 7465 ubdued and tinte\n-0000d4f0: 642e 2020 5468 6520 6869 6464 656e 2073 d. The hidden s\n-0000d500: 7572 6661 6365 2061 6c67 6f72 6974 686d urface algorithm\n-0000d510: 206f 6620 e280 9873 6b65 7463 68e2 8099 of ...sketch...\n-0000d520: 0a77 6f72 6b73 2077 656c 6c20 7769 7468 .works well with\n-0000d530: 2073 7563 6820 7472 616e 7370 6172 656e such transparen\n-0000d540: 7420 706f 6c79 676f 6e73 2e0a 0a20 2020 t polygons... \n-0000d550: 4e6f 7465 2074 6861 7420 6375 6c6c 3d66 Note that cull=f\n-0000d560: 616c 7365 206d 7573 7420 6265 2075 7365 alse must be use\n-0000d570: 6420 666f 7220 7265 6172 2d66 6163 696e d for rear-facin\n-0000d580: 6720 706f 6c79 676f 6e73 2074 6f20 6265 g polygons to be\n-0000d590: 0a76 6973 6962 6c65 2077 6865 6e20 706f .visible when po\n-0000d5a0: 7369 7469 6f6e 6564 2062 6568 696e 6420 sitioned behind \n-0000d5b0: 6f74 6865 7220 7472 616e 7370 6172 656e other transparen\n-0000d5c0: 7420 7375 7266 6163 6573 2e0a 0a1f 0a46 t surfaces.....F\n-0000d5d0: 696c 653a 2073 6b65 7463 682e 696e 666f ile: sketch.info\n-0000d5e0: 2c20 204e 6f64 653a 2049 6e74 6572 6e61 , Node: Interna\n-0000d5f0: 6c20 6f70 7469 6f6e 732c 2020 5072 6576 l options, Prev\n-0000d600: 3a20 5472 616e 7370 6172 656e 6379 2c20 : Transparency, \n-0000d610: 2055 703a 204f 7074 696f 6e73 0a0a 332e Up: Options..3.\n-0000d620: 312e 352e 3620 496e 7465 726e 616c 206f 1.5.6 Internal o\n-0000d630: 7074 696f 6e73 0a2e 2e2e 2e2e 2e2e 2e2e ptions..........\n-0000d640: 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e0a ................\n-0000d650: 0a54 6865 7265 2061 7265 2061 6c73 6f20 .There are also \n-0000d660: 5f69 6e74 6572 6e61 6c5f 206f 7074 696f _internal_ optio\n-0000d670: 6e73 2075 7365 6420 6f6e 6c79 2062 7920 ns used only by \n-0000d680: e280 9873 6b65 7463 68e2 8099 2061 6e64 ...sketch... and\n-0000d690: 206e 6f74 2070 6173 7365 640a 6f6e 2074 not passed.on t\n-0000d6a0: 6f20 e280 9850 5354 7269 636b 73e2 8099 o ...PSTricks...\n-0000d6b0: 2e20 2054 6865 7365 2061 7265 2073 756d . These are sum\n-0000d6c0: 6d61 7269 7a65 6420 696e 2074 6865 2066 marized in the f\n-0000d6d0: 6f6c 6c6f 7769 6e67 2074 6162 6c65 2e0a ollowing table..\n-0000d6e0: 4b65 7920 2020 2020 2020 506f 7373 6962 Key Possib\n-0000d6f0: 6c65 2076 616c 7565 7320 2020 2020 2020 le values \n-0000d700: 2020 4465 7363 7269 7074 696f 6e0a 2d2d Description.--\n-0000d710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ----------------\n+0000cb00: 2e2e 2e2e 2e2e 0a0a e280 9854 696b 5a2f ...........TikZ/\n+0000cb10: 5047 46e2 8099 2061 6c6c 6f77 7320 6e61 PGF... allows na\n+0000cb20: 6d65 6420 7374 796c 6573 2064 6566 696e med styles defin\n+0000cb30: 6564 2062 7920 7468 6520 7573 6572 2c20 ed by the user, \n+0000cb40: 666f 7220 6578 616d 706c 650a 2020 5c74 for example. \\t\n+0000cb50: 696b 7a73 7479 6c65 7b6d 7970 6f6c 7967 ikzstyle{mypolyg\n+0000cb60: 6f6e 7374 796c 657d 203d 205b 6669 6c6c onstyle} = [fill\n+0000cb70: 3d62 6c75 6521 3230 2c66 696c 6c20 6f70 =blue!20,fill op\n+0000cb80: 6163 6974 793d 302e 385d 2020 0a20 205c acity=0.8] . \\\n+0000cb90: 7469 6b7a 7374 796c 657b 6d79 6c69 6e65 tikzstyle{myline\n+0000cba0: 7374 796c 657d 203d 205b 7265 6421 3230 style} = [red!20\n+0000cbb0: 2c64 6173 6865 645d 0a53 696e 6365 20e2 ,dashed].Since .\n+0000cbc0: 8098 736b 6574 6368 e280 9920 6861 7320 ..sketch... has \n+0000cbd0: 6e6f 2069 6e66 6f72 6d61 7469 6f6e 206f no information o\n+0000cbe0: 6e20 7468 6520 636f 6e74 656e 7473 206f n the contents o\n+0000cbf0: 6620 7375 6368 2073 7479 6c65 732c 2069 f such styles, i\n+0000cc00: 740a 6f6d 6974 7320 7468 656d 2065 6e74 t.omits them ent\n+0000cc10: 6972 656c 7920 6672 6f6d 206c 696e 6573 irely from lines\n+0000cc20: 2c20 706f 6c79 676f 6e73 2c20 616e 6420 , polygons, and \n+0000cc30: 7468 6569 7220 6564 6765 7320 6475 7269 their edges duri\n+0000cc40: 6e67 206f 7074 696f 6e0a 7370 6c69 7474 ng option.splitt\n+0000cc50: 696e 672e 2020 466f 7220 6578 616d 706c ing. For exampl\n+0000cc60: 652c 0a20 2070 6f6c 7967 6f6e 5b73 7479 e,. polygon[sty\n+0000cc70: 6c65 3d6d 7970 6f6c 7967 6f6e 7374 796c le=mypolygonstyl\n+0000cc80: 652c 7374 796c 653d 7468 6963 6b5d 2830 e,style=thick](0\n+0000cc90: 2c30 2c31 2928 312c 302c 3029 2830 2c31 ,0,1)(1,0,0)(0,1\n+0000cca0: 2c30 290a 2020 6c69 6e65 5b73 7479 6c65 ,0). line[style\n+0000ccb0: 3d6d 796c 696e 6573 7479 6c65 5d28 2d31 =mylinestyle](-1\n+0000ccc0: 2c2d 312c 2d31 2928 322c 322c 3229 0a70 ,-1,-1)(2,2,2).p\n+0000ccd0: 726f 6475 6365 7320 7468 6520 e280 9854 roduces the ...T\n+0000cce0: 696b 5ae2 8099 206f 7574 7075 740a 2020 ikZ... output. \n+0000ccf0: 5c64 7261 7728 2d31 2c2d 3129 2d2d 282e \\draw(-1,-1)--(.\n+0000cd00: 3333 332c 2e33 3333 293b 0a20 205c 6669 333,.333);. \\fi\n+0000cd10: 6c6c 6472 6177 5b74 6869 636b 2c66 696c lldraw[thick,fil\n+0000cd20: 6c3d 7768 6974 655d 2830 2c30 292d 2d28 l=white](0,0)--(\n+0000cd30: 312c 3029 2d2d 2830 2c31 292d 2d63 7963 1,0)--(0,1)--cyc\n+0000cd40: 6c65 3b0a 2020 5c64 7261 7728 2e33 3333 le;. \\draw(.333\n+0000cd50: 2c2e 3333 3329 2d2d 2832 2c32 293b 0a4e ,.333)--(2,2);.N\n+0000cd60: 6f74 6520 7468 6174 2074 6865 2075 7365 ote that the use\n+0000cd70: 722d 6465 6669 6e65 6420 7374 796c 6573 r-defined styles\n+0000cd80: 2061 7265 206e 6f74 2070 7265 7365 6e74 are not present\n+0000cd90: 2e20 2053 6b65 7463 6820 616c 736f 2069 . Sketch also i\n+0000cda0: 7373 7565 730a 7761 726e 696e 6773 3a0a ssues.warnings:.\n+0000cdb0: 2020 7761 726e 696e 672c 2075 6e6b 6e6f warning, unkno\n+0000cdc0: 776e 2070 6f6c 7967 6f6e 206f 7074 696f wn polygon optio\n+0000cdd0: 6e20 7374 796c 653d 6d79 706f 6c79 676f n style=mypolygo\n+0000cde0: 6e73 7479 6c65 2077 696c 6c20 6265 2069 nstyle will be i\n+0000cdf0: 676e 6f72 6564 0a20 2077 6172 6e69 6e67 gnored. warning\n+0000ce00: 2c20 756e 6b6e 6f77 6e20 6c69 6e65 206f , unknown line o\n+0000ce10: 7074 696f 6e20 7374 796c 653d 6d79 6c69 ption style=myli\n+0000ce20: 6e65 7374 796c 6520 7769 6c6c 2062 6520 nestyle will be \n+0000ce30: 6967 6e6f 7265 640a 0a20 2020 5468 6520 ignored.. The \n+0000ce40: 7265 6d65 6479 2069 7320 746f 2073 7461 remedy is to sta\n+0000ce50: 7465 2065 7870 6c69 6369 746c 7920 7768 te explicitly wh\n+0000ce60: 6574 6865 7220 6120 7573 6572 2d64 6566 ether a user-def\n+0000ce70: 696e 6564 2073 7479 6c65 2073 686f 756c ined style shoul\n+0000ce80: 640a 6265 2061 7474 6368 6564 2074 6f20 d.be attched to \n+0000ce90: 706f 6c79 676f 6e73 206f 7220 6c69 6e65 polygons or line\n+0000cea0: 7320 696e 2074 6865 20e2 8098 5469 6b5a s in the ...TikZ\n+0000ceb0: e280 9920 6f75 7470 7574 2075 7369 6e67 ... output using\n+0000cec0: 0a5f 7073 6575 646f 2d6f 7074 696f 6e73 ._pseudo-options\n+0000ced0: 5f20 e280 9866 696c 6c20 7374 796c 65e2 _ ...fill style.\n+0000cee0: 8099 2061 6e64 20e2 8098 6c69 6e65 2073 .. and ...line s\n+0000cef0: 7479 6c65 e280 992c 0a20 2070 6f6c 7967 tyle...,. polyg\n+0000cf00: 6f6e 5b66 696c 6c20 7374 796c 653d 6d79 on[fill style=my\n+0000cf10: 706f 6c79 676f 6e73 7479 6c65 2c73 7479 polygonstyle,sty\n+0000cf20: 6c65 3d74 6869 636b 5d28 302c 302c 3129 le=thick](0,0,1)\n+0000cf30: 2831 2c30 2c30 2928 302c 312c 3029 0a20 (1,0,0)(0,1,0). \n+0000cf40: 206c 696e 655b 6c69 6e65 2073 7479 6c65 line[line style\n+0000cf50: 3d6d 796c 696e 6573 7479 6c65 5d28 2d31 =mylinestyle](-1\n+0000cf60: 2c2d 312c 2d31 2928 322c 322c 3229 0a4e ,-1,-1)(2,2,2).N\n+0000cf70: 6f77 2c20 7468 6520 6f75 7470 7574 2069 ow, the output i\n+0000cf80: 730a 2020 5c64 7261 775b 6d79 6c69 6e65 s. \\draw[myline\n+0000cf90: 7374 796c 655d 282d 312c 2d31 292d 2d28 style](-1,-1)--(\n+0000cfa0: 2e33 3333 2c2e 3333 3329 3b0a 2020 5c66 .333,.333);. \\f\n+0000cfb0: 696c 6c64 7261 775b 6d79 706f 6c79 676f illdraw[mypolygo\n+0000cfc0: 6e73 7479 6c65 2c74 6869 636b 5d28 302c nstyle,thick](0,\n+0000cfd0: 3029 2d2d 2831 2c30 292d 2d28 302c 3129 0)--(1,0)--(0,1)\n+0000cfe0: 2d2d 6379 636c 653b 0a20 205c 6472 6177 --cycle;. \\draw\n+0000cff0: 5b6d 796c 696e 6573 7479 6c65 5d28 2e33 [mylinestyle](.3\n+0000d000: 3333 2c2e 3333 3329 2d2d 2832 2c32 293b 33,.333)--(2,2);\n+0000d010: 0a0a 2020 2041 2075 7365 6675 6c20 7465 .. A useful te\n+0000d020: 6368 6e69 7175 6520 6973 2074 6f20 696e chnique is to in\n+0000d030: 636c 7564 6520 7573 6572 2d64 6566 696e clude user-defin\n+0000d040: 6564 2073 7479 6c65 2064 6566 696e 6974 ed style definit\n+0000d050: 696f 6e73 2069 6e0a e280 9873 6b65 7463 ions in....sketc\n+0000d060: 68e2 8099 2063 6f64 6520 6173 20e2 8098 h... code as ...\n+0000d070: 7370 6563 6961 6ce2 8099 7320 7769 7468 special...s with\n+0000d080: 206f 7074 696f 6e20 e280 985b 6c61 793d option ...[lay=\n+0000d090: 756e 6465 725d e280 9920 746f 2065 6e73 under]... to ens\n+0000d0a0: 7572 6520 7468 6174 2074 6865 0a73 7479 ure that the.sty\n+0000d0b0: 6c65 7320 6172 6520 656d 6974 7465 6420 les are emitted \n+0000d0c0: 6669 7273 7420 696e 2074 6865 206f 7574 first in the out\n+0000d0d0: 7075 742c 2062 6566 6f72 6520 616e 7920 put, before any \n+0000d0e0: 7573 6573 206f 6620 7468 6520 7374 796c uses of the styl\n+0000d0f0: 650a 6e61 6d65 732e 2020 2831 2920 2046 e.names. (1) F\n+0000d100: 6f72 2065 7861 6d70 6c65 2c0a 2020 7370 or example,. sp\n+0000d110: 6563 6961 6c7c 5c74 696b 7a73 7479 6c65 ecial|\\tikzstyle\n+0000d120: 7b6d 7970 6f6c 7967 6f6e 7374 796c 657d {mypolygonstyle}\n+0000d130: 203d 205b 6669 6c6c 3d62 6c75 6521 3230 = [fill=blue!20\n+0000d140: 2c66 696c 6c20 6f70 6163 6974 793d 302e ,fill opacity=0.\n+0000d150: 385d 7c5b 6c61 793d 756e 6465 725d 0a20 8]|[lay=under]. \n+0000d160: 2073 7065 6369 616c 7c5c 7469 6b7a 7374 special|\\tikzst\n+0000d170: 796c 657b 6d79 6c69 6e65 7374 796c 657d yle{mylinestyle}\n+0000d180: 203d 205b 7265 6421 3230 2c64 6173 6865 = [red!20,dashe\n+0000d190: 645d 7c5b 6c61 793d 756e 6465 725d 0a54 d]|[lay=under].T\n+0000d1a0: 6865 2061 7574 686f 7220 6973 2072 6573 he author is res\n+0000d1b0: 706f 6e73 6962 6c65 2066 6f72 2075 7369 ponsible for usi\n+0000d1c0: 6e67 2074 6865 206b 6579 2c20 e280 986c ng the key, ...l\n+0000d1d0: 696e 6520 7374 796c 65e2 8099 206f 7220 ine style... or \n+0000d1e0: e280 9866 696c 6c0a 7374 796c 65e2 8099 ...fill.style...\n+0000d1f0: 2c20 7468 6174 206d 6174 6368 6573 2074 , that matches t\n+0000d200: 6865 2063 6f6e 7465 6e74 206f 6620 7468 he content of th\n+0000d210: 6520 7374 796c 6520 6465 6669 6e69 7469 e style definiti\n+0000d220: 6f6e 2e0a 0a20 2020 2d2d 2d2d 2d2d 2d2d on... --------\n+0000d230: 2d2d 2046 6f6f 746e 6f74 6573 202d 2d2d -- Footnotes ---\n+0000d240: 2d2d 2d2d 2d2d 2d0a 0a20 2020 2831 2920 -------.. (1) \n+0000d250: 5468 6973 2063 6c65 7665 7220 7472 6963 This clever tric\n+0000d260: 6b20 6973 2064 7565 2074 6f20 4b6a 656c k is due to Kjel\n+0000d270: 6c20 4d61 676e 6520 4661 7573 6b65 2e0a l Magne Fauske..\n+0000d280: 0a1f 0a46 696c 653a 2073 6b65 7463 682e ...File: sketch.\n+0000d290: 696e 666f 2c20 204e 6f64 653a 2054 7261 info, Node: Tra\n+0000d2a0: 6e73 7061 7265 6e63 792c 2020 4e65 7874 nsparency, Next\n+0000d2b0: 3a20 496e 7465 726e 616c 206f 7074 696f : Internal optio\n+0000d2c0: 6e73 2c20 2050 7265 763a 2054 696b 5a2f ns, Prev: TikZ/\n+0000d2d0: 5047 4620 7573 6572 2d64 6566 696e 6564 PGF user-defined\n+0000d2e0: 2073 7479 6c65 732c 2020 5570 3a20 4f70 styles, Up: Op\n+0000d2f0: 7469 6f6e 730a 0a33 2e31 2e35 2e35 2054 tions..3.1.5.5 T\n+0000d300: 7261 6e73 7061 7265 6e63 790a 2e2e 2e2e ransparency.....\n+0000d310: 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e ................\n+0000d320: 0a0a 426f 7468 20e2 8098 5053 5472 6963 ..Both ...PSTric\n+0000d330: 6b73 e280 9920 616e 6420 e280 9854 696b ks... and ...Tik\n+0000d340: 5a2f 5047 46e2 8099 2073 7570 706f 7274 Z/PGF... support\n+0000d350: 2070 6f6c 7967 6f6e 206f 7074 696f 6e73 polygon options\n+0000d360: 2074 6861 7420 6861 7665 2074 6865 0a65 that have the.e\n+0000d370: 6666 6563 7420 6f66 206d 616b 696e 6720 ffect of making \n+0000d380: 7468 6520 706f 6c79 676f 6e20 6170 7065 the polygon appe\n+0000d390: 6172 2074 7261 6e73 7061 7265 6e74 2e20 ar transparent. \n+0000d3a0: 2046 6f72 20e2 8098 5053 5472 6963 6b73 For ...PSTricks\n+0000d3b0: e280 992c 0a6b 6579 776f 7264 20e2 8098 ...,.keyword ...\n+0000d3c0: 7472 616e 7370 616c 7068 61e2 8099 2077 transpalpha... w\n+0000d3d0: 6173 2075 7365 6420 6475 7269 6e67 2069 as used during i\n+0000d3e0: 6e69 7469 616c 2064 6576 656c 6f70 6d65 nitial developme\n+0000d3f0: 6e74 206f 660a 7472 616e 7370 6172 656e nt of.transparen\n+0000d400: 6379 2066 6561 7475 7265 732c 2061 6e64 cy features, and\n+0000d410: 20e2 8098 6f70 6163 6974 79e2 8099 2077 ...opacity... w\n+0000d420: 6173 2061 646f 7074 6564 206c 6174 6572 as adopted later\n+0000d430: 2e20 20e2 8098 536b 6574 6368 e280 9920 . ...Sketch... \n+0000d440: 686f 6e6f 7273 0a62 6f74 682e 2020 e280 honors.both. ..\n+0000d450: 9854 696b 5a2f 5047 46e2 8099 2075 7365 .TikZ/PGF... use\n+0000d460: 7320 e280 986f 7061 6369 7479 e280 9920 s ...opacity... \n+0000d470: 6f6e 6c79 2e20 2057 6865 6e20 7472 616e only. When tran\n+0000d480: 7370 6172 656e 7420 706f 6c79 676f 6e73 sparent polygons\n+0000d490: 2061 7265 2069 6e0a 7468 6520 666f 7265 are in.the fore\n+0000d4a0: 6772 6f75 6e64 2c20 6f62 6a65 6374 7320 ground, objects \n+0000d4b0: 6265 6869 6e64 2074 6865 6d20 2864 7261 behind them (dra\n+0000d4c0: 776e 2065 6172 6c69 6572 2920 6172 6520 wn earlier) are \n+0000d4d0: 7669 7369 626c 6520 7769 7468 0a63 6f6c visible with.col\n+0000d4e0: 6f72 2073 7562 6475 6564 2061 6e64 2074 or subdued and t\n+0000d4f0: 696e 7465 642e 2020 5468 6520 6869 6464 inted. The hidd\n+0000d500: 656e 2073 7572 6661 6365 2061 6c67 6f72 en surface algor\n+0000d510: 6974 686d 206f 6620 e280 9873 6b65 7463 ithm of ...sketc\n+0000d520: 68e2 8099 0a77 6f72 6b73 2077 656c 6c20 h....works well \n+0000d530: 7769 7468 2073 7563 6820 7472 616e 7370 with such transp\n+0000d540: 6172 656e 7420 706f 6c79 676f 6e73 2e0a arent polygons..\n+0000d550: 0a20 2020 4e6f 7465 2074 6861 7420 6375 . Note that cu\n+0000d560: 6c6c 3d66 616c 7365 206d 7573 7420 6265 ll=false must be\n+0000d570: 2075 7365 6420 666f 7220 7265 6172 2d66 used for rear-f\n+0000d580: 6163 696e 6720 706f 6c79 676f 6e73 2074 acing polygons t\n+0000d590: 6f20 6265 0a76 6973 6962 6c65 2077 6865 o be.visible whe\n+0000d5a0: 6e20 706f 7369 7469 6f6e 6564 2062 6568 n positioned beh\n+0000d5b0: 696e 6420 6f74 6865 7220 7472 616e 7370 ind other transp\n+0000d5c0: 6172 656e 7420 7375 7266 6163 6573 2e0a arent surfaces..\n+0000d5d0: 0a1f 0a46 696c 653a 2073 6b65 7463 682e ...File: sketch.\n+0000d5e0: 696e 666f 2c20 204e 6f64 653a 2049 6e74 info, Node: Int\n+0000d5f0: 6572 6e61 6c20 6f70 7469 6f6e 732c 2020 ernal options, \n+0000d600: 5072 6576 3a20 5472 616e 7370 6172 656e Prev: Transparen\n+0000d610: 6379 2c20 2055 703a 204f 7074 696f 6e73 cy, Up: Options\n+0000d620: 0a0a 332e 312e 352e 3620 496e 7465 726e ..3.1.5.6 Intern\n+0000d630: 616c 206f 7074 696f 6e73 0a2e 2e2e 2e2e al options......\n+0000d640: 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e ................\n+0000d650: 2e2e 2e0a 0a54 6865 7265 2061 7265 2061 .....There are a\n+0000d660: 6c73 6f20 5f69 6e74 6572 6e61 6c5f 206f lso _internal_ o\n+0000d670: 7074 696f 6e73 2075 7365 6420 6f6e 6c79 ptions used only\n+0000d680: 2062 7920 e280 9873 6b65 7463 68e2 8099 by ...sketch...\n+0000d690: 2061 6e64 206e 6f74 2070 6173 7365 640a and not passed.\n+0000d6a0: 6f6e 2074 6f20 e280 9850 5354 7269 636b on to ...PSTrick\n+0000d6b0: 73e2 8099 2e20 2054 6865 7365 2061 7265 s.... 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Tur\n-0000d7a0: 6e20 6375 6c6c 696e 6720 6f66 2062 6163 n culling of bac\n-0000d7b0: 6b66 6163 6573 206f 6e20 616e 6420 6f66 kfaces on and of\n-0000d7c0: 6620 7265 7370 6563 7469 7665 6c79 2066 f respectively f\n-0000d7d0: 6f72 2074 6869 730a 2020 2020 2020 2020 or this. \n+0000d760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ----------------\n+0000d770: 2d2d 0ae2 8098 6375 6c6c e280 9920 2020 --....cull... \n+0000d780: 20e2 8098 7472 7565 e280 992c 20e2 8098 ...true..., ...\n+0000d790: 6661 6c73 65e2 8099 2020 2020 2020 2020 false... \n+0000d7a0: 2054 7572 6e20 6375 6c6c 696e 6720 6f66 Turn culling of\n+0000d7b0: 2062 6163 6b66 6163 6573 206f 6e20 616e backfaces on an\n+0000d7c0: 6420 6f66 6620 7265 7370 6563 7469 7665 d off respective\n+0000d7d0: 6c79 2066 6f72 2074 6869 730a 2020 2020 ly for this. \n 0000d7e0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0000d7f0: 2020 2020 2020 2020 2020 6f62 6a65 6374 object\n-0000d800: 2e20 2054 6865 2064 6566 6175 6c74 2076 . The default v\n-0000d810: 616c 7565 2069 7320 e280 9874 7275 65e2 alue is ...true.\n-0000d820: 8099 2e0a e280 986c 6179 e280 9920 2020 .......lay... \n-0000d830: 2020 e280 986f 7665 72e2 8099 2c20 e280 ...over..., ..\n-0000d840: 9869 6ee2 8099 2c20 e280 9875 6e64 6572 .in..., ...under\n-0000d850: e280 9920 2020 466f 7263 6520 7468 6973 ... Force this\n-0000d860: 206f 626a 6563 7420 746f 2062 6520 e280 object to be ..\n-0000d870: 9875 6e64 6572 e280 9920 6f72 20e2 8098 .under... or ...\n-0000d880: 6f76 6572 e280 9920 616c 6c20 6f74 6865 over... all othe\n-0000d890: 7220 6f62 6a65 6374 730a 2020 2020 2020 r objects. \n+0000d7f0: 2020 2020 2020 2020 2020 2020 2020 6f62 ob\n+0000d800: 6a65 6374 2e20 2054 6865 2064 6566 6175 ject. The defau\n+0000d810: 6c74 2076 616c 7565 2069 7320 e280 9874 lt value is ...t\n+0000d820: 7275 65e2 8099 2e0a e280 986c 6179 e280 rue........lay..\n+0000d830: 9920 2020 2020 e280 986f 7665 72e2 8099 . ...over...\n+0000d840: 2c20 e280 9869 6ee2 8099 2c20 e280 9875 , ...in..., ...u\n+0000d850: 6e64 6572 e280 9920 2020 466f 7263 6520 nder... Force \n+0000d860: 7468 6973 206f 626a 6563 7420 746f 2062 this object to b\n+0000d870: 6520 e280 9875 6e64 6572 e280 9920 6f72 e ...under... or\n+0000d880: 20e2 8098 6f76 6572 e280 9920 616c 6c20 ...over... all \n+0000d890: 6f74 6865 7220 6f62 6a65 6374 730a 2020 other objects. \n 0000d8a0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0000d8b0: 2020 2020 2020 2020 2020 2020 696e 2074 in t\n-0000d8c0: 6865 2064 6570 7468 2073 6f72 7420 6f72 he depth sort or\n-0000d8d0: 6465 7220 6372 6561 7465 6420 6279 2074 der created by t\n-0000d8e0: 6865 2068 6964 6465 6e20 7375 7266 6163 he hidden surfac\n-0000d8f0: 650a 2020 2020 2020 2020 2020 2020 2020 e. \n+0000d8b0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0000d8c0: 696e 2074 6865 2064 6570 7468 2073 6f72 in the depth sor\n+0000d8d0: 7420 6f72 6465 7220 6372 6561 7465 6420 t order created \n+0000d8e0: 6279 2074 6865 2068 6964 6465 6e20 7375 by the hidden su\n+0000d8f0: 7266 6163 650a 2020 2020 2020 2020 2020 rface. \n 0000d900: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0000d910: 2020 2020 616c 676f 7269 7468 6d2e 2020 algorithm. \n-0000d920: 5468 6520 6465 6661 756c 7420 7661 6c75 The default valu\n-0000d930: 6520 e280 986f 7665 72e2 8099 2067 7561 e ...over... gua\n-0000d940: 7261 6e74 6565 7320 7468 6174 206f 7574 rantees that out\n-0000d950: 7075 740a 2020 2020 2020 2020 2020 2020 put. \n+0000d910: 2020 2020 2020 2020 616c 676f 7269 7468 algorith\n+0000d920: 6d2e 2020 5468 6520 6465 6661 756c 7420 m. The default \n+0000d930: 7661 6c75 6520 e280 986f 7665 72e2 8099 value ...over...\n+0000d940: 2067 7561 7261 6e74 6565 7320 7468 6174 guarantees that\n+0000d950: 206f 7574 7075 740a 2020 2020 2020 2020 output. \n 0000d960: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0000d970: 2020 2020 2020 6475 6520 746f 2074 6865 due to the\n-0000d980: 20e2 8098 7370 6563 6961 6ce2 8099 2077 ...special... w\n-0000d990: 696c 6c20 6265 2076 6973 6962 6c65 2e0a ill be visible..\n-0000d9a0: e280 9873 706c 6974 e280 9920 2020 e280 ...split... ..\n-0000d9b0: 9874 7275 65e2 8099 2c20 e280 9866 616c .true..., ...fal\n-0000d9c0: 7365 e280 9920 2020 2020 2020 2020 5475 se... Tu\n-0000d9d0: 726e 2073 706c 6974 7469 6e67 206f 6620 rn splitting of \n-0000d9e0: 7377 6565 702d 6765 6e65 7261 7465 6420 sweep-generated \n-0000d9f0: 626f 6479 2070 6f6c 7967 6f6e 7320 6f6e body polygons on\n-0000da00: 2061 6e64 206f 6666 0a20 2020 2020 2020 and off. \n+0000d970: 2020 2020 2020 2020 2020 6475 6520 746f due to\n+0000d980: 2074 6865 20e2 8098 7370 6563 6961 6ce2 the ...special.\n+0000d990: 8099 2077 696c 6c20 6265 2076 6973 6962 .. will be visib\n+0000d9a0: 6c65 2e0a e280 9873 706c 6974 e280 9920 le.....split... \n+0000d9b0: 2020 e280 9874 7275 65e2 8099 2c20 e280 ...true..., ..\n+0000d9c0: 9866 616c 7365 e280 9920 2020 2020 2020 .false... \n+0000d9d0: 2020 5475 726e 2073 706c 6974 7469 6e67 Turn splitting\n+0000d9e0: 206f 6620 7377 6565 702d 6765 6e65 7261 of sweep-genera\n+0000d9f0: 7465 6420 626f 6479 2070 6f6c 7967 6f6e ted body polygon\n+0000da00: 7320 6f6e 2061 6e64 206f 6666 0a20 2020 s on and off. \n 0000da10: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0000da20: 2020 2020 2020 2020 2020 2072 6573 7065 respe\n-0000da30: 6374 6976 656c 792e 2020 2a4e 6f74 6520 ctively. *Note \n-0000da40: 5377 6565 7073 3a3a 2e20 2054 6865 2064 Sweeps::. The d\n-0000da50: 6566 6175 6c74 2076 616c 7565 20e2 8098 efault value ...\n-0000da60: 7472 7565 e280 990a 2020 2020 2020 2020 true.... \n+0000da20: 2020 2020 2020 2020 2020 2020 2020 2072 r\n+0000da30: 6573 7065 6374 6976 656c 792e 2020 2a4e espectively. *N\n+0000da40: 6f74 6520 5377 6565 7073 3a3a 2e20 2054 ote Sweeps::. T\n+0000da50: 6865 2064 6566 6175 6c74 2076 616c 7565 he default value\n+0000da60: 20e2 8098 7472 7565 e280 990a 2020 2020 ...true.... \n 0000da70: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0000da80: 2020 2020 2020 2020 2020 6361 7573 6573 causes\n-0000da90: 2022 7761 7270 6564 2220 706f 6c79 676f \"warped\" polygo\n-0000daa0: 6e73 2074 6f20 6265 2073 706c 6974 2069 ns to be split i\n-0000dab0: 6e74 6f20 7472 6961 6e67 6c65 732c 2077 nto triangles, w\n-0000dac0: 6869 6368 0a20 2020 2020 2020 2020 2020 hich. \n+0000da80: 2020 2020 2020 2020 2020 2020 2020 6361 ca\n+0000da90: 7573 6573 2022 7761 7270 6564 2220 706f uses \"warped\" po\n+0000daa0: 6c79 676f 6e73 2074 6f20 6265 2073 706c lygons to be spl\n+0000dab0: 6974 2069 6e74 6f20 7472 6961 6e67 6c65 it into triangle\n+0000dac0: 732c 2077 6869 6368 0a20 2020 2020 2020 s, which. \n 0000dad0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0000dae0: 2020 2020 2020 2061 766f 6964 7320 6d69 avoids mi\n-0000daf0: 7374 616b 6573 2062 7920 7468 6520 6869 stakes by the hi\n-0000db00: 6464 656e 2073 7572 6661 6365 2061 6c67 dden surface alg\n-0000db10: 6f72 6974 686d 2e0a 0a33 2e31 2e36 2050 orithm...3.1.6 P\n-0000db20: 6f69 6e74 206c 6973 7473 0a2d 2d2d 2d2d oint lists.-----\n-0000db30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 2a53 ------------..*S\n-0000db40: 796e 7461 783a 2a0a 2020 2020 2028 5831 yntax:*. (X1\n-0000db50: 2c59 312c 5a31 2928 5832 2c59 322c 5a32 ,Y1,Z1)(X2,Y2,Z2\n-0000db60: 292e 2e2e 0a41 2073 6571 7565 6e63 6520 )....A sequence \n-0000db70: 6f66 206f 6e65 206f 7220 6d6f 7265 2070 of one or more p\n-0000db80: 6f69 6e74 7320 6d61 6b65 7320 6120 706f oints makes a po\n-0000db90: 696e 7420 6c69 7374 2c20 6120 6665 6174 int list, a feat\n-0000dba0: 7572 6520 636f 6d6d 6f6e 2074 6f0a 616c ure common to.al\n-0000dbb0: 6c20 6472 6177 6162 6c65 732e 2020 4561 l drawables. Ea\n-0000dbc0: 6368 206f 6620 7468 6520 706f 696e 7420 ch of the point \n-0000dbd0: 636f 6d70 6f6e 656e 7473 2069 7320 6120 components is a \n-0000dbe0: 7363 616c 6172 2061 7269 7468 6d65 7469 scalar arithmeti\n-0000dbf0: 630a 6578 7072 6573 7369 6f6e 2e20 2041 c.expression. A\n-0000dc00: 6e79 2070 6f69 6e74 206d 6179 2068 6176 ny point may hav\n-0000dc10: 6520 7468 6520 7a2d 636f 6d70 6f6e 656e e the z-componen\n-0000dc20: 7420 6f6d 6974 7465 643b 2069 7420 7769 t omitted; it wi\n-0000dc30: 6c6c 2064 6566 6175 6c74 0a74 6f20 7a3d ll default.to z=\n-0000dc40: 302e 0a0a 2a20 4d65 6e75 3a0a 0a2a 2044 0...* Menu:..* D\n-0000dc50: 7261 7761 626c 6573 3a3a 2020 2020 2020 rawables:: \n-0000dc60: 2020 2020 2020 2020 2020 2020 2054 6869 Thi\n-0000dc70: 6e67 7320 7468 6174 2061 7265 2064 7261 ngs that are dra\n-0000dc80: 776e 2e0a 2a20 4465 6669 6e69 7469 6f6e wn..* Definition\n-0000dc90: 733a 3a20 2020 2020 2020 2020 2020 2020 s:: \n-0000dca0: 2020 2020 5468 696e 6773 2077 6974 6820 Things with \n-0000dcb0: 6e61 6d65 732e 0a0a 1f0a 4669 6c65 3a20 names.....File: \n-0000dcc0: 736b 6574 6368 2e69 6e66 6f2c 2020 4e6f sketch.info, No\n-0000dcd0: 6465 3a20 4472 6177 6162 6c65 732c 2020 de: Drawables, \n-0000dce0: 4e65 7874 3a20 4465 6669 6e69 7469 6f6e Next: Definition\n-0000dcf0: 732c 2020 5072 6576 3a20 4c61 6e67 7561 s, Prev: Langua\n-0000dd00: 6765 2062 6173 6963 732c 2020 5570 3a20 ge basics, Up: \n-0000dd10: 496e 7075 7420 6c61 6e67 7561 6765 0a0a Input language..\n-0000dd20: 332e 3220 4472 6177 6162 6c65 730a 3d3d 3.2 Drawables.==\n-0000dd30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a44 7261 ===========..Dra\n-0000dd40: 7761 626c 6573 2061 7265 2073 696d 706c wables are simpl\n-0000dd50: 7920 e280 9873 6b65 7463 68e2 8099 206f y ...sketch... o\n-0000dd60: 626a 6563 7473 2074 6861 7420 6d69 6768 bjects that migh\n-0000dd70: 7420 6170 7065 6172 2069 6e20 7468 6520 t appear in the \n-0000dd80: 6472 6177 696e 672e 0a54 6865 7920 696e drawing..They in\n-0000dd90: 636c 7564 6520 646f 7473 2c20 706f 6c79 clude dots, poly\n-0000dda0: 6c69 6e65 732c 2063 7572 7665 732c 2070 lines, curves, p\n-0000ddb0: 6f6c 7967 6f6e 732c 2061 6e64 206d 6f72 olygons, and mor\n-0000ddc0: 6520 636f 6d70 6c65 7820 6f62 6a65 6374 e complex object\n-0000ddd0: 730a 7468 6174 2061 7265 2062 7569 6c74 s.that are built\n-0000dde0: 2075 7020 6672 6f6d 2073 696d 706c 6572 up from simpler\n-0000ddf0: 206f 6e65 7320 696e 2076 6172 696f 7573 ones in various\n-0000de00: 2077 6179 732e 2020 4669 6e61 6c6c 792c ways. Finally,\n-0000de10: 20e2 809c 7370 6563 6961 6ce2 809d 0a6f ...special....o\n-0000de20: 626a 6563 7473 2061 7265 2074 686f 7365 bjects are those\n-0000de30: 2063 6f6d 706f 7365 6420 6f66 204c 6154 composed of LaT\n-0000de40: 6558 206f 7220 e280 9850 5354 7269 636b eX or ...PSTrick\n-0000de50: 73e2 8099 2063 6f64 652c 2070 6572 6861 s... code, perha\n-0000de60: 7073 0a69 6e63 6c75 6469 6e67 2063 6f6f ps.including coo\n-0000de70: 7264 696e 6174 6573 2061 6e64 2061 6e67 rdinates and ang\n-0000de80: 6c65 7320 636f 6d70 7574 6564 2062 7920 les computed by \n-0000de90: e280 9873 6b65 7463 68e2 8099 2e0a 0a2a ...sketch......*\n-0000dea0: 204d 656e 753a 0a0a 2a20 446f 7473 3a3a Menu:..* Dots::\n-0000deb0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0000dec0: 2020 2020 2020 2020 4472 6177 2064 6f74 Draw dot\n-0000ded0: 732e 0a2a 204c 696e 6573 3a3a 2020 2020 s..* Lines:: \n+0000dae0: 2020 2020 2020 2020 2020 2061 766f 6964 avoid\n+0000daf0: 7320 6d69 7374 616b 6573 2062 7920 7468 s mistakes by th\n+0000db00: 6520 6869 6464 656e 2073 7572 6661 6365 e hidden surface\n+0000db10: 2061 6c67 6f72 6974 686d 2e0a 0a33 2e31 algorithm...3.1\n+0000db20: 2e36 2050 6f69 6e74 206c 6973 7473 0a2d .6 Point lists.-\n+0000db30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ----------------\n+0000db40: 0a0a 2a53 796e 7461 783a 2a0a 2020 2020 ..*Syntax:*. \n+0000db50: 2028 5831 2c59 312c 5a31 2928 5832 2c59 (X1,Y1,Z1)(X2,Y\n+0000db60: 322c 5a32 292e 2e2e 0a41 2073 6571 7565 2,Z2)....A seque\n+0000db70: 6e63 6520 6f66 206f 6e65 206f 7220 6d6f nce of one or mo\n+0000db80: 7265 2070 6f69 6e74 7320 6d61 6b65 7320 re points makes \n+0000db90: 6120 706f 696e 7420 6c69 7374 2c20 6120 a point list, a \n+0000dba0: 6665 6174 7572 6520 636f 6d6d 6f6e 2074 feature common t\n+0000dbb0: 6f0a 616c 6c20 6472 6177 6162 6c65 732e o.all drawables.\n+0000dbc0: 2020 4561 6368 206f 6620 7468 6520 706f Each of the po\n+0000dbd0: 696e 7420 636f 6d70 6f6e 656e 7473 2069 int components i\n+0000dbe0: 7320 6120 7363 616c 6172 2061 7269 7468 s a scalar arith\n+0000dbf0: 6d65 7469 630a 6578 7072 6573 7369 6f6e metic.expression\n+0000dc00: 2e20 2041 6e79 2070 6f69 6e74 206d 6179 . Any point may\n+0000dc10: 2068 6176 6520 7468 6520 7a2d 636f 6d70 have the z-comp\n+0000dc20: 6f6e 656e 7420 6f6d 6974 7465 643b 2069 onent omitted; i\n+0000dc30: 7420 7769 6c6c 2064 6566 6175 6c74 0a74 t will default.t\n+0000dc40: 6f20 7a3d 302e 0a0a 2a20 4d65 6e75 3a0a o z=0...* Menu:.\n+0000dc50: 0a2a 2044 7261 7761 626c 6573 3a3a 2020 .* Drawables:: \n+0000dc60: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0000dc70: 2054 6869 6e67 7320 7468 6174 2061 7265 Things that are\n+0000dc80: 2064 7261 776e 2e0a 2a20 4465 6669 6e69 drawn..* Defini\n+0000dc90: 7469 6f6e 733a 3a20 2020 2020 2020 2020 tions:: \n+0000dca0: 2020 2020 2020 2020 5468 696e 6773 2077 Things w\n+0000dcb0: 6974 6820 6e61 6d65 732e 0a0a 1f0a 4669 ith names.....Fi\n+0000dcc0: 6c65 3a20 736b 6574 6368 2e69 6e66 6f2c le: sketch.info,\n+0000dcd0: 2020 4e6f 6465 3a20 4472 6177 6162 6c65 Node: Drawable\n+0000dce0: 732c 2020 4e65 7874 3a20 4465 6669 6e69 s, Next: Defini\n+0000dcf0: 7469 6f6e 732c 2020 5072 6576 3a20 4c61 tions, Prev: La\n+0000dd00: 6e67 7561 6765 2062 6173 6963 732c 2020 nguage basics, \n+0000dd10: 5570 3a20 496e 7075 7420 6c61 6e67 7561 Up: Input langua\n+0000dd20: 6765 0a0a 332e 3220 4472 6177 6162 6c65 ge..3.2 Drawable\n+0000dd30: 730a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a s.=============.\n+0000dd40: 0a44 7261 7761 626c 6573 2061 7265 2073 .Drawables are s\n+0000dd50: 696d 706c 7920 e280 9873 6b65 7463 68e2 imply ...sketch.\n+0000dd60: 8099 206f 626a 6563 7473 2074 6861 7420 .. objects that \n+0000dd70: 6d69 6768 7420 6170 7065 6172 2069 6e20 might appear in \n+0000dd80: 7468 6520 6472 6177 696e 672e 0a54 6865 the drawing..The\n+0000dd90: 7920 696e 636c 7564 6520 646f 7473 2c20 y include dots, \n+0000dda0: 706f 6c79 6c69 6e65 732c 2063 7572 7665 polylines, curve\n+0000ddb0: 732c 2070 6f6c 7967 6f6e 732c 2061 6e64 s, polygons, and\n+0000ddc0: 206d 6f72 6520 636f 6d70 6c65 7820 6f62 more complex ob\n+0000ddd0: 6a65 6374 730a 7468 6174 2061 7265 2062 jects.that are b\n+0000dde0: 7569 6c74 2075 7020 6672 6f6d 2073 696d uilt up from sim\n+0000ddf0: 706c 6572 206f 6e65 7320 696e 2076 6172 pler ones in var\n+0000de00: 696f 7573 2077 6179 732e 2020 4669 6e61 ious ways. Fina\n+0000de10: 6c6c 792c 20e2 809c 7370 6563 6961 6ce2 lly, ...special.\n+0000de20: 809d 0a6f 626a 6563 7473 2061 7265 2074 ...objects are t\n+0000de30: 686f 7365 2063 6f6d 706f 7365 6420 6f66 hose composed of\n+0000de40: 204c 6154 6558 206f 7220 e280 9850 5354 LaTeX or ...PST\n+0000de50: 7269 636b 73e2 8099 2063 6f64 652c 2070 ricks... code, p\n+0000de60: 6572 6861 7073 0a69 6e63 6c75 6469 6e67 erhaps.including\n+0000de70: 2063 6f6f 7264 696e 6174 6573 2061 6e64 coordinates and\n+0000de80: 2061 6e67 6c65 7320 636f 6d70 7574 6564 angles computed\n+0000de90: 2062 7920 e280 9873 6b65 7463 68e2 8099 by ...sketch...\n+0000dea0: 2e0a 0a2a 204d 656e 753a 0a0a 2a20 446f ...* Menu:..* Do\n+0000deb0: 7473 3a3a 2020 2020 2020 2020 2020 2020 ts:: \n+0000dec0: 2020 2020 2020 2020 2020 2020 4472 6177 Draw\n+0000ded0: 2064 6f74 732e 0a2a 204c 696e 6573 3a3a dots..* Lines::\n 0000dee0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0000def0: 2020 2044 7261 7720 706f 6c79 6c69 6e65 Draw polyline\n-0000df00: 732e 0a2a 2043 7572 7665 733a 3a20 2020 s..* Curves:: \n-0000df10: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0000df20: 2020 2044 7261 7720 6375 7276 6573 2e0a Draw curves..\n-0000df30: 2a20 506f 6c79 676f 6e73 3a3a 2020 2020 * Polygons:: \n+0000def0: 2020 2020 2020 2044 7261 7720 706f 6c79 Draw poly\n+0000df00: 6c69 6e65 732e 0a2a 2043 7572 7665 733a lines..* Curves:\n+0000df10: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+0000df20: 2020 2020 2020 2044 7261 7720 6375 7276 Draw curv\n+0000df30: 6573 2e0a 2a20 506f 6c79 676f 6e73 3a3a es..* Polygons::\n 0000df40: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0000df50: 4472 6177 2070 6f6c 7967 6f6e 732e 0a2a Draw polygons..*\n-0000df60: 2053 7065 6369 616c 733a 3a20 2020 2020 Specials:: \n-0000df70: 2020 2020 2020 2020 2020 2020 2020 2045 E\n-0000df80: 6d62 6564 2072 6177 204c 6154 6558 2061 mbed raw LaTeX a\n-0000df90: 6e64 20e2 8098 5053 5472 6963 6b73 e280 nd ...PSTricks..\n-0000dfa0: 992e 0a2a 2053 7765 6570 733a 3a20 2020 ...* Sweeps:: \n-0000dfb0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0000dfc0: 2020 2044 7261 7720 7377 6565 7073 206f Draw sweeps o\n-0000dfd0: 6620 646f 7473 2061 6e64 2070 6f6c 796c f dots and polyl\n-0000dfe0: 696e 6573 2e0a 2a20 426c 6f63 6b73 3a3a ines..* Blocks::\n-0000dff0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0000e000: 2020 2020 2020 4772 6f75 7020 6f74 6865 Group othe\n-0000e010: 7220 6472 6177 6162 6c65 732e 0a2a 2052 r drawables..* R\n-0000e020: 6570 6561 7473 3a3a 2020 2020 2020 2020 epeats:: \n-0000e030: 2020 2020 2020 2020 2020 2020 2044 7261 Dra\n-0000e040: 7720 7472 616e 7366 6f72 6d65 6420 636f w transformed co\n-0000e050: 7069 6573 206f 6620 6f62 6a65 6374 732e pies of objects.\n-0000e060: 0a2a 2050 7574 733a 3a20 2020 2020 2020 .* Puts:: \n+0000df50: 2020 2020 4472 6177 2070 6f6c 7967 6f6e Draw polygon\n+0000df60: 732e 0a2a 2053 7065 6369 616c 733a 3a20 s..* Specials:: \n+0000df70: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0000df80: 2020 2045 6d62 6564 2072 6177 204c 6154 Embed raw LaT\n+0000df90: 6558 2061 6e64 20e2 8098 5053 5472 6963 eX and ...PSTric\n+0000dfa0: 6b73 e280 992e 0a2a 2053 7765 6570 733a ks.....* Sweeps:\n+0000dfb0: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+0000dfc0: 2020 2020 2020 2044 7261 7720 7377 6565 Draw swee\n+0000dfd0: 7073 206f 6620 646f 7473 2061 6e64 2070 ps of dots and p\n+0000dfe0: 6f6c 796c 696e 6573 2e0a 2a20 426c 6f63 olylines..* Bloc\n+0000dff0: 6b73 3a3a 2020 2020 2020 2020 2020 2020 ks:: \n+0000e000: 2020 2020 2020 2020 2020 4772 6f75 7020 Group \n+0000e010: 6f74 6865 7220 6472 6177 6162 6c65 732e other drawables.\n+0000e020: 0a2a 2052 6570 6561 7473 3a3a 2020 2020 .* Repeats:: \n+0000e030: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0000e040: 2044 7261 7720 7472 616e 7366 6f72 6d65 Draw transforme\n+0000e050: 6420 636f 7069 6573 206f 6620 6f62 6a65 d copies of obje\n+0000e060: 6374 732e 0a2a 2050 7574 733a 3a20 2020 cts..* Puts:: \n 0000e070: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0000e080: 2044 7261 7720 6f6e 6520 6f62 6a65 6374 Draw one object\n-0000e090: 2074 7261 6e73 666f 726d 6564 2e0a 0a1f transformed....\n-0000e0a0: 0a46 696c 653a 2073 6b65 7463 682e 696e .File: sketch.in\n-0000e0b0: 666f 2c20 204e 6f64 653a 2044 6f74 732c fo, Node: Dots,\n-0000e0c0: 2020 4e65 7874 3a20 4c69 6e65 732c 2020 Next: Lines, \n-0000e0d0: 5072 6576 3a20 4472 6177 6162 6c65 732c Prev: Drawables,\n-0000e0e0: 2020 5570 3a20 4472 6177 6162 6c65 730a Up: Drawables.\n-0000e0f0: 0a33 2e32 2e31 2044 6f74 730a 2d2d 2d2d .3.2.1 Dots.----\n-0000e100: 2d2d 2d2d 2d2d 0a0a 2a53 796e 7461 783a ------..*Syntax:\n-0000e110: 2a0a 2020 2020 2064 6f74 735b 4f50 5449 *. dots[OPTI\n-0000e120: 4f4e 535d 2050 4f49 4e54 5f4c 4953 540a ONS] POINT_LIST.\n-0000e130: 5468 6973 2063 6f6d 6d61 6e64 2069 7320 This command is \n-0000e140: 7468 6520 7468 7265 652d 6469 6d65 6e73 the three-dimens\n-0000e150: 696f 6e61 6c20 6571 7569 7661 6c65 6e74 ional equivalent\n-0000e160: 206f 6620 7468 6520 e280 9850 5354 7269 of the ...PSTri\n-0000e170: 636b 73e2 8099 0a63 6f6d 6d61 6e64 20e2 cks....command .\n-0000e180: 8098 5c70 7364 6f74 73e2 8099 2e0a 0a1f ..\\psdots.......\n-0000e190: 0a46 696c 653a 2073 6b65 7463 682e 696e .File: sketch.in\n-0000e1a0: 666f 2c20 204e 6f64 653a 204c 696e 6573 fo, Node: Lines\n-0000e1b0: 2c20 204e 6578 743a 2043 7572 7665 732c , Next: Curves,\n-0000e1c0: 2020 5072 6576 3a20 446f 7473 2c20 2055 Prev: Dots, U\n-0000e1d0: 703a 2044 7261 7761 626c 6573 0a0a 332e p: Drawables..3.\n-0000e1e0: 322e 3220 4c69 6e65 730a 2d2d 2d2d 2d2d 2.2 Lines.------\n-0000e1f0: 2d2d 2d2d 2d0a 0a2a 5379 6e74 6178 3a2a -----..*Syntax:*\n-0000e200: 0a20 2020 2020 6c69 6e65 5b4f 5054 494f . line[OPTIO\n-0000e210: 4e53 5d20 504f 494e 545f 4c49 5354 0a54 NS] POINT_LIST.T\n-0000e220: 6869 7320 636f 6d6d 616e 6420 6973 2074 his command is t\n-0000e230: 6865 2074 6872 6565 2d64 696d 656e 7369 he three-dimensi\n-0000e240: 6f6e 616c 2065 7175 6976 616c 656e 7420 onal equivalent \n-0000e250: 6f66 2074 6865 20e2 8098 5053 5472 6963 of the ...PSTric\n-0000e260: 6b73 e280 990a 636f 6d6d 616e 6420 e280 ks....command ..\n-0000e270: 985c 7073 6c69 6e65 e280 992e 0a0a 1f0a .\\psline........\n-0000e280: 4669 6c65 3a20 736b 6574 6368 2e69 6e66 File: sketch.inf\n-0000e290: 6f2c 2020 4e6f 6465 3a20 4375 7276 6573 o, Node: Curves\n-0000e2a0: 2c20 204e 6578 743a 2050 6f6c 7967 6f6e , Next: Polygon\n-0000e2b0: 732c 2020 5072 6576 3a20 4c69 6e65 732c s, Prev: Lines,\n-0000e2c0: 2020 5570 3a20 4472 6177 6162 6c65 730a Up: Drawables.\n-0000e2d0: 0a33 2e32 2e33 2043 7572 7665 730a 2d2d .3.2.3 Curves.--\n-0000e2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 2a53 796e ----------..*Syn\n-0000e2f0: 7461 783a 2a0a 2020 2020 2063 7572 7665 tax:*. curve\n-0000e300: 5b4f 5054 494f 4e53 5d20 504f 494e 545f [OPTIONS] POINT_\n-0000e310: 4c49 5354 0a54 6869 7320 636f 6d6d 616e LIST.This comman\n-0000e320: 6420 6973 2074 6865 2074 6872 6565 2d64 d is the three-d\n-0000e330: 696d 656e 7369 6f6e 616c 2065 7175 6976 imensional equiv\n-0000e340: 616c 656e 7420 6f66 2074 6865 20e2 8098 alent of the ...\n-0000e350: 5053 5472 6963 6b73 e280 990a 636f 6d6d PSTricks....comm\n-0000e360: 616e 6420 e280 985c 7073 6375 7276 65e2 and ...\\pscurve.\n-0000e370: 8099 2e20 2049 7420 6973 206e 6f74 2069 ... It is not i\n-0000e380: 6d70 6c65 6d65 6e74 6564 2069 6e20 7468 mplemented in th\n-0000e390: 6520 6375 7272 656e 7420 7665 7273 696f e current versio\n-0000e3a0: 6e20 6f66 0ae2 8098 736b 6574 6368 e280 n of....sketch..\n-0000e3b0: 992e 0a0a 1f0a 4669 6c65 3a20 736b 6574 ......File: sket\n-0000e3c0: 6368 2e69 6e66 6f2c 2020 4e6f 6465 3a20 ch.info, Node: \n-0000e3d0: 506f 6c79 676f 6e73 2c20 204e 6578 743a Polygons, Next:\n-0000e3e0: 2053 7065 6369 616c 732c 2020 5072 6576 Specials, Prev\n-0000e3f0: 3a20 4375 7276 6573 2c20 2055 703a 2044 : Curves, Up: D\n-0000e400: 7261 7761 626c 6573 0a0a 332e 322e 3420 rawables..3.2.4 \n-0000e410: 506f 6c79 676f 6e73 0a2d 2d2d 2d2d 2d2d Polygons.-------\n-0000e420: 2d2d 2d2d 2d2d 2d0a 0a2a 5379 6e74 6178 -------..*Syntax\n-0000e430: 3a2a 0a20 2020 2020 706f 6c79 676f 6e5b :*. polygon[\n-0000e440: 4f50 5449 4f4e 535d 2050 4f49 4e54 5f4c OPTIONS] POINT_L\n-0000e450: 4953 540a 5468 6973 2063 6f6d 6d61 6e64 IST.This command\n-0000e460: 2069 7320 7468 6520 7468 7265 652d 6469 is the three-di\n-0000e470: 6d65 6e73 696f 6e61 6c20 6571 7569 7661 mensional equiva\n-0000e480: 6c65 6e74 206f 6620 7468 6520 e280 9850 lent of the ...P\n-0000e490: 5354 7269 636b 73e2 8099 0a63 6f6d 6d61 STricks....comma\n-0000e4a0: 6e64 20e2 8098 5c70 7370 6f6c 7967 6f6e nd ...\\pspolygon\n-0000e4b0: e280 992e 2020 5468 6520 e280 9873 6b65 .... The ...ske\n-0000e4c0: 7463 68e2 8099 2068 6964 6465 6e20 7375 tch... hidden su\n-0000e4d0: 7266 6163 6520 616c 676f 7269 7468 6d20 rface algorithm \n-0000e4e0: 6173 7375 6d65 730a 7468 6174 2070 6f6c assumes.that pol\n-0000e4f0: 7967 6f6e 7320 6172 6520 636f 6e76 6578 ygons are convex\n-0000e500: 2061 6e64 2070 6c61 6e61 722e 2020 496e and planar. In\n-0000e510: 2070 7261 6374 6963 652c 2064 7261 7769 practice, drawi\n-0000e520: 6e67 7320 6d61 7920 7765 6c6c 0a74 7572 ngs may well.tur\n-0000e530: 6e20 6f75 7420 636f 7272 6563 746c 7920 n out correctly \n-0000e540: 6576 656e 2069 6620 7468 6573 6520 6173 even if these as\n-0000e550: 7375 6d70 7469 6f6e 7320 6172 6520 7669 sumptions are vi\n-0000e560: 6f6c 6174 6564 2e0a 0a1f 0a46 696c 653a olated.....File:\n-0000e570: 2073 6b65 7463 682e 696e 666f 2c20 204e sketch.info, N\n-0000e580: 6f64 653a 2053 7065 6369 616c 732c 2020 ode: Specials, \n-0000e590: 4e65 7874 3a20 5377 6565 7073 2c20 2050 Next: Sweeps, P\n-0000e5a0: 7265 763a 2050 6f6c 7967 6f6e 732c 2020 rev: Polygons, \n-0000e5b0: 5570 3a20 4472 6177 6162 6c65 730a 0a33 Up: Drawables..3\n-0000e5c0: 2e32 2e35 2053 7065 6369 616c 730a 2d2d .2.5 Specials.--\n-0000e5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 2a53 ------------..*S\n-0000e5e0: 796e 7461 783a 2a0a 2020 2020 2073 7065 yntax:*. spe\n-0000e5f0: 6369 616c 2024 5241 575f 5445 5854 245b cial $RAW_TEXT$[\n-0000e600: 4f50 5449 4f4e 535d 2041 5247 5f4c 4953 OPTIONS] ARG_LIS\n-0000e610: 540a 4865 7265 20e2 8098 24e2 8099 2063 T.Here ...$... c\n-0000e620: 616e 2062 6520 616e 7920 6368 6172 6163 an be any charac\n-0000e630: 7465 7220 616e 6420 6973 2075 7365 6420 ter and is used \n-0000e640: 746f 2064 656c 696d 6974 2074 6865 2073 to delimit the s\n-0000e650: 7461 7274 2061 6e64 2065 6e64 0a6f 6620 tart and end.of \n-0000e660: 5241 575f 5445 5854 2e20 2054 6865 2063 RAW_TEXT. The c\n-0000e670: 6f6d 6d61 6e64 2065 6d62 6564 7320 5241 ommand embeds RA\n-0000e680: 575f 5445 5854 2069 6e20 7468 6520 e280 W_TEXT in the ..\n-0000e690: 9873 6b65 7463 68e2 8099 206f 7574 7075 .sketch... outpu\n-0000e6a0: 7420 6166 7465 720a 7065 7266 6f72 6d69 t after.performi\n-0000e6b0: 6e67 2073 7562 7374 6974 7574 696f 6e73 ng substitutions\n-0000e6c0: 2061 7320 666f 6c6c 6f77 732e 0a20 2020 as follows.. \n-0000e6d0: e280 a220 e280 9823 49e2 8099 2077 6865 ... ...#I... whe\n-0000e6e0: 7265 2049 2069 7320 6120 706f 7369 7469 re I is a positi\n-0000e6f0: 7665 2069 6e74 6567 6572 2069 7320 7265 ve integer is re\n-0000e700: 706c 6163 6564 2062 7920 7468 6520 4927 placed by the I'\n-0000e710: 7468 2076 616c 7565 2069 6e0a 2020 2020 th value in. \n-0000e720: 2041 5247 5f4c 4953 542e 2020 506f 696e ARG_LIST. Poin\n-0000e730: 7420 616e 6420 7665 6374 6f72 2061 7267 t and vector arg\n-0000e740: 756d 656e 7473 2062 6563 6f6d 6520 7477 uments become tw\n-0000e750: 6f2d 6469 6d65 6e73 696f 6e61 6c0a 2020 o-dimensional. \n-0000e760: 2020 2070 6f69 6e74 732c 2077 6869 6368 points, which\n-0000e770: 2061 7265 2074 6865 2074 7261 6e73 666f are the transfo\n-0000e780: 726d 6564 2033 6420 6172 6775 6d65 6e74 rmed 3d argument\n-0000e790: 7320 7072 6f6a 6563 7465 6420 6f6e 746f s projected onto\n-0000e7a0: 2074 6865 0a20 2020 2020 582d 5920 706c the. X-Y pl\n-0000e7b0: 616e 652e 2020 5468 6973 2061 6c6c 6f77 ane. This allow\n-0000e7c0: 7320 7477 6f2d 6469 6d65 6e74 696f 6e61 s two-dimentiona\n-0000e7d0: 6c20 6f75 7470 7574 2065 6c65 6d65 6e74 l output element\n-0000e7e0: 7320 7375 6368 2061 730a 2020 2020 206c s such as. l\n-0000e7f0: 6162 656c 7320 746f 2062 6520 6561 7369 abels to be easi\n-0000e800: 6c79 2070 6f73 6974 696f 6e65 6420 7769 ly positioned wi\n-0000e810: 7468 2072 6573 7065 6374 2074 6f20 7468 th respect to th\n-0000e820: 7265 652d 6469 6d65 6e73 696f 6e61 6c0a ree-dimensional.\n-0000e830: 2020 2020 2066 6561 7475 7265 7320 696e features in\n-0000e840: 2074 6865 2064 7261 7769 6e67 2e20 2053 the drawing. S\n-0000e850: 6361 6c61 7220 6172 6775 6d65 6e74 7320 calar arguments \n-0000e860: 6172 6520 7375 6273 6974 7574 6564 2064 are subsituted d\n-0000e870: 6972 6563 746c 792e 0a20 2020 2020 4e6f irectly.. No\n-0000e880: 2074 7261 6e73 666f 726d 6174 696f 6e20 transformation \n-0000e890: 6973 2061 7070 6c69 6564 2e0a 2020 20e2 is applied.. .\n-0000e8a0: 80a2 20e2 8098 237b 497d e280 9920 6973 .. ...#{I}... is\n-0000e8b0: 2061 6c73 6f20 7265 706c 6163 6564 2061 also replaced a\n-0000e8c0: 7320 6162 6f76 652e 0a20 2020 e280 a220 s above.. ... \n-0000e8d0: e280 9823 4927 32e2 8099 2069 7320 7265 ...#I'2... is re\n-0000e8e0: 706c 6163 6564 2061 7320 6162 6f76 6520 placed as above \n-0000e8f0: 666f 7220 706f 696e 7473 206f 7220 7665 for points or ve\n-0000e900: 6374 6f72 732e 2020 4974 2069 7320 616e ctors. It is an\n-0000e910: 2065 7272 6f72 0a20 2020 2020 666f 7220 error. for \n-0000e920: 7468 6520 4927 7468 2061 7267 756d 656e the I'th argumen\n-0000e930: 7420 746f 2062 6520 6120 7363 616c 6172 t to be a scalar\n-0000e940: 2e0a 2020 20e2 80a2 20e2 8098 2349 2778 .. ... ...#I'x\n-0000e950: e280 992c 20e2 8098 2349 2779 e280 992c ..., ...#I'y...,\n-0000e960: 206f 7220 e280 9823 4927 7ae2 8099 2069 or ...#I'z... i\n-0000e970: 7320 7265 706c 6163 6564 2c20 7265 7370 s replaced, resp\n-0000e980: 6563 7469 7665 6c79 2c20 6279 2074 6865 ectively, by the\n-0000e990: 2073 6361 6c61 720a 2020 2020 2058 2c20 scalar. X, \n-0000e9a0: 592c 206f 7220 5a2d 636f 6f72 6469 6e61 Y, or Z-coordina\n-0000e9b0: 7465 206f 6620 7468 6520 6172 6775 6d65 te of the argume\n-0000e9c0: 6e74 2070 6f69 6e74 206f 7220 7665 6374 nt point or vect\n-0000e9d0: 6f72 2e20 2049 7420 6973 2061 6e0a 2020 or. It is an. \n-0000e9e0: 2020 2065 7272 6f72 2066 6f72 2074 6865 error for the\n-0000e9f0: 2049 2774 6820 6172 6775 6d65 6e74 2074 I'th argument t\n-0000ea00: 6f20 6265 2061 2073 6361 6c61 722e 0a20 o be a scalar.. \n-0000ea10: 2020 e280 a220 e280 9823 4927 33e2 8099 ... ...#I'3...\n-0000ea20: 2069 7320 7265 706c 6163 6564 2062 7920 is replaced by \n-0000ea30: 7468 6520 7468 7265 652d 6469 6d65 6e73 the three-dimens\n-0000ea40: 696f 6e61 6c20 7472 616e 7366 6f72 6d65 ional transforme\n-0000ea50: 6420 6172 6775 6d65 6e74 2e0a 2020 2020 d argument.. \n-0000ea60: 204e 6f74 6520 7468 6174 2069 6620 6120 Note that if a \n-0000ea70: 7065 7273 7065 6374 6976 6520 7472 616e perspective tran\n-0000ea80: 7366 6f72 6d61 7469 6f6e 2068 6173 2062 sformation has b\n-0000ea90: 6565 6e20 6170 706c 6965 642c 2074 6865 een applied, the\n-0000eaa0: 0a20 2020 2020 5a2d 636f 6f72 6469 6e61 . Z-coordina\n-0000eab0: 7465 2068 6173 206c 6974 746c 6520 6765 te has little ge\n-0000eac0: 6f6d 6574 7269 6320 7369 676e 6966 6963 ometric signific\n-0000ead0: 616e 6365 2c20 7468 6f75 6768 2069 740a ance, though it.\n-0000eae0: 2020 2020 2061 6363 7572 6174 656c 7920 accurately \n-0000eaf0: 696e 6469 6361 7465 7320 7265 6c61 7469 indicates relati\n-0000eb00: 7665 2064 6570 7468 2e0a 2020 20e2 80a2 ve depth.. ...\n-0000eb10: 20e2 8098 2349 2d4a e280 9920 7768 6572 ...#I-J... wher\n-0000eb20: 6520 4920 616e 6420 4a20 6172 6520 706f e I and J are po\n-0000eb30: 7369 7469 7665 2069 6e74 6567 6572 7320 sitive integers \n-0000eb40: 6973 2072 6570 6c61 6365 6420 6279 2061 is replaced by a\n-0000eb50: 2073 7472 696e 670a 2020 2020 20e2 8098 string. ...\n-0000eb60: 7b41 4e47 4c45 7de2 8099 2077 6865 7265 {ANGLE}... where\n-0000eb70: 2041 4e47 4c45 2069 7320 7468 6520 706f ANGLE is the po\n-0000eb80: 6c61 7220 616e 676c 6520 6f66 2061 2076 lar angle of a v\n-0000eb90: 6563 746f 7220 6672 6f6d 2074 6865 2049 ector from the I\n-0000eba0: 2774 680a 2020 2020 2070 6f69 6e74 2069 'th. point i\n-0000ebb0: 6e20 504f 494e 545f 4c49 5354 2074 6f20 n POINT_LIST to \n-0000ebc0: 7468 6520 4a27 7468 2070 6f69 6e74 2070 the J'th point p\n-0000ebd0: 726f 6a65 6374 6564 2069 6e74 6f20 7468 rojected into th\n-0000ebe0: 6520 782d 7920 706c 616e 652e 0a20 2020 e x-y plane.. \n-0000ebf0: 2020 4974 2069 7320 616e 2065 7272 6f72 It is an error\n-0000ec00: 2066 6f72 2074 6865 2049 2774 6820 6f72 for the I'th or\n-0000ec10: 204a 2774 6820 6172 6775 6d65 6e74 2074 J'th argument t\n-0000ec20: 6f20 6265 2061 2073 6361 6c61 7220 6f72 o be a scalar or\n-0000ec30: 2061 0a20 2020 2020 7665 6374 6f72 2e0a a. vector..\n-0000ec40: 2020 20e2 80a2 20e2 8098 237b 492d 4a7d ... ...#{I-J}\n-0000ec50: e280 9920 6973 2061 6c73 6f20 7265 706c ... is also repl\n-0000ec60: 6163 6564 2061 7320 6162 6f76 652e 0a20 aced as above.. \n-0000ec70: 2020 e280 a220 e280 9823 23e2 8099 2069 ... ...##... i\n-0000ec80: 7320 7265 706c 6163 6564 2077 6974 6820 s replaced with \n-0000ec90: e280 9823 e280 992e 0a54 6865 2066 6f72 ...#.....The for\n-0000eca0: 6d73 2077 6974 6820 6272 6163 6573 20e2 ms with braces .\n-0000ecb0: 8098 7b20 7de2 8099 2061 7265 2075 7365 ..{ }... are use\n-0000ecc0: 6675 6c20 7768 656e 2074 6865 2061 7267 ful when the arg\n-0000ecd0: 756d 656e 7420 6973 2069 6d6d 6564 6961 ument is immedia\n-0000ece0: 7465 6c79 0a66 6f6c 6c6f 7765 6420 6279 tely.followed by\n-0000ecf0: 2061 2064 6967 6974 206f 7220 6120 7469 a digit or a ti\n-0000ed00: 636b 2063 6861 7261 6374 6572 2074 6861 ck character tha\n-0000ed10: 7420 6973 2070 6172 7420 6f66 2074 6865 t is part of the\n-0000ed20: 2054 6558 2063 6f64 652e 0a0a 2020 2054 TeX code... T\n-0000ed30: 6865 206f 6e6c 7920 7573 6566 756c 206f he only useful o\n-0000ed40: 7074 696f 6e20 6f66 20e2 8098 7370 6563 ption of ...spec\n-0000ed50: 6961 6ce2 8099 2069 7320 e280 986c 6179 ial... is ...lay\n-0000ed60: e280 992c 2077 6869 6368 2064 6574 6572 ..., which deter\n-0000ed70: 6d69 6e65 7320 6966 2074 6865 0a73 7562 mines if the.sub\n-0000ed80: 7374 6974 7565 6420 7261 7720 7465 7874 stitued raw text\n-0000ed90: 2069 7320 656d 6974 7465 6420 6265 666f is emitted befo\n-0000eda0: 7265 2c20 6166 7465 722c 206f 7220 7573 re, after, or us\n-0000edb0: 696e 6720 7468 6520 6669 7273 7420 706f ing the first po\n-0000edc0: 696e 740a 696e 2050 4f49 4e54 5f4c 4953 int.in POINT_LIS\n-0000edd0: 5420 6173 2061 6e20 696e 6469 6361 746f T as an indicato\n-0000ede0: 7220 6f66 2064 6570 7468 2e20 2054 6865 r of depth. The\n-0000edf0: 7365 206f 6363 7572 2c20 7265 7370 6563 se occur, respec\n-0000ee00: 7469 7665 6c79 2c20 7769 7468 0ae2 8098 tively, with....\n-0000ee10: 6c61 793d 756e 6465 72e2 8099 2c20 e280 lay=under..., ..\n-0000ee20: 986c 6179 3d6f 7665 72e2 8099 2c20 616e .lay=over..., an\n-0000ee30: 6420 e280 986c 6179 3d69 6ee2 8099 2e20 d ...lay=in.... \n-0000ee40: 2053 6565 202a 6e6f 7465 2053 7065 6369 See *note Speci\n-0000ee50: 616c 206f 626a 6563 7473 3a3a 2061 6e64 al objects:: and\n-0000ee60: 0a2a 6e6f 7465 2054 696b 5a2f 5047 4620 .*note TikZ/PGF \n-0000ee70: 7573 6572 2d64 6566 696e 6564 2073 7479 user-defined sty\n-0000ee80: 6c65 733a 3a20 666f 7220 6578 616d 706c les:: for exampl\n-0000ee90: 6573 2e20 202a 4e6f 7465 2049 6e74 6572 es. *Note Inter\n-0000eea0: 6e61 6c0a 6f70 7469 6f6e 733a 3a2e 0a0a nal.options::...\n-0000eeb0: 1f0a 4669 6c65 3a20 736b 6574 6368 2e69 ..File: sketch.i\n-0000eec0: 6e66 6f2c 2020 4e6f 6465 3a20 5377 6565 nfo, Node: Swee\n-0000eed0: 7073 2c20 204e 6578 743a 2042 6c6f 636b ps, Next: Block\n-0000eee0: 732c 2020 5072 6576 3a20 5370 6563 6961 s, Prev: Specia\n-0000eef0: 6c73 2c20 2055 703a 2044 7261 7761 626c ls, Up: Drawabl\n-0000ef00: 6573 0a0a 332e 322e 3620 5377 6565 7073 es..3.2.6 Sweeps\n-0000ef10: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a2a .------------..*\n-0000ef20: 5379 6e74 6178 3a2a 0a20 2020 2020 7377 Syntax:*. sw\n-0000ef30: 6565 7020 7b20 4e2c 2054 5f31 2c20 545f eep { N, T_1, T_\n-0000ef40: 322c 202e 2e2e 2c20 545f 5220 7d5b 4f50 2, ..., T_R }[OP\n-0000ef50: 5449 4f4e 535d 2053 5745 5054 5f4f 424a TIONS] SWEPT_OBJ\n-0000ef60: 4543 540a 2020 2020 2073 7765 6570 207b ECT. sweep {\n-0000ef70: 204e 3c3e 2c20 545f 312c 2054 5f32 2c20 N<>, T_1, T_2, \n-0000ef80: 2e2e 2e2c 2054 5f52 207d 5b4f 5054 494f ..., T_R }[OPTIO\n-0000ef90: 4e53 5d20 5357 4550 545f 4f42 4a45 4354 NS] SWEPT_OBJECT\n-0000efa0: 0a54 6865 2073 7765 6570 2063 6f6e 6e65 .The sweep conne\n-0000efb0: 6374 7320 4e20 286f 7220 7065 7268 6170 cts N (or perhap\n-0000efc0: 7320 4e2b 3129 2063 6f70 6965 7320 6f66 s N+1) copies of\n-0000efd0: 2053 5745 5054 5f4f 424a 4543 5420 696e SWEPT_OBJECT in\n-0000efe0: 206f 7264 6572 2074 6f0a 6372 6561 7465 order to.create\n-0000eff0: 2061 206e 6577 206f 626a 6563 7420 6f66 a new object of\n-0000f000: 2068 6967 6865 7220 6469 6d65 6e73 696f higher dimensio\n-0000f010: 6e2e 2020 5468 6520 545f 4920 2866 6f72 n. The T_I (for\n-0000f020: 2069 2062 6574 7765 656e 2031 2061 6e64 i between 1 and\n-0000f030: 0a72 2920 6172 6520 7472 616e 7366 6f72 .r) are transfor\n-0000f040: 6d73 2e20 2054 6865 206b 2774 6820 636f ms. The k'th co\n-0000f050: 7079 206f 6620 5357 4550 545f 4f42 4a45 py of SWEPT_OBJE\n-0000f060: 4354 2069 7320 7072 6f64 7563 6564 2062 CT is produced b\n-0000f070: 790a 6170 706c 7969 6e67 2074 6865 2066 y.applying the f\n-0000f080: 6f6c 6c6f 7769 6e67 2074 7261 6e73 666f ollowing transfo\n-0000f090: 726d 2074 6f20 7468 6520 6f72 6967 696e rm to the origin\n-0000f0a0: 616c 2e0a 2020 2020 2054 5f31 5e6b 2074 al.. T_1^k t\n-0000f0b0: 6865 6e20 545f 325e 6b20 7468 656e 202e hen T_2^k then .\n-0000f0c0: 2e2e 2074 6865 6e20 545f 525e 6b0a 4865 .. then T_R^k.He\n-0000f0d0: 7265 2054 5e6b 206d 6561 6e73 2022 7472 re T^k means \"tr\n-0000f0e0: 616e 7366 6f72 6d20 5420 6170 706c 6965 ansform T applie\n-0000f0f0: 6420 6b20 7469 6d65 732e 2220 2054 6865 d k times.\" The\n-0000f100: 206f 7269 6769 6e61 6c20 6f62 6a65 6374 original object\n-0000f110: 2069 730a 7468 6520 7a65 726f 2774 6820 is.the zero'th \n-0000f120: 636f 7079 2c20 7769 7468 206b 3d30 2061 copy, with k=0 a\n-0000f130: 6e64 2065 6666 6563 7469 7665 6c79 206e nd effectively n\n-0000f140: 6f20 7472 616e 7366 6f72 6d20 6170 706c o transform appl\n-0000f150: 6965 6420 2854 5e30 3d49 2c0a 7468 6520 ied (T^0=I,.the \n-0000f160: 6964 656e 7469 7479 2074 7261 6e73 666f identity transfo\n-0000f170: 726d 292e 0a0a 2020 2054 6865 206d 6574 rm)... The met\n-0000f180: 686f 6420 6f66 2063 6f6e 6e65 6374 696e hod of connectin\n-0000f190: 6720 7468 6520 636f 7069 6573 2064 6570 g the copies dep\n-0000f1a0: 656e 6473 206f 6e20 7468 6520 7479 7065 ends on the type\n-0000f1b0: 206f 660a 5357 4550 545f 4f42 4a45 4354 of.SWEPT_OBJECT\n-0000f1c0: 2061 6e64 206f 6e20 7768 6574 6865 7220 and on whether \n-0000f1d0: 7468 6520 636c 6f73 7572 6520 7461 6720 the closure tag \n-0000f1e0: e280 983c 3ee2 8099 2069 7320 7072 6573 ...<>... is pres\n-0000f1f0: 656e 7420 6f72 206e 6f74 2e0a 0a20 2020 ent or not... \n-0000f200: 416e 2065 7861 6d70 6c65 206f 6620 6120 An example of a \n-0000f210: 7377 6565 7020 7768 6572 6520 723d 3220 sweep where r=2 \n-0000f220: 6973 2074 6865 204d 6f62 6975 7320 6669 is the Mobius fi\n-0000f230: 6775 7265 2061 7420 2a6e 6f74 6520 4d6f gure at *note Mo\n-0000f240: 7265 2074 6f0a 6c65 6172 6e3a 3a2e 0a0a re to.learn::...\n-0000f250: 2a20 4d65 6e75 3a0a 0a2a 2053 7765 7074 * Menu:..* Swept\n-0000f260: 2070 6f69 6e74 733a 3a20 2020 2020 2020 points:: \n-0000f270: 2020 2020 2020 2020 2053 7765 7074 2070 Swept p\n-0000f280: 6f69 6e74 7320 6d61 6b65 206c 696e 6573 oints make lines\n-0000f290: 206f 7220 706f 6c79 676f 6e73 2e0a 2a20 or polygons..* \n-0000f2a0: 5377 6570 7420 6c69 6e65 733a 3a20 2020 Swept lines:: \n-0000f2b0: 2020 2020 2020 2020 2020 2020 2020 5377 Sw\n-0000f2c0: 6570 7420 6c69 6e65 7320 6d61 6b65 206f ept lines make o\n-0000f2d0: 7065 6e20 6f72 2063 6c6f 7365 6420 7375 pen or closed su\n-0000f2e0: 7266 6163 6573 2e0a 2a20 5377 6570 7420 rfaces..* Swept \n-0000f2f0: 706f 6c79 676f 6e73 3a3a 2020 2020 2020 polygons:: \n-0000f300: 2020 2020 2020 2020 5377 6570 7420 706f Swept po\n-0000f310: 6c79 676f 6e73 206d 616b 6520 636c 6f73 lygons make clos\n-0000f320: 6564 2073 7572 6661 6365 732e 0a2a 2053 ed surfaces..* S\n-0000f330: 7765 7074 2062 6c6f 636b 733a 3a20 2020 wept blocks:: \n-0000f340: 2020 2020 2020 2020 2020 2020 2053 7765 Swe\n-0000f350: 7074 2062 6c6f 636b 20e2 89a1 2062 6c6f pt block ... blo\n-0000f360: 636b 206f 6620 7377 6565 7073 2e0a 2a20 ck of sweeps..* \n-0000f370: 5377 6565 7020 6661 6365 2073 706c 6974 Sweep face split\n-0000f380: 7469 6e67 3a3a 2020 2020 2020 2020 4669 ting:: Fi\n-0000f390: 7869 6e67 2077 6172 7065 6420 6661 6365 xing warped face\n-0000f3a0: 7320 7769 7468 2074 7269 616e 676c 6573 s with triangles\n-0000f3b0: 2e0a 0a1f 0a46 696c 653a 2073 6b65 7463 .....File: sketc\n-0000f3c0: 682e 696e 666f 2c20 204e 6f64 653a 2053 h.info, Node: S\n-0000f3d0: 7765 7074 2070 6f69 6e74 732c 2020 4e65 wept points, Ne\n-0000f3e0: 7874 3a20 5377 6570 7420 6c69 6e65 732c xt: Swept lines,\n-0000f3f0: 2020 5072 6576 3a20 5377 6565 7073 2c20 Prev: Sweeps, \n-0000f400: 2055 703a 2053 7765 6570 730a 0a33 2e32 Up: Sweeps..3.2\n-0000f410: 2e36 2e31 2053 7765 7074 2070 6f69 6e74 .6.1 Swept point\n-0000f420: 730a 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e s...............\n-0000f430: 2e2e 2e2e 2e2e 0a0a 4966 2053 5745 5054 ........If SWEPT\n-0000f440: 5f4f 424a 4543 5420 6973 2061 2070 6f69 _OBJECT is a poi\n-0000f450: 6e74 206c 6973 7420 616e 6420 7468 6572 nt list and ther\n-0000f460: 6520 6973 206e 6f20 636c 6f73 7572 6520 e is no closure \n-0000f470: 7461 672c 2074 6865 6e0a e280 9873 7765 tag, then....swe\n-0000f480: 6570 e280 9920 636f 6e6e 6563 7473 204e ep... connects N\n-0000f490: 2b31 2073 7563 6365 7373 6976 6520 636f +1 successive co\n-0000f4a0: 7069 6573 206f 6620 6561 6368 2070 6f69 pies of each poi\n-0000f4b0: 6e74 2028 696e 636c 7564 696e 6720 7468 nt (including th\n-0000f4c0: 650a 6f72 6967 696e 616c 2920 7769 7468 e.original) with\n-0000f4d0: 2073 7472 6169 6768 7420 6c69 6e65 2073 straight line s\n-0000f4e0: 6567 6d65 6e74 7320 746f 2066 6f72 6d20 egments to form \n-0000f4f0: 6120 706f 6c79 6c69 6e65 2e20 2049 6620 a polyline. If \n-0000f500: 7468 6572 6520 6172 650a 6d20 706f 696e there are.m poin\n-0000f510: 7473 2069 6e20 7468 6520 6f72 6967 696e ts in the origin\n-0000f520: 616c 2070 6f69 6e74 206c 6973 742c 2074 al point list, t\n-0000f530: 6865 6e20 6d20 706f 6c79 6c69 6e65 7320 hen m polylines \n-0000f540: 7769 7468 204e 2073 6567 6d65 6e74 730a with N segments.\n-0000f550: 6561 6368 2061 7265 2066 6f72 6d65 6420 each are formed \n-0000f560: 6279 2074 6865 2073 7765 6570 2e20 2049 by the sweep. I\n-0000f570: 6e20 7468 6973 206d 616e 6e65 722c 20e2 n this manner, .\n-0000f580: 8098 7377 6565 70e2 8099 2066 6f72 6d73 ..sweep... forms\n-0000f590: 2061 2073 6574 206f 660a 6f6e 652d 6469 a set of.one-di\n-0000f5a0: 6d65 6e73 696f 6e61 6c20 6f62 6a65 6374 mensional object\n-0000f5b0: 7320 2870 6f6c 796c 696e 6573 2920 6672 s (polylines) fr\n-0000f5c0: 6f6d 207a 6572 6f2d 6469 6d65 6e73 696f om zero-dimensio\n-0000f5d0: 6e61 6c20 6f6e 6573 2028 706f 696e 7473 nal ones (points\n-0000f5e0: 292e 0a0a 2020 2057 6865 6e20 7468 6572 )... When ther\n-0000f5f0: 6520 5f69 735f 2061 2063 6c6f 7375 7265 e _is_ a closure\n-0000f600: 2074 6167 2c20 e280 9873 7765 6570 e280 tag, ...sweep..\n-0000f610: 9920 636f 6e6e 6563 7473 204e 2073 7563 . connects N suc\n-0000f620: 6365 7373 6976 6520 636f 7069 6573 0a6f cessive copies.o\n-0000f630: 6620 6561 6368 2070 6f69 6e74 2028 696e f each point (in\n-0000f640: 636c 7564 696e 6720 7468 6520 6f72 6967 cluding the orig\n-0000f650: 696e 616c 2920 7769 7468 2073 7472 6169 inal) with strai\n-0000f660: 6768 7420 6c69 6e65 2073 6567 6d65 6e74 ght line segment\n-0000f670: 7320 616e 640a 6669 6e61 6c6c 7920 636f s and.finally co\n-0000f680: 6e6e 6563 7473 2074 6865 206c 6173 7420 nnects the last \n-0000f690: 636f 7079 2062 6163 6b20 746f 2074 6865 copy back to the\n-0000f6a0: 206f 7269 6769 6e61 6c20 746f 2066 6f72 original to for\n-0000f6b0: 6d20 6120 706f 6c79 676f 6e0a 7769 7468 m a polygon.with\n-0000f6c0: 204e 2073 6964 6573 2e20 2049 6620 7468 N sides. If th\n-0000f6d0: 6572 6520 6172 6520 6d20 706f 696e 7473 ere are m points\n-0000f6e0: 2069 6e20 7468 6520 6f72 6967 696e 616c in the original\n-0000f6f0: 2070 6f69 6e74 206c 6973 742c 2074 6865 point list, the\n-0000f700: 6e20 6d0a 706f 6c79 676f 6e73 2077 6974 n m.polygons wit\n-0000f710: 6820 4e20 7369 6465 7320 6561 6368 2061 h N sides each a\n-0000f720: 7265 2066 6f72 6d65 6420 6279 2074 6865 re formed by the\n-0000f730: 2073 7765 6570 2e20 2049 6e20 7468 6973 sweep. In this\n-0000f740: 206d 616e 6e65 722c 0ae2 8098 7377 6565 manner,....swee\n-0000f750: 70e2 8099 2066 6f72 6d73 2061 2073 6574 p... forms a set\n-0000f760: 206f 6620 7477 6f2d 6469 6d65 6e73 696f of two-dimensio\n-0000f770: 6e61 6c20 6f62 6a65 6374 7320 2870 6f6c nal objects (pol\n-0000f780: 7967 6f6e 7329 2066 726f 6d0a 7a65 726f ygons) from.zero\n-0000f790: 2d64 696d 656e 7369 6f6e 616c 206f 6e65 -dimensional one\n-0000f7a0: 7320 2870 6f69 6e74 7329 2e0a 0a20 2020 s (points)... \n-0000f7b0: 4f70 7469 6f6e 7320 6f66 2074 6865 20e2 Options of the .\n-0000f7c0: 8098 7377 6565 70e2 8099 2061 7265 2063 ..sweep... are c\n-0000f7d0: 6f70 6965 6420 6469 7265 6374 6c79 2074 opied directly t\n-0000f7e0: 6f20 7468 6520 7265 7375 6c74 696e 670a o the resulting.\n-0000f7f0: 706f 6c79 6c69 6e65 2873 292e 0a0a 1f0a polyline(s).....\n-0000f800: 4669 6c65 3a20 736b 6574 6368 2e69 6e66 File: sketch.inf\n-0000f810: 6f2c 2020 4e6f 6465 3a20 5377 6570 7420 o, Node: Swept \n-0000f820: 6c69 6e65 732c 2020 4e65 7874 3a20 5377 lines, Next: Sw\n-0000f830: 6570 7420 706f 6c79 676f 6e73 2c20 2050 ept polygons, P\n-0000f840: 7265 763a 2053 7765 7074 2070 6f69 6e74 rev: Swept point\n-0000f850: 732c 2020 5570 3a20 5377 6565 7073 0a0a s, Up: Sweeps..\n-0000f860: 332e 322e 362e 3220 5377 6570 7420 6c69 3.2.6.2 Swept li\n-0000f870: 6e65 730a 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e nes.............\n-0000f880: 2e2e 2e2e 2e2e 2e0a 0a49 6620 5357 4550 .........If SWEP\n-0000f890: 545f 4f42 4a45 4354 2069 7320 6120 706f T_OBJECT is a po\n-0000f8a0: 6c79 6c69 6e65 2061 6e64 2074 6865 7265 lyline and there\n-0000f8b0: 2069 7320 6e6f 2063 6c6f 7375 7265 2074 is no closure t\n-0000f8c0: 6167 2c20 7468 656e 20e2 8098 7377 6565 ag, then ...swee\n-0000f8d0: 70e2 8099 0a63 6f6e 6e65 6374 7320 4e2b p....connects N+\n-0000f8e0: 3120 7375 6363 6573 7369 7665 2063 6f70 1 successive cop\n-0000f8f0: 6965 7320 6f66 2074 6865 2070 6f6c 796c ies of the polyl\n-0000f900: 696e 6520 2869 6e63 6c75 6469 6e67 2074 ine (including t\n-0000f910: 6865 206f 7269 6769 6e61 6c29 0a77 6974 he original).wit\n-0000f920: 6820 666f 7572 2d73 6964 6564 2070 6f6c h four-sided pol\n-0000f930: 7967 6f6e 732c 2065 6163 6820 7061 6972 ygons, each pair\n-0000f940: 206f 6620 636f 7069 6573 2067 6976 696e of copies givin\n-0000f950: 6720 7269 7365 2074 6f20 6120 2270 6f6c g rise to a \"pol\n-0000f960: 7967 6f6e 0a73 7472 6970 2e22 2020 4966 ygon.strip.\" If\n-0000f970: 2074 6865 7265 2061 7265 206d 2070 6f69 there are m poi\n-0000f980: 6e74 7320 696e 2074 6865 206f 7269 6769 nts in the origi\n-0000f990: 6e61 6c20 706f 6c79 6c69 6e65 2c20 7468 nal polyline, th\n-0000f9a0: 656e 2028 6d2d 3129 4e0a 706f 6c79 676f en (m-1)N.polygo\n-0000f9b0: 6e73 2061 7265 2066 6f72 6d65 6420 6279 ns are formed by\n-0000f9c0: 2074 6865 2073 7765 6570 2e20 2057 6520 the sweep. We \n-0000f9d0: 6361 6c6c 2074 6865 7365 20e2 809c 626f call these ...bo\n-0000f9e0: 6479 2070 6f6c 7967 6f6e 73e2 809d 2e20 dy polygons.... \n-0000f9f0: 2049 6e0a 7468 6973 206d 616e 6e65 722c In.this manner,\n-0000fa00: 20e2 8098 7377 6565 70e2 8099 2066 6f72 ...sweep... for\n-0000fa10: 6d73 2061 2074 776f 2d64 696d 656e 7369 ms a two-dimensi\n-0000fa20: 6f6e 616c 2073 7572 6661 6365 2066 726f onal surface fro\n-0000fa30: 6d20 6672 6f6d 2061 0a6f 6e65 2d64 696d m from a.one-dim\n-0000fa40: 656e 7369 6f6e 616c 2070 6f6c 796c 696e ensional polylin\n-0000fa50: 652e 0a0a 2020 2054 6865 206f 7264 6572 e... The order\n-0000fa60: 206f 6620 7665 7274 6963 6573 2070 726f of vertices pro\n-0000fa70: 6475 6365 6420 6279 20e2 8098 7377 6565 duced by ...swee\n-0000fa80: 70e2 8099 2069 7320 696d 706f 7274 616e p... is importan\n-0000fa90: 742e 2020 4966 2061 0a70 6f6c 7967 6f6e t. If a.polygon\n-0000faa0: 2773 2076 6572 7469 6365 7320 646f 206e 's vertices do n\n-0000fab0: 6f74 2061 7070 6561 7220 696e 2063 6f75 ot appear in cou\n-0000fac0: 6e74 6572 2d63 6c6f 636b 7769 7365 206f nter-clockwise o\n-0000fad0: 7264 6572 2069 6e20 7468 6520 6669 6e61 rder in the fina\n-0000fae0: 6c0a 696d 6167 652c 2074 6865 2070 6f6c l.image, the pol\n-0000faf0: 7967 6f6e 2077 696c 6c20 6265 2063 756c ygon will be cul\n-0000fb00: 6c65 6420 2875 6e6c 6573 7320 e280 9863 led (unless ...c\n-0000fb10: 756c 6c3d 6661 6c73 65e2 8099 2069 7320 ull=false... is \n-0000fb20: 7365 7429 2e20 2049 6620 7468 650a 706f set). If the.po\n-0000fb30: 696e 7473 2069 6e20 7468 6520 6b27 7468 ints in the k'th\n-0000fb40: 2063 6f70 7920 6f66 2074 6865 2070 6f6c copy of the pol\n-0000fb50: 796c 696e 6520 6172 6520 505f 312c 2050 yline are P_1, P\n-0000fb60: 5f32 2c20 2e2e 2e2c 2050 5f6d 2c20 616e _2, ..., P_m, an\n-0000fb70: 6420 7468 650a 706f 696e 7473 2069 6e20 d the.points in \n-0000fb80: 7468 6520 6e65 7874 2063 6f70 792c 2074 the next copy, t\n-0000fb90: 6865 2028 6b2b 3129 7374 2c20 6172 6520 he (k+1)st, are \n-0000fba0: 505f 3127 2c20 505f 3227 2c20 2e2e 2e2c P_1', P_2', ...,\n-0000fbb0: 2050 5f6d 272c 2074 6865 6e0a 7468 6520 P_m', then.the \n-0000fbc0: 7665 7274 6578 206f 7264 6572 206f 6620 vertex order of \n-0000fbd0: 7468 6520 6765 6e65 7261 7465 6420 706f the generated po\n-0000fbe0: 6c79 676f 6e73 2069 730a 2020 2020 2042 lygons is. B\n-0000fbf0: 6f64 7920 706f 6c79 676f 6e20 313a 2050 ody polygon 1: P\n-0000fc00: 5f32 2050 5f31 2050 5f31 2720 505f 3227 _2 P_1 P_1' P_2'\n-0000fc10: 0a20 2020 2020 426f 6479 2070 6f6c 7967 . Body polyg\n-0000fc20: 6f6e 2032 3a20 505f 3320 505f 3220 505f on 2: P_3 P_2 P_\n-0000fc30: 3227 2050 5f33 270a 2020 2020 202e 2e2e 2' P_3'. ...\n-0000fc40: 0a20 2020 2020 426f 6479 2070 6f6c 7967 . Body polyg\n-0000fc50: 6f6e 206d 2d31 3a20 505f 6d20 505f 7b6d on m-1: P_m P_{m\n-0000fc60: 2d31 7d20 505f 7b6d 2d31 7d27 2050 5f6d -1} P_{m-1}' P_m\n-0000fc70: 270a 0a20 2020 4f70 7469 6f6e 7320 6f66 '.. Options of\n-0000fc80: 2075 6e63 6c6f 7365 6420 6c69 6e65 2073 unclosed line s\n-0000fc90: 7765 6570 7320 6172 6520 636f 7069 6564 weeps are copied\n-0000fca0: 2074 6f20 6561 6368 206f 7574 7075 7420 to each output \n-0000fcb0: 706f 6c79 676f 6e2e 0a4f 7074 696f 6e73 polygon..Options\n-0000fcc0: 206f 6620 7468 6520 7377 6570 7420 6c69 of the swept li\n-0000fcd0: 6e65 2061 7265 2069 676e 6f72 6564 2e0a ne are ignored..\n-0000fce0: 0a20 2020 5768 656e 2074 6865 7265 205f . When there _\n-0000fcf0: 6973 5f20 6120 636c 6f73 7572 6520 7461 is_ a closure ta\n-0000fd00: 672c 2074 6865 6e20 e280 9873 7765 6570 g, then ...sweep\n-0000fd10: e280 9920 636f 6e6e 6563 7473 204e 2073 ... connects N s\n-0000fd20: 7563 6365 7373 6976 650a 636f 7069 6573 uccessive.copies\n-0000fd30: 206f 6620 7468 6520 706f 6c79 6c69 6e65 of the polyline\n-0000fd40: 2028 696e 636c 7564 696e 6720 7468 6520 (including the \n-0000fd50: 6f72 6967 696e 616c 2920 7769 7468 2066 original) with f\n-0000fd60: 6f75 722d 7369 6465 6420 626f 6479 0a70 our-sided body.p\n-0000fd70: 6f6c 7967 6f6e 7320 6a75 7374 2061 7320 olygons just as \n-0000fd80: 7468 6520 6361 7365 2077 6974 6820 6e6f the case with no\n-0000fd90: 2063 6c6f 7375 7265 2074 6167 2e20 2049 closure tag. I\n-0000fda0: 7420 7468 656e 2063 6f6e 6e65 6374 7320 t then connects \n-0000fdb0: 7468 650a 6c61 7374 2063 6f70 7920 6261 the.last copy ba\n-0000fdc0: 636b 2074 6f20 7468 6520 6f72 6967 696e ck to the origin\n-0000fdd0: 616c 2074 6f20 666f 726d 2061 2072 6962 al to form a rib\n-0000fde0: 626f 6e2d 7368 6170 6564 2073 7572 6661 bon-shaped surfa\n-0000fdf0: 6365 2074 6861 740a 636c 6f73 6573 206f ce that.closes o\n-0000fe00: 6e20 6974 7365 6c66 2077 6974 6820 7477 n itself with tw\n-0000fe10: 6f20 686f 6c65 7320 7265 6d61 696e 696e o holes remainin\n-0000fe20: 672e 0a0a 2020 2046 696e 616c 6c79 2c20 g... Finally, \n-0000fe30: 7468 6520 7377 6565 7020 6164 6473 2074 the sweep adds t\n-0000fe40: 776f 206d 6f72 6520 706f 6c79 676f 6e73 wo more polygons\n-0000fe50: 2074 6f20 7365 616c 2074 6865 2068 6f6c to seal the hol\n-0000fe60: 6573 2061 6e64 2066 6f72 6d0a 6120 636c es and form.a cl\n-0000fe70: 6f73 6564 2073 7572 6661 6365 2074 6861 osed surface tha\n-0000fe80: 742c 2064 6570 656e 6469 6e67 206f 6e20 t, depending on \n-0000fe90: 7468 6520 7377 6565 7020 7472 616e 7366 the sweep transf\n-0000fea0: 6f72 6d73 2c20 6d61 7920 7265 7072 6573 orms, may repres\n-0000feb0: 656e 740a 7468 6520 626f 756e 6461 7279 ent.the boundary\n-0000fec0: 206f 6620 6120 736f 6c69 642e 2020 496e of a solid. In\n-0000fed0: 2074 6869 7320 6d61 6e6e 6572 2c20 e280 this manner, ..\n-0000fee0: 9873 7765 6570 e280 9920 666f 726d 7320 .sweep... forms \n-0000fef0: 7468 6520 626f 756e 6461 7279 206f 660a the boundary of.\n-0000ff00: 6120 7468 7265 652d 6469 6d65 6e73 696f a three-dimensio\n-0000ff10: 6e61 6c20 6f62 6a65 6374 2066 726f 6d20 nal object from \n-0000ff20: 6120 6f6e 652d 6469 6d65 6e73 696f 6e61 a one-dimensiona\n-0000ff30: 6c20 706f 6c79 6c69 6e65 2e20 2057 6520 l polyline. We \n-0000ff40: 6361 6c6c 0a74 6865 7365 2068 6f6c 652d call.these hole-\n-0000ff50: 6669 6c6c 696e 6720 706f 6c79 676f 6e73 filling polygons\n-0000ff60: 20e2 809c 656e 6473 e280 9d2e 0a0a 2020 ...ends...... \n-0000ff70: 2054 6865 206f 7264 6572 206f 6620 7665 The order of ve\n-0000ff80: 7274 6963 6573 206f 6620 656e 6420 706f rtices of end po\n-0000ff90: 6c79 676f 6e73 2069 7320 696d 706f 7274 lygons is import\n-0000ffa0: 616e 7420 666f 7220 636f 7272 6563 740a ant for correct.\n-0000ffb0: 6375 6c6c 696e 6720 6173 2064 6573 6372 culling as descr\n-0000ffc0: 6962 6564 2061 626f 7665 2e20 2049 6620 ibed above. If \n-0000ffd0: 505f 315e 312c 2050 5f31 5e32 2c20 2e2e P_1^1, P_1^2, ..\n-0000ffe0: 2e2c 2050 5f31 5e6e 2061 7265 2074 6865 ., P_1^n are the\n-0000fff0: 204e 0a63 6f70 6965 7320 6f66 2074 6865 N.copies of the\n-00010000: 2066 6972 7374 2070 6f6c 796c 696e 6520 first polyline \n-00010010: 706f 696e 7420 616e 6420 505f 6d5e 312c point and P_m^1,\n-00010020: 2050 5f6d 5e32 2c20 2e2e 2e20 2c50 5f6d P_m^2, ... ,P_m\n-00010030: 5e6e 2061 7265 2074 6865 0a4e 2063 6f70 ^n are the.N cop\n-00010040: 6965 7320 6f66 2074 6865 206c 6173 7420 ies of the last \n-00010050: 706f 6c79 6c69 6e65 2070 6f69 6e74 2c20 polyline point, \n-00010060: 7468 656e 2074 6865 2065 6e64 2070 6f6c then the end pol\n-00010070: 7967 6f6e 2076 6572 7465 7820 6f72 6465 ygon vertex orde\n-00010080: 720a 6973 0a20 2020 2020 456e 6420 706f r.is. End po\n-00010090: 6c79 676f 6e20 313a 2050 5f31 5e6e 2c20 lygon 1: P_1^n, \n-000100a0: 505f 315e 7b6e 2d31 7d2c 202e 2e2e 202c P_1^{n-1}, ... ,\n-000100b0: 505f 315e 310a 2020 2020 2045 6e64 2070 P_1^1. End p\n-000100c0: 6f6c 7967 6f6e 2032 3a20 505f 6d5e 312c olygon 2: P_m^1,\n-000100d0: 2050 5f6d 5e32 2c20 2e2e 2e20 2c50 5f6d P_m^2, ... ,P_m\n-000100e0: 5e6e 0a0a 2020 2049 6620 7468 6572 6520 ^n.. If there \n-000100f0: 6172 6520 6e6f 206f 7074 696f 6e73 206f are no options o\n-00010100: 6e20 7468 6520 7377 6570 7420 6c69 6e65 n the swept line\n-00010110: 2c20 7468 656e 2074 6865 20e2 8098 7377 , then the ...sw\n-00010120: 6565 70e2 8099 206f 7074 696f 6e73 0a61 eep... options.a\n-00010130: 7265 2063 6f70 6965 6420 746f 2065 6163 re copied to eac\n-00010140: 6820 6f75 7470 7574 2070 6f6c 7967 6f6e h output polygon\n-00010150: 2e20 2049 6620 7468 6520 7377 6570 7420 . If the swept \n-00010160: 6c69 6e65 2064 6f65 7320 6861 7665 206f line does have o\n-00010170: 7074 696f 6e73 2c0a 7468 6573 6520 6172 ptions,.these ar\n-00010180: 6520 636f 7069 6564 2074 6f20 636f 7272 e copied to corr\n-00010190: 6573 706f 6e64 696e 6720 626f 6479 2070 esponding body p\n-000101a0: 6f6c 7967 6f6e 733b 2074 6865 2073 7765 olygons; the swe\n-000101b0: 6570 206f 7074 696f 6e73 2061 7265 0a63 ep options are.c\n-000101c0: 6f70 6965 6420 746f 2074 6865 2065 6e64 opied to the end\n-000101d0: 2070 6f6c 7967 6f6e 732e 2020 496e 2074 polygons. In t\n-000101e0: 6869 7320 6d61 6e6e 6572 2c20 626f 6479 his manner, body\n-000101f0: 2061 6e64 2065 6e64 7320 6d61 7920 6265 and ends may be\n-00010200: 2064 7261 776e 0a77 6974 6820 6469 6666 drawn.with diff\n-00010210: 6572 656e 7420 6368 6172 6163 7465 7269 erent characteri\n-00010220: 7374 6963 7320 7375 6368 2061 7320 e280 stics such as ..\n-00010230: 9866 696c 6c63 6f6c 6f72 e280 992e 0a0a .fillcolor......\n-00010240: 1f0a 4669 6c65 3a20 736b 6574 6368 2e69 ..File: sketch.i\n-00010250: 6e66 6f2c 2020 4e6f 6465 3a20 5377 6570 nfo, Node: Swep\n-00010260: 7420 706f 6c79 676f 6e73 2c20 204e 6578 t polygons, Nex\n-00010270: 743a 2053 7765 7074 2062 6c6f 636b 732c t: Swept blocks,\n-00010280: 2020 5072 6576 3a20 5377 6570 7420 6c69 Prev: Swept li\n-00010290: 6e65 732c 2020 5570 3a20 5377 6565 7073 nes, Up: Sweeps\n-000102a0: 0a0a 332e 322e 362e 3320 5377 6570 7420 ..3.2.6.3 Swept \n-000102b0: 706f 6c79 676f 6e73 0a2e 2e2e 2e2e 2e2e polygons........\n-000102c0: 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e0a ................\n-000102d0: 0a49 6620 5357 4550 545f 4f42 4a45 4354 .If SWEPT_OBJECT\n-000102e0: 2069 7320 6120 706f 6c79 676f 6e2c 2074 is a polygon, t\n-000102f0: 6865 20e2 8098 7377 6565 70e2 8099 2063 he ...sweep... c\n-00010300: 6f6e 6e65 6374 7320 4e2b 3120 7375 6363 onnects N+1 succ\n-00010310: 6573 7369 7665 2063 6f70 6965 730a 6f66 essive copies.of\n-00010320: 2074 6865 2063 6c6f 7365 6420 706f 6c79 the closed poly\n-00010330: 6c69 6e65 2062 6f72 6465 7220 6f66 2074 line border of t\n-00010340: 6865 2070 6f6c 7967 6f6e 2074 6f20 666f he polygon to fo\n-00010350: 726d 2062 6f64 7920 706f 6c79 676f 6e73 rm body polygons\n-00010360: 0a65 7861 6374 6c79 2061 7320 7468 6f75 .exactly as thou\n-00010370: 6768 2074 6865 2062 6f72 6465 7220 7765 gh the border we\n-00010380: 7265 2061 2073 7765 7074 2070 6f6c 796c re a swept polyl\n-00010390: 696e 6520 6173 2064 6573 6372 6962 6564 ine as described\n-000103a0: 2069 6e20 2a6e 6f74 650a 5377 6570 7420 in *note.Swept \n-000103b0: 6c69 6e65 733a 3a2e 2020 4966 2074 6865 lines::. If the\n-000103c0: 7265 2061 7265 206d 2070 6f69 6e74 7320 re are m points \n-000103d0: 696e 2074 6865 206f 7269 6769 6e61 6c20 in the original \n-000103e0: 706f 6c79 676f 6e2c 2074 6865 6e20 6d4e polygon, then mN\n-000103f0: 0a62 6f64 7920 706f 6c79 676f 6e73 2061 .body polygons a\n-00010400: 7265 2066 6f72 6d65 6420 6279 2074 6869 re formed by thi\n-00010410: 7320 7377 6565 702e 2020 5468 6520 626f s sweep. The bo\n-00010420: 6479 2070 6f6c 7967 6f6e 7320 666f 726d dy polygons form\n-00010430: 2061 6e0a e280 9c65 7874 7275 7369 6f6e an....extrusion\n-00010440: e280 9d20 6f66 2074 6865 2062 6f75 6e64 ... of the bound\n-00010450: 6172 7920 6f66 2074 6865 206f 7269 6769 ary of the origi\n-00010460: 6e61 6c20 706f 6c79 676f 6e20 7769 7468 nal polygon with\n-00010470: 2074 776f 2068 6f6c 6573 2061 740a 7468 two holes at.th\n-00010480: 6520 6f70 656e 2065 6e64 732e 0a0a 2020 e open ends... \n-00010490: 2046 696e 616c 6c79 2c20 7468 6520 7377 Finally, the sw\n-000104a0: 6565 7020 6164 6473 2074 776f 2063 6f70 eep adds two cop\n-000104b0: 6965 7320 6f66 2074 6865 206f 7269 6769 ies of the origi\n-000104c0: 6e61 6c20 706f 6c79 676f 6e20 746f 2063 nal polygon to c\n-000104d0: 6f76 6572 0a74 6865 2068 6f6c 6573 2e20 over.the holes. \n-000104e0: 2057 6520 6361 6c6c 2074 6865 7365 2068 We call these h\n-000104f0: 6f6c 652d 6669 6c6c 696e 6720 706f 6c79 ole-filling poly\n-00010500: 676f 6e73 20e2 809c 656e 6473 e280 9d2e gons ...ends....\n-00010510: 2020 496e 2074 6869 7320 6d61 6e6e 6572 In this manner\n-00010520: 2c0a e280 9873 7765 6570 e280 9920 666f ,....sweep... fo\n-00010530: 726d 7320 7468 6520 626f 756e 6461 7279 rms the boundary\n-00010540: 206f 6620 6120 7468 7265 652d 6469 6d65 of a three-dime\n-00010550: 6e73 696f 6e61 6c20 6f62 6a65 6374 2066 nsional object f\n-00010560: 726f 6d20 610a 7477 6f2d 6469 6d65 6e73 rom a.two-dimens\n-00010570: 696f 6e61 6c20 706f 6c79 676f 6e2e 0a0a ional polygon...\n-00010580: 2020 2054 6865 206f 7264 6572 206f 6620 The order of \n-00010590: 7665 7274 6963 6573 206f 6620 656e 6420 vertices of end \n-000105a0: 706f 6c79 676f 6e73 2069 7320 696d 706f polygons is impo\n-000105b0: 7274 616e 7420 666f 7220 636f 7272 6563 rtant for correc\n-000105c0: 740a 6375 6c6c 696e 6720 6173 2064 6573 t.culling as des\n-000105d0: 6372 6962 6564 2061 626f 7665 2e20 2041 cribed above. A\n-000105e0: 6e20 6578 6163 7420 636f 7079 206f 6620 n exact copy of \n-000105f0: 7468 6520 6f72 6967 696e 616c 2070 6f6c the original pol\n-00010600: 7967 6f6e 2077 6974 680a 7665 7274 6578 ygon with.vertex\n-00010610: 206f 7264 6572 2069 6e74 6163 7420 666f order intact fo\n-00010620: 726d 7320 7468 6520 6669 7273 7420 656e rms the first en\n-00010630: 6420 706f 6c79 676f 6e2e 2020 5468 6520 d polygon. The \n-00010640: 6f74 6865 7220 656e 6420 706f 6c79 676f other end polygo\n-00010650: 6e0a 7265 7375 6c74 7320 6672 6f6d 2074 n.results from t\n-00010660: 7261 6e73 666f 726d 696e 6720 616e 6420 ransforming and \n-00010670: 7468 6520 7265 7665 7273 696e 6720 7468 the reversing th\n-00010680: 6520 6f72 6465 7220 6f66 2076 6572 7469 e order of verti\n-00010690: 6365 7320 696e 2074 6865 0a6f 7269 6769 ces in the.origi\n-000106a0: 6e61 6c2e 2020 5468 6520 7472 616e 7366 nal. The transf\n-000106b0: 6f72 6d20 706c 6163 6573 2074 6865 206f orm places the o\n-000106c0: 7269 6769 6e61 6c20 706f 6c79 676f 6e20 riginal polygon \n-000106d0: 6174 2074 6865 2075 6e63 6f76 6572 6564 at the uncovered\n-000106e0: 0a68 6f6c 653b 2069 7420 6973 0a20 2020 .hole; it is. \n-000106f0: 2020 545f 315e 6e20 7468 656e 2054 5f32 T_1^n then T_2\n-00010700: 5e6e 2074 6865 6e20 2e2e 2e20 7468 656e ^n then ... then\n-00010710: 2054 5f52 5e6e 2e0a 4966 2074 6865 7265 T_R^n..If there\n-00010720: 2061 7265 206e 6f20 6f70 7469 6f6e 7320 are no options \n-00010730: 6f6e 2074 6865 2073 7765 7074 2070 6f6c on the swept pol\n-00010740: 7967 6f6e 2c20 7468 656e 2074 6865 20e2 ygon, then the .\n-00010750: 8098 7377 6565 70e2 8099 206f 7074 696f ..sweep... optio\n-00010760: 6e73 0a61 7265 2063 6f70 6965 6420 746f ns.are copied to\n-00010770: 2065 6163 6820 6f75 7470 7574 2070 6f6c each output pol\n-00010780: 7967 6f6e 2e20 2049 6620 7468 6520 7377 ygon. If the sw\n-00010790: 6570 7420 706f 6c79 676f 6e20 646f 6573 ept polygon does\n-000107a0: 2068 6176 650a 6f70 7469 6f6e 732c 2074 have.options, t\n-000107b0: 6865 7365 2061 7265 2063 6f70 6965 6420 hese are copied \n-000107c0: 746f 2074 6865 2065 6e64 733b 2074 6865 to the ends; the\n-000107d0: 2073 7765 6570 206f 7074 696f 6e73 2061 sweep options a\n-000107e0: 7265 2063 6f70 6965 6420 746f 0a74 6865 re copied to.the\n-000107f0: 2062 6f64 7920 706f 6c79 676f 6e73 2e20 body polygons. \n-00010800: 2049 6e20 7468 6973 206d 616e 6e65 722c In this manner,\n-00010810: 2062 6f64 7920 616e 6420 656e 6473 206d body and ends m\n-00010820: 6179 2062 6520 6472 6177 6e20 7769 7468 ay be drawn with\n-00010830: 0a64 6966 6665 7265 6e74 2063 6861 7261 .different chara\n-00010840: 6374 6572 6973 7469 6373 2073 7563 6820 cteristics such \n-00010850: 6173 20e2 8098 6669 6c6c 636f 6c6f 72e2 as ...fillcolor.\n-00010860: 8099 2e0a 0a1f 0a46 696c 653a 2073 6b65 .......File: ske\n-00010870: 7463 682e 696e 666f 2c20 204e 6f64 653a tch.info, Node:\n-00010880: 2053 7765 7074 2062 6c6f 636b 732c 2020 Swept blocks, \n-00010890: 4e65 7874 3a20 5377 6565 7020 6661 6365 Next: Sweep face\n-000108a0: 2073 706c 6974 7469 6e67 2c20 2050 7265 splitting, Pre\n-000108b0: 763a 2053 7765 7074 2070 6f6c 7967 6f6e v: Swept polygon\n-000108c0: 732c 2020 5570 3a20 5377 6565 7073 0a0a s, Up: Sweeps..\n-000108d0: 332e 322e 362e 3420 5377 6570 7420 626c 3.2.6.4 Swept bl\n-000108e0: 6f63 6b73 0a2e 2e2e 2e2e 2e2e 2e2e 2e2e ocks............\n-000108f0: 2e2e 2e2e 2e2e 2e2e 2e0a 0a54 6865 2073 ...........The s\n-00010900: 7765 7074 206f 626a 6563 7420 5357 4550 wept object SWEP\n-00010910: 545f 4f42 4a45 4354 206d 6179 2061 6c73 T_OBJECT may als\n-00010920: 6f20 6265 2061 6e79 2063 6f6c 6c65 6374 o be any collect\n-00010930: 696f 6e20 6f66 2070 6f6c 796c 696e 6573 ion of polylines\n-00010940: 0a61 6e64 2070 6f6c 7967 6f6e 732e 2020 .and polygons. \n-00010950: 5468 6973 206d 6179 2062 6520 6120 626c This may be a bl\n-00010960: 6f63 6b20 636f 6d70 6f73 6564 206f 6620 ock composed of \n-00010970: e280 986c 696e 65e2 8099 2061 6e64 2f6f ...line... and/o\n-00010980: 7220 e280 9870 6f6c 7967 6f6e e280 990a r ...polygon....\n-00010990: 636f 6d6d 616e 6473 2069 6e20 6272 6163 commands in brac\n-000109a0: 6573 20e2 8098 7b20 7de2 8099 2c20 6f72 es ...{ }..., or\n-000109b0: 2069 7420 6d61 7920 6265 2074 6865 2072 it may be the r\n-000109c0: 6573 756c 7420 6f66 2061 20e2 8098 7265 esult of a ...re\n-000109d0: 7065 6174 e280 992c 2061 6e6f 7468 6572 peat..., another\n-000109e0: 0ae2 8098 7377 6565 70e2 8099 2c20 6574 ....sweep..., et\n-000109f0: 632e 2020 5468 6520 7377 6565 7020 6163 c. The sweep ac\n-00010a00: 7473 2069 6e64 6570 656e 6465 6e74 6c79 ts independently\n-00010a10: 206f 6e20 6561 6368 206f 626a 6563 7420 on each object \n-00010a20: 696e 2074 6865 2062 6c6f 636b 0a65 7861 in the block.exa\n-00010a30: 6374 6c79 2061 7320 6966 2069 7420 7765 ctly as if it we\n-00010a40: 7265 2061 2073 696e 676c 6520 7377 6570 re a single swep\n-00010a50: 7420 6f62 6a65 6374 2064 6573 6372 6962 t object describ\n-00010a60: 6564 2061 626f 7665 2069 6e20 2a6e 6f74 ed above in *not\n-00010a70: 650a 5377 6570 7420 6c69 6e65 733a 3a20 e.Swept lines:: \n-00010a80: 616e 6420 2a6e 6f74 6520 5377 6570 7420 and *note Swept \n-00010a90: 706f 6c79 676f 6e73 3a3a 2e0a 0a1f 0a46 polygons::.....F\n-00010aa0: 696c 653a 2073 6b65 7463 682e 696e 666f ile: sketch.info\n-00010ab0: 2c20 204e 6f64 653a 2053 7765 6570 2066 , Node: Sweep f\n-00010ac0: 6163 6520 7370 6c69 7474 696e 672c 2020 ace splitting, \n-00010ad0: 5072 6576 3a20 5377 6570 7420 626c 6f63 Prev: Swept bloc\n-00010ae0: 6b73 2c20 2055 703a 2053 7765 6570 730a ks, Up: Sweeps.\n-00010af0: 0a33 2e32 2e36 2e35 2053 7765 6570 2066 .3.2.6.5 Sweep f\n-00010b00: 6163 6520 7370 6c69 7474 696e 670a 2e2e ace splitting...\n-00010b10: 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e ................\n-00010b20: 2e2e 2e2e 2e2e 2e2e 2e2e 0a0a 4265 666f ............Befo\n-00010b30: 7265 2073 656e 6469 6e67 2065 6163 6820 re sending each \n-00010b40: 666f 7572 2d73 6964 6564 2062 6f64 7920 four-sided body \n-00010b50: 706f 6c79 676f 6e20 6f66 2061 20e2 8098 polygon of a ...\n-00010b60: 7377 6565 70e2 8099 2074 6f20 7468 6520 sweep... to the \n-00010b70: 6f75 7470 7574 2c0a e280 9873 6b65 7463 output,....sketc\n-00010b80: 68e2 8099 2074 6573 7473 2074 6f20 7365 h... tests to se\n-00010b90: 6520 6966 2069 7420 6973 2072 6f75 6768 e if it is rough\n-00010ba0: 6c79 2070 6c61 6e61 722e 2020 5369 6e63 ly planar. Sinc\n-00010bb0: 6520 706c 616e 6172 6974 7920 6973 0a6e e planarity is.n\n-00010bc0: 6563 6573 7361 7279 2066 6f72 2070 726f ecessary for pro\n-00010bd0: 7065 7220 6675 6e63 7469 6f6e 696e 6720 per functioning \n-00010be0: 6f66 2074 6865 2068 6964 6465 6e20 7375 of the hidden su\n-00010bf0: 7266 6163 6520 616c 676f 7269 7468 6d2c rface algorithm,\n-00010c00: 0a22 7761 7270 6564 2220 706f 6c79 676f .\"warped\" polygo\n-00010c10: 6e73 2061 7265 2061 7574 6f6d 6174 6963 ns are automatic\n-00010c20: 616c 6c79 2073 706c 6974 2069 6e74 6f20 ally split into \n-00010c30: 7477 6f20 7472 6961 6e67 6c65 732e 0a0a two triangles...\n-00010c40: 2020 2048 6f6c 652d 6669 6c6c 696e 6720 Hole-filling \n-00010c50: 706f 6c79 676f 6e73 2070 726f 6475 6365 polygons produce\n-00010c60: 6420 6279 2063 6c6f 7375 7265 2d74 6167 d by closure-tag\n-00010c70: 6765 6420 6c69 6e65 2073 7765 6570 7320 ged line sweeps \n-00010c80: 6172 6520 6e6f 740a 7370 6c69 742e 2020 are not.split. \n-00010c90: 4e6f 7220 6172 6520 6f72 6967 696e 616c Nor are original\n-00010ca0: 2070 6f6c 7967 6f6e 7320 696e 2070 6f6c polygons in pol\n-00010cb0: 7967 6f6e 2073 7765 6570 732e 2020 4974 ygon sweeps. It\n-00010cc0: 2069 7320 7468 6520 7573 6572 2773 0a72 is the user's.r\n-00010cd0: 6573 706f 6e73 6962 696c 6974 7920 746f esponsibility to\n-00010ce0: 2065 6e73 7572 6520 7468 6573 6520 6172 ensure these ar\n-00010cf0: 6520 706c 616e 6172 2e0a 0a1f 0a46 696c e planar.....Fil\n-00010d00: 653a 2073 6b65 7463 682e 696e 666f 2c20 e: sketch.info, \n-00010d10: 204e 6f64 653a 2042 6c6f 636b 732c 2020 Node: Blocks, \n-00010d20: 4e65 7874 3a20 5265 7065 6174 732c 2020 Next: Repeats, \n-00010d30: 5072 6576 3a20 5377 6565 7073 2c20 2055 Prev: Sweeps, U\n-00010d40: 703a 2044 7261 7761 626c 6573 0a0a 332e p: Drawables..3.\n-00010d50: 322e 3720 426c 6f63 6b73 0a2d 2d2d 2d2d 2.7 Blocks.-----\n-00010d60: 2d2d 2d2d 2d2d 2d0a 0a41 6e79 2073 6571 -------..Any seq\n-00010d70: 7565 6e63 6520 6f66 2064 7261 7761 626c uence of drawabl\n-00010d80: 6573 206d 6179 2062 6520 6772 6f75 7065 es may be groupe\n-00010d90: 6420 696e 2061 20e2 809c 626c 6f63 6be2 d in a ...block.\n-00010da0: 809d 206d 6572 656c 7920 6279 0a65 6e63 .. merely by.enc\n-00010db0: 6c6f 7369 6e67 2074 6865 6d20 696e 2062 losing them in b\n-00010dc0: 7261 6365 7320 e280 987b 207d e280 992e races ...{ }....\n-00010dd0: 2020 4120 626c 6f63 6b20 6973 2069 7473 A block is its\n-00010de0: 656c 6620 6472 6177 6162 6c65 2e20 2041 elf drawable. A\n-00010df0: 206b 6579 2075 7365 0a6f 6620 626c 6f63 key use.of bloc\n-00010e00: 6b73 2069 7320 746f 2065 7874 656e 6420 ks is to extend \n-00010e10: 7468 6520 6566 6665 6374 206f 6620 6120 the effect of a \n-00010e20: 7369 6e67 6c65 20e2 8098 6465 66e2 8099 single ...def...\n-00010e30: 2c20 2a6e 6f74 650a 4465 6669 6e69 7469 , *note.Definiti\n-00010e40: 6f6e 733a 3a2c 20e2 8098 7075 74e2 8099 ons::, ...put...\n-00010e50: 202a 6e6f 7465 2050 7574 733a 3a2c 20e2 *note Puts::, .\n-00010e60: 8098 7377 6565 70e2 8099 202a 6e6f 7465 ..sweep... *note\n-00010e70: 2053 7765 6570 733a 3a2c 206f 7220 e280 Sweeps::, or ..\n-00010e80: 9872 6570 6561 74e2 8099 0a2a 6e6f 7465 .repeat....*note\n-00010e90: 2052 6570 6561 7473 3a3a 2074 6f20 696e Repeats:: to in\n-00010ea0: 636c 7564 6520 7365 7665 7261 6c20 6f62 clude several ob\n-00010eb0: 6a65 6374 7320 7261 7468 6572 2074 6861 jects rather tha\n-00010ec0: 6e20 6f6e 652e 0a0a 2020 2044 6566 696e n one... Defin\n-00010ed0: 6974 696f 6e73 2028 2a4e 6f74 6520 4465 itions (*Note De\n-00010ee0: 6669 6e69 7469 6f6e 733a 3a2e 2920 2069 finitions::.) i\n-00010ef0: 6e73 6964 6520 6120 626c 6f63 6b20 6861 nside a block ha\n-00010f00: 7665 20e2 809c 6c65 7869 6361 6c0a 7363 ve ...lexical.sc\n-00010f10: 6f70 65e2 809d 2065 7874 656e 6469 6e67 ope... extending\n-00010f20: 2066 726f 6d20 7468 6520 706c 6163 6520 from the place \n-00010f30: 6f66 2064 6566 696e 6974 696f 6e20 746f of definition to\n-00010f40: 2074 6865 2065 6e64 206f 6620 7468 6520 the end of the \n-00010f50: 626c 6f63 6b2e 0a0a 1f0a 4669 6c65 3a20 block.....File: \n-00010f60: 736b 6574 6368 2e69 6e66 6f2c 2020 4e6f sketch.info, No\n-00010f70: 6465 3a20 5265 7065 6174 732c 2020 4e65 de: Repeats, Ne\n-00010f80: 7874 3a20 5075 7473 2c20 2050 7265 763a xt: Puts, Prev:\n-00010f90: 2042 6c6f 636b 732c 2020 5570 3a20 4472 Blocks, Up: Dr\n-00010fa0: 6177 6162 6c65 730a 0a33 2e32 2e38 2052 awables..3.2.8 R\n-00010fb0: 6570 6561 7473 0a2d 2d2d 2d2d 2d2d 2d2d epeats.---------\n-00010fc0: 2d2d 2d2d 0a0a 2a53 796e 7461 783a 2a0a ----..*Syntax:*.\n-00010fd0: 2020 2020 2072 6570 6561 7420 7b20 4e2c repeat { N,\n-00010fe0: 2054 5f31 2c20 545f 322c 202e 2e2e 2c20 T_1, T_2, ..., \n-00010ff0: 545f 5220 7d20 5245 5045 4154 4544 5f4f T_R } REPEATED_O\n-00011000: 424a 4543 540a 5468 6520 7265 7065 6174 BJECT.The repeat\n-00011010: 206d 616b 6573 204e 2074 7261 6e73 666f makes N transfo\n-00011020: 726d 6564 2063 6f70 6965 7320 6f66 2052 rmed copies of R\n-00011030: 4550 4541 5445 445f 4f42 4a45 4354 2028 EPEATED_OBJECT (\n-00011040: 696e 636c 7564 696e 6720 7468 650a 6f72 including the.or\n-00011050: 6967 696e 616c 292e 2020 5468 6520 545f iginal). The T_\n-00011060: 4920 6172 6520 7472 616e 7366 6f72 6d73 I are transforms\n-00011070: 2e20 2054 6865 206b 2774 6820 636f 7079 . The k'th copy\n-00011080: 206f 6620 7468 650a 5245 5045 4154 4544 of the.REPEATED\n-00011090: 5f4f 424a 4543 5420 2866 6f72 206b 3d30 _OBJECT (for k=0\n-000110a0: 2c31 2c2e 2e2e 2c6e 2d31 2920 6973 2070 ,1,...,n-1) is p\n-000110b0: 726f 6475 6365 6420 696e 2074 6865 2073 roduced in the s\n-000110c0: 616d 6520 6d61 6e6e 6572 2061 730a 666f ame manner as.fo\n-000110d0: 7220 e280 9873 7765 6570 e280 9973 2064 r ...sweep...s d\n-000110e0: 6573 6372 6962 6564 2069 6e20 2a6e 6f74 escribed in *not\n-000110f0: 6520 5377 6565 7073 3a3a 2e20 2054 6869 e Sweeps::. Thi\n-00011100: 7320 6973 2072 6570 6561 7465 6420 6865 s is repeated he\n-00011110: 7265 2028 6e6f 2070 756e 0a69 6e74 656e re (no pun.inten\n-00011120: 6465 6429 2066 6f72 2063 6f6e 7665 6e69 ded) for conveni\n-00011130: 656e 6365 2e20 2054 6f20 6d61 6b65 2074 ence. To make t\n-00011140: 6865 206b 2774 6820 636f 7079 2c20 7468 he k'th copy, th\n-00011150: 6520 666f 6c6c 6f77 696e 670a 7472 616e e following.tran\n-00011160: 7366 6f72 6d20 6973 2061 7070 6c69 6564 sform is applied\n-00011170: 2074 6f20 7468 6520 6f72 6967 696e 616c to the original\n-00011180: 206f 626a 6563 742e 0a20 2020 2020 545f object.. T_\n-00011190: 315e 6b20 7468 656e 2054 5f32 5e6b 2074 1^k then T_2^k t\n-000111a0: 6865 6e20 2e2e 2e20 7468 656e 2054 5f52 hen ... then T_R\n-000111b0: 5e6b 0a48 6572 6520 545e 6b20 6d65 616e ^k.Here T^k mean\n-000111c0: 7320 2274 7261 6e73 666f 726d 2054 2061 s \"transform T a\n-000111d0: 7070 6c69 6564 206b 2074 696d 6573 2e22 pplied k times.\"\n-000111e0: 0a0a 1f0a 4669 6c65 3a20 736b 6574 6368 ....File: sketch\n-000111f0: 2e69 6e66 6f2c 2020 4e6f 6465 3a20 5075 .info, Node: Pu\n-00011200: 7473 2c20 2050 7265 763a 2052 6570 6561 ts, Prev: Repea\n-00011210: 7473 2c20 2055 703a 2044 7261 7761 626c ts, Up: Drawabl\n-00011220: 6573 0a0a 332e 322e 3920 5075 7473 0a2d es..3.2.9 Puts.-\n-00011230: 2d2d 2d2d 2d2d 2d2d 2d0a 0a2a 5379 6e74 ---------..*Synt\n-00011240: 6178 3a2a 0a20 2020 2020 7075 7420 7b20 ax:*. put { \n-00011250: 5420 7d20 5055 545f 4f42 4a45 4354 0a50 T } PUT_OBJECT.P\n-00011260: 7574 206d 6572 656c 7920 6170 706c 6965 ut merely applie\n-00011270: 7320 7472 616e 7366 6f72 6d20 5420 746f s transform T to\n-00011280: 2074 6865 2064 7261 7761 626c 6520 5055 the drawable PU\n-00011290: 545f 4f42 4a45 4354 2e0a 0a1f 0a46 696c T_OBJECT.....Fil\n-000112a0: 653a 2073 6b65 7463 682e 696e 666f 2c20 e: sketch.info, \n-000112b0: 204e 6f64 653a 2044 6566 696e 6974 696f Node: Definitio\n-000112c0: 6e73 2c20 204e 6578 743a 2047 6c6f 6261 ns, Next: Globa\n-000112d0: 6c20 656e 7669 726f 6e6d 656e 742c 2020 l environment, \n-000112e0: 5072 6576 3a20 4472 6177 6162 6c65 732c Prev: Drawables,\n-000112f0: 2020 5570 3a20 496e 7075 7420 6c61 6e67 Up: Input lang\n-00011300: 7561 6765 0a0a 332e 3320 4465 6669 6e69 uage..3.3 Defini\n-00011310: 7469 6f6e 730a 3d3d 3d3d 3d3d 3d3d 3d3d tions.==========\n-00011320: 3d3d 3d3d 3d0a 0a44 6566 696e 6974 696f =====..Definitio\n-00011330: 6e73 2067 6976 6520 6e61 6d65 7320 746f ns give names to\n-00011340: 20e2 8098 736b 6574 6368 e280 9920 6f62 ...sketch... ob\n-00011350: 6a65 6374 732e 2020 4465 6669 6e69 7469 jects. Definiti\n-00011360: 6f6e 7320 616c 6f6e 6520 6172 650a 6265 ons alone are.be\n-00011370: 6e69 676e 2e20 2041 20e2 8098 736b 6574 nign. A ...sket\n-00011380: 6368 e280 9920 696e 7075 7420 6669 6c65 ch... input file\n-00011390: 2063 6f6e 7369 7374 696e 6720 656e 7469 consisting enti\n-000113a0: 7265 6c79 206f 6620 6465 6669 6e69 7469 rely of definiti\n-000113b0: 6f6e 7320 7769 6c6c 0a67 656e 6572 6174 ons will.generat\n-000113c0: 6520 6e6f 2064 7261 7769 6e67 2e20 204f e no drawing. O\n-000113d0: 6e6c 7920 7768 656e 2064 6566 696e 6974 nly when definit\n-000113e0: 696f 6e73 2061 7265 20e2 809c 7265 6665 ions are ...refe\n-000113f0: 7265 6e63 6564 e280 9d20 646f 2074 6865 renced... do the\n-00011400: 790a 706f 7465 6e74 6961 6c6c 7920 6c65 y.potentially le\n-00011410: 6164 2074 6f20 696e 6b20 6f6e 2074 6865 ad to ink on the\n-00011420: 2064 7261 7769 6e67 2e0a 0a20 2020 5468 drawing... Th\n-00011430: 6520 696e 7465 6e74 206f 6620 6465 6669 e intent of defi\n-00011440: 6e69 7469 6f6e 7320 6973 2074 6f20 6d61 nitions is to ma\n-00011450: 6b65 20e2 8098 736b 6574 6368 e280 9920 ke ...sketch... \n-00011460: 636f 6465 206d 6f72 6520 636f 6e63 6973 code more concis\n-00011470: 6520 616e 640a 7265 6164 6162 6c65 2e20 e and.readable. \n-00011480: 2054 6865 7265 2069 7320 6e6f 2069 6e70 There is no inp\n-00011490: 7574 2066 696c 6520 656d 706c 6f79 696e ut file employin\n-000114a0: 6720 6465 6669 6e69 7469 6f6e 7320 7468 g definitions th\n-000114b0: 6174 2063 6f75 6c64 206e 6f74 0a62 6520 at could not.be \n-000114c0: 7265 2d77 7269 7474 656e 2077 6974 686f re-written witho\n-000114d0: 7574 2074 6865 6d2e 0a0a 2020 2044 6566 ut them... Def\n-000114e0: 696e 6162 6c65 206f 626a 6563 7473 2069 inable objects i\n-000114f0: 6e63 6c75 6465 2061 6e79 2072 6573 756c nclude any resul\n-00011500: 7420 6f66 2061 6e20 6166 6669 6e65 2061 t of an affine a\n-00011510: 7269 7468 6d65 7469 630a 6578 7072 6573 rithmetic.expres\n-00011520: 7369 6f6e 2028 7363 616c 6172 2c20 706f sion (scalar, po\n-00011530: 696e 742c 2076 6563 746f 722c 206f 7220 int, vector, or \n-00011540: 7472 616e 7366 6f72 6d29 2c20 616e 7920 transform), any \n-00011550: 6472 6177 6162 6c65 206f 626a 6563 740a drawable object.\n-00011560: 2864 6f74 732c 206c 696e 652c 2063 7572 (dots, line, cur\n-00011570: 7665 2c20 706f 6c79 676f 6e2c 2062 6c6f ve, polygon, blo\n-00011580: 636b 2c20 7377 6565 702c 2070 7574 2c20 ck, sweep, put, \n-00011590: 7265 7065 6174 2c20 6f72 2073 7065 6369 repeat, or speci\n-000115a0: 616c 292c 2061 6e64 0a6f 7074 696f 6e20 al), and.option \n-000115b0: 7374 7269 6e67 732e 2020 496e 2061 6464 strings. In add\n-000115c0: 6974 696f 6e2c 20e2 809c 7461 6720 6465 ition, ...tag de\n-000115d0: 6669 6e69 7469 6f6e 73e2 809d 2c20 7768 finitions..., wh\n-000115e0: 6963 6820 6861 7665 206e 6f0a 6173 736f ich have no.asso\n-000115f0: 6369 6174 6564 206f 626a 6563 7420 6174 ciated object at\n-00011600: 2061 6c6c 2c20 616c 6c6f 7720 7468 6520 all, allow the \n-00011610: 6d65 616e 696e 6720 6f66 206f 7468 6572 meaning of other\n-00011620: 2064 6566 696e 6974 696f 6e73 2074 6f20 definitions to \n-00011630: 6265 0a73 656c 6563 7465 6420 6672 6f6d be.selected from\n-00011640: 2061 2073 6574 206f 6620 616c 7465 726e a set of altern\n-00011650: 6174 6976 6573 2e20 2053 696e 6365 2074 atives. Since t\n-00011660: 6167 7320 6d61 7920 6265 2064 6566 696e ags may be defin\n-00011670: 6564 2028 616e 640a 756e 6465 6669 6e65 ed (and.undefine\n-00011680: 6429 2069 6e20 7468 6520 636f 6d6d 616e d) in the comman\n-00011690: 6420 6c69 6e65 206f 6620 e280 9873 6b65 d line of ...ske\n-000116a0: 7463 68e2 8099 2c20 7468 6579 2063 616e tch..., they can\n-000116b0: 2062 6520 616e 2061 6964 2069 6e20 7468 be an aid in th\n-000116c0: 650a 7363 7269 7074 2d64 7269 7665 6e20 e.script-driven \n-000116d0: 7072 6570 6172 6174 696f 6e20 6f66 2064 preparation of d\n-000116e0: 6f63 756d 656e 7473 2e0a 0a2a 204d 656e ocuments...* Men\n-000116f0: 753a 0a0a 2a20 466f 726d 7320 6f66 2064 u:..* Forms of d\n-00011700: 6566 696e 6974 696f 6e73 3a3a 2020 2020 efinitions:: \n-00011710: 2020 2020 4469 6666 6572 656e 7420 6465 Different de\n-00011720: 6673 2066 6f72 2064 6966 6665 7265 6e74 fs for different\n-00011730: 2070 7572 706f 7365 732e 0a2a 2046 6f72 purposes..* For\n-00011740: 6d73 206f 6620 7265 6665 7265 6e63 6573 ms of references\n-00011750: 3a3a 2020 2020 2020 2020 2048 6f77 2072 :: How r\n-00011760: 6566 6572 656e 6365 7320 6465 6e6f 7465 eferences denote\n-00011770: 2074 7970 6573 2e0a 0a1f 0a46 696c 653a types.....File:\n-00011780: 2073 6b65 7463 682e 696e 666f 2c20 204e sketch.info, N\n-00011790: 6f64 653a 2046 6f72 6d73 206f 6620 6465 ode: Forms of de\n-000117a0: 6669 6e69 7469 6f6e 732c 2020 4e65 7874 finitions, Next\n-000117b0: 3a20 466f 726d 7320 6f66 2072 6566 6572 : Forms of refer\n-000117c0: 656e 6365 732c 2020 5072 6576 3a20 4465 ences, Prev: De\n-000117d0: 6669 6e69 7469 6f6e 732c 2020 5570 3a20 finitions, Up: \n-000117e0: 4465 6669 6e69 7469 6f6e 730a 0a33 2e33 Definitions..3.3\n-000117f0: 2e31 2046 6f72 6d73 206f 6620 6465 6669 .1 Forms of defi\n-00011800: 6e69 7469 6f6e 730a 2d2d 2d2d 2d2d 2d2d nitions.--------\n+0000e080: 2020 2020 2044 7261 7720 6f6e 6520 6f62 Draw one ob\n+0000e090: 6a65 6374 2074 7261 6e73 666f 726d 6564 ject transformed\n+0000e0a0: 2e0a 0a1f 0a46 696c 653a 2073 6b65 7463 .....File: sketc\n+0000e0b0: 682e 696e 666f 2c20 204e 6f64 653a 2044 h.info, Node: D\n+0000e0c0: 6f74 732c 2020 4e65 7874 3a20 4c69 6e65 ots, Next: Line\n+0000e0d0: 732c 2020 5072 6576 3a20 4472 6177 6162 s, Prev: Drawab\n+0000e0e0: 6c65 732c 2020 5570 3a20 4472 6177 6162 les, Up: Drawab\n+0000e0f0: 6c65 730a 0a33 2e32 2e31 2044 6f74 730a les..3.2.1 Dots.\n+0000e100: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 2a53 796e ----------..*Syn\n+0000e110: 7461 783a 2a0a 2020 2020 2064 6f74 735b tax:*. dots[\n+0000e120: 4f50 5449 4f4e 535d 2050 4f49 4e54 5f4c OPTIONS] POINT_L\n+0000e130: 4953 540a 5468 6973 2063 6f6d 6d61 6e64 IST.This command\n+0000e140: 2069 7320 7468 6520 7468 7265 652d 6469 is the three-di\n+0000e150: 6d65 6e73 696f 6e61 6c20 6571 7569 7661 mensional equiva\n+0000e160: 6c65 6e74 206f 6620 7468 6520 e280 9850 lent of the ...P\n+0000e170: 5354 7269 636b 73e2 8099 0a63 6f6d 6d61 STricks....comma\n+0000e180: 6e64 20e2 8098 5c70 7364 6f74 73e2 8099 nd ...\\psdots...\n+0000e190: 2e0a 0a1f 0a46 696c 653a 2073 6b65 7463 .....File: sketc\n+0000e1a0: 682e 696e 666f 2c20 204e 6f64 653a 204c h.info, Node: L\n+0000e1b0: 696e 6573 2c20 204e 6578 743a 2043 7572 ines, Next: Cur\n+0000e1c0: 7665 732c 2020 5072 6576 3a20 446f 7473 ves, Prev: Dots\n+0000e1d0: 2c20 2055 703a 2044 7261 7761 626c 6573 , Up: Drawables\n+0000e1e0: 0a0a 332e 322e 3220 4c69 6e65 730a 2d2d ..3.2.2 Lines.--\n+0000e1f0: 2d2d 2d2d 2d2d 2d2d 2d0a 0a2a 5379 6e74 ---------..*Synt\n+0000e200: 6178 3a2a 0a20 2020 2020 6c69 6e65 5b4f ax:*. line[O\n+0000e210: 5054 494f 4e53 5d20 504f 494e 545f 4c49 PTIONS] POINT_LI\n+0000e220: 5354 0a54 6869 7320 636f 6d6d 616e 6420 ST.This command \n+0000e230: 6973 2074 6865 2074 6872 6565 2d64 696d is the three-dim\n+0000e240: 656e 7369 6f6e 616c 2065 7175 6976 616c ensional equival\n+0000e250: 656e 7420 6f66 2074 6865 20e2 8098 5053 ent of the ...PS\n+0000e260: 5472 6963 6b73 e280 990a 636f 6d6d 616e Tricks....comman\n+0000e270: 6420 e280 985c 7073 6c69 6e65 e280 992e d ...\\psline....\n+0000e280: 0a0a 1f0a 4669 6c65 3a20 736b 6574 6368 ....File: sketch\n+0000e290: 2e69 6e66 6f2c 2020 4e6f 6465 3a20 4375 .info, Node: Cu\n+0000e2a0: 7276 6573 2c20 204e 6578 743a 2050 6f6c rves, Next: Pol\n+0000e2b0: 7967 6f6e 732c 2020 5072 6576 3a20 4c69 ygons, Prev: Li\n+0000e2c0: 6e65 732c 2020 5570 3a20 4472 6177 6162 nes, Up: Drawab\n+0000e2d0: 6c65 730a 0a33 2e32 2e33 2043 7572 7665 les..3.2.3 Curve\n+0000e2e0: 730a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a s.------------..\n+0000e2f0: 2a53 796e 7461 783a 2a0a 2020 2020 2063 *Syntax:*. c\n+0000e300: 7572 7665 5b4f 5054 494f 4e53 5d20 504f urve[OPTIONS] PO\n+0000e310: 494e 545f 4c49 5354 0a54 6869 7320 636f INT_LIST.This co\n+0000e320: 6d6d 616e 6420 6973 2074 6865 2074 6872 mmand is the thr\n+0000e330: 6565 2d64 696d 656e 7369 6f6e 616c 2065 ee-dimensional e\n+0000e340: 7175 6976 616c 656e 7420 6f66 2074 6865 quivalent of the\n+0000e350: 20e2 8098 5053 5472 6963 6b73 e280 990a ...PSTricks....\n+0000e360: 636f 6d6d 616e 6420 e280 985c 7073 6375 command ...\\pscu\n+0000e370: 7276 65e2 8099 2e20 2049 7420 6973 206e rve.... It is n\n+0000e380: 6f74 2069 6d70 6c65 6d65 6e74 6564 2069 ot implemented i\n+0000e390: 6e20 7468 6520 6375 7272 656e 7420 7665 n the current ve\n+0000e3a0: 7273 696f 6e20 6f66 0ae2 8098 736b 6574 rsion of....sket\n+0000e3b0: 6368 e280 992e 0a0a 1f0a 4669 6c65 3a20 ch........File: \n+0000e3c0: 736b 6574 6368 2e69 6e66 6f2c 2020 4e6f sketch.info, No\n+0000e3d0: 6465 3a20 506f 6c79 676f 6e73 2c20 204e de: Polygons, N\n+0000e3e0: 6578 743a 2053 7065 6369 616c 732c 2020 ext: Specials, \n+0000e3f0: 5072 6576 3a20 4375 7276 6573 2c20 2055 Prev: Curves, U\n+0000e400: 703a 2044 7261 7761 626c 6573 0a0a 332e p: Drawables..3.\n+0000e410: 322e 3420 506f 6c79 676f 6e73 0a2d 2d2d 2.4 Polygons.---\n+0000e420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a2a 5379 -----------..*Sy\n+0000e430: 6e74 6178 3a2a 0a20 2020 2020 706f 6c79 ntax:*. poly\n+0000e440: 676f 6e5b 4f50 5449 4f4e 535d 2050 4f49 gon[OPTIONS] POI\n+0000e450: 4e54 5f4c 4953 540a 5468 6973 2063 6f6d NT_LIST.This com\n+0000e460: 6d61 6e64 2069 7320 7468 6520 7468 7265 mand is the thre\n+0000e470: 652d 6469 6d65 6e73 696f 6e61 6c20 6571 e-dimensional eq\n+0000e480: 7569 7661 6c65 6e74 206f 6620 7468 6520 uivalent of the \n+0000e490: e280 9850 5354 7269 636b 73e2 8099 0a63 ...PSTricks....c\n+0000e4a0: 6f6d 6d61 6e64 20e2 8098 5c70 7370 6f6c ommand ...\\pspol\n+0000e4b0: 7967 6f6e e280 992e 2020 5468 6520 e280 ygon.... The ..\n+0000e4c0: 9873 6b65 7463 68e2 8099 2068 6964 6465 .sketch... hidde\n+0000e4d0: 6e20 7375 7266 6163 6520 616c 676f 7269 n surface algori\n+0000e4e0: 7468 6d20 6173 7375 6d65 730a 7468 6174 thm assumes.that\n+0000e4f0: 2070 6f6c 7967 6f6e 7320 6172 6520 636f polygons are co\n+0000e500: 6e76 6578 2061 6e64 2070 6c61 6e61 722e nvex and planar.\n+0000e510: 2020 496e 2070 7261 6374 6963 652c 2064 In practice, d\n+0000e520: 7261 7769 6e67 7320 6d61 7920 7765 6c6c rawings may well\n+0000e530: 0a74 7572 6e20 6f75 7420 636f 7272 6563 .turn out correc\n+0000e540: 746c 7920 6576 656e 2069 6620 7468 6573 tly even if thes\n+0000e550: 6520 6173 7375 6d70 7469 6f6e 7320 6172 e assumptions ar\n+0000e560: 6520 7669 6f6c 6174 6564 2e0a 0a1f 0a46 e violated.....F\n+0000e570: 696c 653a 2073 6b65 7463 682e 696e 666f ile: sketch.info\n+0000e580: 2c20 204e 6f64 653a 2053 7065 6369 616c , Node: Special\n+0000e590: 732c 2020 4e65 7874 3a20 5377 6565 7073 s, Next: Sweeps\n+0000e5a0: 2c20 2050 7265 763a 2050 6f6c 7967 6f6e , Prev: Polygon\n+0000e5b0: 732c 2020 5570 3a20 4472 6177 6162 6c65 s, Up: Drawable\n+0000e5c0: 730a 0a33 2e32 2e35 2053 7065 6369 616c s..3.2.5 Special\n+0000e5d0: 730a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d s.--------------\n+0000e5e0: 0a0a 2a53 796e 7461 783a 2a0a 2020 2020 ..*Syntax:*. \n+0000e5f0: 2073 7065 6369 616c 2024 5241 575f 5445 special $RAW_TE\n+0000e600: 5854 245b 4f50 5449 4f4e 535d 2041 5247 XT$[OPTIONS] ARG\n+0000e610: 5f4c 4953 540a 4865 7265 20e2 8098 24e2 _LIST.Here ...$.\n+0000e620: 8099 2063 616e 2062 6520 616e 7920 6368 .. can be any ch\n+0000e630: 6172 6163 7465 7220 616e 6420 6973 2075 aracter and is u\n+0000e640: 7365 6420 746f 2064 656c 696d 6974 2074 sed to delimit t\n+0000e650: 6865 2073 7461 7274 2061 6e64 2065 6e64 he start and end\n+0000e660: 0a6f 6620 5241 575f 5445 5854 2e20 2054 .of RAW_TEXT. T\n+0000e670: 6865 2063 6f6d 6d61 6e64 2065 6d62 6564 he command embed\n+0000e680: 7320 5241 575f 5445 5854 2069 6e20 7468 s RAW_TEXT in th\n+0000e690: 6520 e280 9873 6b65 7463 68e2 8099 206f e ...sketch... o\n+0000e6a0: 7574 7075 7420 6166 7465 720a 7065 7266 utput after.perf\n+0000e6b0: 6f72 6d69 6e67 2073 7562 7374 6974 7574 orming substitut\n+0000e6c0: 696f 6e73 2061 7320 666f 6c6c 6f77 732e ions as follows.\n+0000e6d0: 0a20 2020 e280 a220 e280 9823 49e2 8099 . ... ...#I...\n+0000e6e0: 2077 6865 7265 2049 2069 7320 6120 706f where I is a po\n+0000e6f0: 7369 7469 7665 2069 6e74 6567 6572 2069 sitive integer i\n+0000e700: 7320 7265 706c 6163 6564 2062 7920 7468 s replaced by th\n+0000e710: 6520 4927 7468 2076 616c 7565 2069 6e0a e I'th value in.\n+0000e720: 2020 2020 2041 5247 5f4c 4953 542e 2020 ARG_LIST. \n+0000e730: 506f 696e 7420 616e 6420 7665 6374 6f72 Point and vector\n+0000e740: 2061 7267 756d 656e 7473 2062 6563 6f6d arguments becom\n+0000e750: 6520 7477 6f2d 6469 6d65 6e73 696f 6e61 e two-dimensiona\n+0000e760: 6c0a 2020 2020 2070 6f69 6e74 732c 2077 l. points, w\n+0000e770: 6869 6368 2061 7265 2074 6865 2074 7261 hich are the tra\n+0000e780: 6e73 666f 726d 6564 2033 6420 6172 6775 nsformed 3d argu\n+0000e790: 6d65 6e74 7320 7072 6f6a 6563 7465 6420 ments projected \n+0000e7a0: 6f6e 746f 2074 6865 0a20 2020 2020 582d onto the. X-\n+0000e7b0: 5920 706c 616e 652e 2020 5468 6973 2061 Y plane. This a\n+0000e7c0: 6c6c 6f77 7320 7477 6f2d 6469 6d65 6e74 llows two-diment\n+0000e7d0: 696f 6e61 6c20 6f75 7470 7574 2065 6c65 ional output ele\n+0000e7e0: 6d65 6e74 7320 7375 6368 2061 730a 2020 ments such as. \n+0000e7f0: 2020 206c 6162 656c 7320 746f 2062 6520 labels to be \n+0000e800: 6561 7369 6c79 2070 6f73 6974 696f 6e65 easily positione\n+0000e810: 6420 7769 7468 2072 6573 7065 6374 2074 d with respect t\n+0000e820: 6f20 7468 7265 652d 6469 6d65 6e73 696f o three-dimensio\n+0000e830: 6e61 6c0a 2020 2020 2066 6561 7475 7265 nal. feature\n+0000e840: 7320 696e 2074 6865 2064 7261 7769 6e67 s in the drawing\n+0000e850: 2e20 2053 6361 6c61 7220 6172 6775 6d65 . Scalar argume\n+0000e860: 6e74 7320 6172 6520 7375 6273 6974 7574 nts are subsitut\n+0000e870: 6564 2064 6972 6563 746c 792e 0a20 2020 ed directly.. \n+0000e880: 2020 4e6f 2074 7261 6e73 666f 726d 6174 No transformat\n+0000e890: 696f 6e20 6973 2061 7070 6c69 6564 2e0a ion is applied..\n+0000e8a0: 2020 20e2 80a2 20e2 8098 237b 497d e280 ... ...#{I}..\n+0000e8b0: 9920 6973 2061 6c73 6f20 7265 706c 6163 . is also replac\n+0000e8c0: 6564 2061 7320 6162 6f76 652e 0a20 2020 ed as above.. \n+0000e8d0: e280 a220 e280 9823 4927 32e2 8099 2069 ... ...#I'2... i\n+0000e8e0: 7320 7265 706c 6163 6564 2061 7320 6162 s replaced as ab\n+0000e8f0: 6f76 6520 666f 7220 706f 696e 7473 206f ove for points o\n+0000e900: 7220 7665 6374 6f72 732e 2020 4974 2069 r vectors. It i\n+0000e910: 7320 616e 2065 7272 6f72 0a20 2020 2020 s an error. \n+0000e920: 666f 7220 7468 6520 4927 7468 2061 7267 for the I'th arg\n+0000e930: 756d 656e 7420 746f 2062 6520 6120 7363 ument to be a sc\n+0000e940: 616c 6172 2e0a 2020 20e2 80a2 20e2 8098 alar.. ... ...\n+0000e950: 2349 2778 e280 992c 20e2 8098 2349 2779 #I'x..., ...#I'y\n+0000e960: e280 992c 206f 7220 e280 9823 4927 7ae2 ..., or ...#I'z.\n+0000e970: 8099 2069 7320 7265 706c 6163 6564 2c20 .. is replaced, \n+0000e980: 7265 7370 6563 7469 7665 6c79 2c20 6279 respectively, by\n+0000e990: 2074 6865 2073 6361 6c61 720a 2020 2020 the scalar. \n+0000e9a0: 2058 2c20 592c 206f 7220 5a2d 636f 6f72 X, Y, or Z-coor\n+0000e9b0: 6469 6e61 7465 206f 6620 7468 6520 6172 dinate of the ar\n+0000e9c0: 6775 6d65 6e74 2070 6f69 6e74 206f 7220 gument point or \n+0000e9d0: 7665 6374 6f72 2e20 2049 7420 6973 2061 vector. It is a\n+0000e9e0: 6e0a 2020 2020 2065 7272 6f72 2066 6f72 n. error for\n+0000e9f0: 2074 6865 2049 2774 6820 6172 6775 6d65 the I'th argume\n+0000ea00: 6e74 2074 6f20 6265 2061 2073 6361 6c61 nt to be a scala\n+0000ea10: 722e 0a20 2020 e280 a220 e280 9823 4927 r.. ... ...#I'\n+0000ea20: 33e2 8099 2069 7320 7265 706c 6163 6564 3... is replaced\n+0000ea30: 2062 7920 7468 6520 7468 7265 652d 6469 by the three-di\n+0000ea40: 6d65 6e73 696f 6e61 6c20 7472 616e 7366 mensional transf\n+0000ea50: 6f72 6d65 6420 6172 6775 6d65 6e74 2e0a ormed argument..\n+0000ea60: 2020 2020 204e 6f74 6520 7468 6174 2069 Note that i\n+0000ea70: 6620 6120 7065 7273 7065 6374 6976 6520 f a perspective \n+0000ea80: 7472 616e 7366 6f72 6d61 7469 6f6e 2068 transformation h\n+0000ea90: 6173 2062 6565 6e20 6170 706c 6965 642c as been applied,\n+0000eaa0: 2074 6865 0a20 2020 2020 5a2d 636f 6f72 the. Z-coor\n+0000eab0: 6469 6e61 7465 2068 6173 206c 6974 746c dinate has littl\n+0000eac0: 6520 6765 6f6d 6574 7269 6320 7369 676e e geometric sign\n+0000ead0: 6966 6963 616e 6365 2c20 7468 6f75 6768 ificance, though\n+0000eae0: 2069 740a 2020 2020 2061 6363 7572 6174 it. accurat\n+0000eaf0: 656c 7920 696e 6469 6361 7465 7320 7265 ely indicates re\n+0000eb00: 6c61 7469 7665 2064 6570 7468 2e0a 2020 lative depth.. \n+0000eb10: 20e2 80a2 20e2 8098 2349 2d4a e280 9920 ... ...#I-J... \n+0000eb20: 7768 6572 6520 4920 616e 6420 4a20 6172 where I and J ar\n+0000eb30: 6520 706f 7369 7469 7665 2069 6e74 6567 e positive integ\n+0000eb40: 6572 7320 6973 2072 6570 6c61 6365 6420 ers is replaced \n+0000eb50: 6279 2061 2073 7472 696e 670a 2020 2020 by a string. \n+0000eb60: 20e2 8098 7b41 4e47 4c45 7de2 8099 2077 ...{ANGLE}... w\n+0000eb70: 6865 7265 2041 4e47 4c45 2069 7320 7468 here ANGLE is th\n+0000eb80: 6520 706f 6c61 7220 616e 676c 6520 6f66 e polar angle of\n+0000eb90: 2061 2076 6563 746f 7220 6672 6f6d 2074 a vector from t\n+0000eba0: 6865 2049 2774 680a 2020 2020 2070 6f69 he I'th. poi\n+0000ebb0: 6e74 2069 6e20 504f 494e 545f 4c49 5354 nt in POINT_LIST\n+0000ebc0: 2074 6f20 7468 6520 4a27 7468 2070 6f69 to the J'th poi\n+0000ebd0: 6e74 2070 726f 6a65 6374 6564 2069 6e74 nt projected int\n+0000ebe0: 6f20 7468 6520 782d 7920 706c 616e 652e o the x-y plane.\n+0000ebf0: 0a20 2020 2020 4974 2069 7320 616e 2065 . It is an e\n+0000ec00: 7272 6f72 2066 6f72 2074 6865 2049 2774 rror for the I't\n+0000ec10: 6820 6f72 204a 2774 6820 6172 6775 6d65 h or J'th argume\n+0000ec20: 6e74 2074 6f20 6265 2061 2073 6361 6c61 nt to be a scala\n+0000ec30: 7220 6f72 2061 0a20 2020 2020 7665 6374 r or a. vect\n+0000ec40: 6f72 2e0a 2020 20e2 80a2 20e2 8098 237b or.. ... ...#{\n+0000ec50: 492d 4a7d e280 9920 6973 2061 6c73 6f20 I-J}... is also \n+0000ec60: 7265 706c 6163 6564 2061 7320 6162 6f76 replaced as abov\n+0000ec70: 652e 0a20 2020 e280 a220 e280 9823 23e2 e.. ... ...##.\n+0000ec80: 8099 2069 7320 7265 706c 6163 6564 2077 .. is replaced w\n+0000ec90: 6974 6820 e280 9823 e280 992e 0a54 6865 ith ...#.....The\n+0000eca0: 2066 6f72 6d73 2077 6974 6820 6272 6163 forms with brac\n+0000ecb0: 6573 20e2 8098 7b20 7de2 8099 2061 7265 es ...{ }... are\n+0000ecc0: 2075 7365 6675 6c20 7768 656e 2074 6865 useful when the\n+0000ecd0: 2061 7267 756d 656e 7420 6973 2069 6d6d argument is imm\n+0000ece0: 6564 6961 7465 6c79 0a66 6f6c 6c6f 7765 ediately.followe\n+0000ecf0: 6420 6279 2061 2064 6967 6974 206f 7220 d by a digit or \n+0000ed00: 6120 7469 636b 2063 6861 7261 6374 6572 a tick character\n+0000ed10: 2074 6861 7420 6973 2070 6172 7420 6f66 that is part of\n+0000ed20: 2074 6865 2054 6558 2063 6f64 652e 0a0a the TeX code...\n+0000ed30: 2020 2054 6865 206f 6e6c 7920 7573 6566 The only usef\n+0000ed40: 756c 206f 7074 696f 6e20 6f66 20e2 8098 ul option of ...\n+0000ed50: 7370 6563 6961 6ce2 8099 2069 7320 e280 special... is ..\n+0000ed60: 986c 6179 e280 992c 2077 6869 6368 2064 .lay..., which d\n+0000ed70: 6574 6572 6d69 6e65 7320 6966 2074 6865 etermines if the\n+0000ed80: 0a73 7562 7374 6974 7565 6420 7261 7720 .substitued raw \n+0000ed90: 7465 7874 2069 7320 656d 6974 7465 6420 text is emitted \n+0000eda0: 6265 666f 7265 2c20 6166 7465 722c 206f before, after, o\n+0000edb0: 7220 7573 696e 6720 7468 6520 6669 7273 r using the firs\n+0000edc0: 7420 706f 696e 740a 696e 2050 4f49 4e54 t point.in POINT\n+0000edd0: 5f4c 4953 5420 6173 2061 6e20 696e 6469 _LIST as an indi\n+0000ede0: 6361 746f 7220 6f66 2064 6570 7468 2e20 cator of depth. \n+0000edf0: 2054 6865 7365 206f 6363 7572 2c20 7265 These occur, re\n+0000ee00: 7370 6563 7469 7665 6c79 2c20 7769 7468 spectively, with\n+0000ee10: 0ae2 8098 6c61 793d 756e 6465 72e2 8099 ....lay=under...\n+0000ee20: 2c20 e280 986c 6179 3d6f 7665 72e2 8099 , ...lay=over...\n+0000ee30: 2c20 616e 6420 e280 986c 6179 3d69 6ee2 , and ...lay=in.\n+0000ee40: 8099 2e20 2053 6565 202a 6e6f 7465 2053 ... See *note S\n+0000ee50: 7065 6369 616c 206f 626a 6563 7473 3a3a pecial objects::\n+0000ee60: 2061 6e64 0a2a 6e6f 7465 2054 696b 5a2f and.*note TikZ/\n+0000ee70: 5047 4620 7573 6572 2d64 6566 696e 6564 PGF user-defined\n+0000ee80: 2073 7479 6c65 733a 3a20 666f 7220 6578 styles:: for ex\n+0000ee90: 616d 706c 6573 2e20 202a 4e6f 7465 2049 amples. *Note I\n+0000eea0: 6e74 6572 6e61 6c0a 6f70 7469 6f6e 733a nternal.options:\n+0000eeb0: 3a2e 0a0a 1f0a 4669 6c65 3a20 736b 6574 :.....File: sket\n+0000eec0: 6368 2e69 6e66 6f2c 2020 4e6f 6465 3a20 ch.info, Node: \n+0000eed0: 5377 6565 7073 2c20 204e 6578 743a 2042 Sweeps, Next: B\n+0000eee0: 6c6f 636b 732c 2020 5072 6576 3a20 5370 locks, Prev: Sp\n+0000eef0: 6563 6961 6c73 2c20 2055 703a 2044 7261 ecials, Up: Dra\n+0000ef00: 7761 626c 6573 0a0a 332e 322e 3620 5377 wables..3.2.6 Sw\n+0000ef10: 6565 7073 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d eeps.-----------\n+0000ef20: 2d0a 0a2a 5379 6e74 6178 3a2a 0a20 2020 -..*Syntax:*. \n+0000ef30: 2020 7377 6565 7020 7b20 4e2c 2054 5f31 sweep { N, T_1\n+0000ef40: 2c20 545f 322c 202e 2e2e 2c20 545f 5220 , T_2, ..., T_R \n+0000ef50: 7d5b 4f50 5449 4f4e 535d 2053 5745 5054 }[OPTIONS] SWEPT\n+0000ef60: 5f4f 424a 4543 540a 2020 2020 2073 7765 _OBJECT. swe\n+0000ef70: 6570 207b 204e 3c3e 2c20 545f 312c 2054 ep { N<>, T_1, T\n+0000ef80: 5f32 2c20 2e2e 2e2c 2054 5f52 207d 5b4f _2, ..., T_R }[O\n+0000ef90: 5054 494f 4e53 5d20 5357 4550 545f 4f42 PTIONS] SWEPT_OB\n+0000efa0: 4a45 4354 0a54 6865 2073 7765 6570 2063 JECT.The sweep c\n+0000efb0: 6f6e 6e65 6374 7320 4e20 286f 7220 7065 onnects N (or pe\n+0000efc0: 7268 6170 7320 4e2b 3129 2063 6f70 6965 rhaps N+1) copie\n+0000efd0: 7320 6f66 2053 5745 5054 5f4f 424a 4543 s of SWEPT_OBJEC\n+0000efe0: 5420 696e 206f 7264 6572 2074 6f0a 6372 T in order to.cr\n+0000eff0: 6561 7465 2061 206e 6577 206f 626a 6563 eate a new objec\n+0000f000: 7420 6f66 2068 6967 6865 7220 6469 6d65 t of higher dime\n+0000f010: 6e73 696f 6e2e 2020 5468 6520 545f 4920 nsion. The T_I \n+0000f020: 2866 6f72 2069 2062 6574 7765 656e 2031 (for i between 1\n+0000f030: 2061 6e64 0a72 2920 6172 6520 7472 616e and.r) are tran\n+0000f040: 7366 6f72 6d73 2e20 2054 6865 206b 2774 sforms. The k't\n+0000f050: 6820 636f 7079 206f 6620 5357 4550 545f h copy of SWEPT_\n+0000f060: 4f42 4a45 4354 2069 7320 7072 6f64 7563 OBJECT is produc\n+0000f070: 6564 2062 790a 6170 706c 7969 6e67 2074 ed by.applying t\n+0000f080: 6865 2066 6f6c 6c6f 7769 6e67 2074 7261 he following tra\n+0000f090: 6e73 666f 726d 2074 6f20 7468 6520 6f72 nsform to the or\n+0000f0a0: 6967 696e 616c 2e0a 2020 2020 2054 5f31 iginal.. T_1\n+0000f0b0: 5e6b 2074 6865 6e20 545f 325e 6b20 7468 ^k then T_2^k th\n+0000f0c0: 656e 202e 2e2e 2074 6865 6e20 545f 525e en ... then T_R^\n+0000f0d0: 6b0a 4865 7265 2054 5e6b 206d 6561 6e73 k.Here T^k means\n+0000f0e0: 2022 7472 616e 7366 6f72 6d20 5420 6170 \"transform T ap\n+0000f0f0: 706c 6965 6420 6b20 7469 6d65 732e 2220 plied k times.\" \n+0000f100: 2054 6865 206f 7269 6769 6e61 6c20 6f62 The original ob\n+0000f110: 6a65 6374 2069 730a 7468 6520 7a65 726f ject is.the zero\n+0000f120: 2774 6820 636f 7079 2c20 7769 7468 206b 'th copy, with k\n+0000f130: 3d30 2061 6e64 2065 6666 6563 7469 7665 =0 and effective\n+0000f140: 6c79 206e 6f20 7472 616e 7366 6f72 6d20 ly no transform \n+0000f150: 6170 706c 6965 6420 2854 5e30 3d49 2c0a applied (T^0=I,.\n+0000f160: 7468 6520 6964 656e 7469 7479 2074 7261 the identity tra\n+0000f170: 6e73 666f 726d 292e 0a0a 2020 2054 6865 nsform)... The\n+0000f180: 206d 6574 686f 6420 6f66 2063 6f6e 6e65 method of conne\n+0000f190: 6374 696e 6720 7468 6520 636f 7069 6573 cting the copies\n+0000f1a0: 2064 6570 656e 6473 206f 6e20 7468 6520 depends on the \n+0000f1b0: 7479 7065 206f 660a 5357 4550 545f 4f42 type of.SWEPT_OB\n+0000f1c0: 4a45 4354 2061 6e64 206f 6e20 7768 6574 JECT and on whet\n+0000f1d0: 6865 7220 7468 6520 636c 6f73 7572 6520 her the closure \n+0000f1e0: 7461 6720 e280 983c 3ee2 8099 2069 7320 tag ...<>... is \n+0000f1f0: 7072 6573 656e 7420 6f72 206e 6f74 2e0a present or not..\n+0000f200: 0a20 2020 416e 2065 7861 6d70 6c65 206f . An example o\n+0000f210: 6620 6120 7377 6565 7020 7768 6572 6520 f a sweep where \n+0000f220: 723d 3220 6973 2074 6865 204d 6f62 6975 r=2 is the Mobiu\n+0000f230: 7320 6669 6775 7265 2061 7420 2a6e 6f74 s figure at *not\n+0000f240: 6520 4d6f 7265 2074 6f0a 6c65 6172 6e3a e More to.learn:\n+0000f250: 3a2e 0a0a 2a20 4d65 6e75 3a0a 0a2a 2053 :...* Menu:..* S\n+0000f260: 7765 7074 2070 6f69 6e74 733a 3a20 2020 wept points:: \n+0000f270: 2020 2020 2020 2020 2020 2020 2053 7765 Swe\n+0000f280: 7074 2070 6f69 6e74 7320 6d61 6b65 206c pt points make l\n+0000f290: 696e 6573 206f 7220 706f 6c79 676f 6e73 ines or polygons\n+0000f2a0: 2e0a 2a20 5377 6570 7420 6c69 6e65 733a ..* Swept lines:\n+0000f2b0: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+0000f2c0: 2020 5377 6570 7420 6c69 6e65 7320 6d61 Swept lines ma\n+0000f2d0: 6b65 206f 7065 6e20 6f72 2063 6c6f 7365 ke open or close\n+0000f2e0: 6420 7375 7266 6163 6573 2e0a 2a20 5377 d surfaces..* Sw\n+0000f2f0: 6570 7420 706f 6c79 676f 6e73 3a3a 2020 ept polygons:: \n+0000f300: 2020 2020 2020 2020 2020 2020 5377 6570 Swep\n+0000f310: 7420 706f 6c79 676f 6e73 206d 616b 6520 t polygons make \n+0000f320: 636c 6f73 6564 2073 7572 6661 6365 732e closed surfaces.\n+0000f330: 0a2a 2053 7765 7074 2062 6c6f 636b 733a .* Swept blocks:\n+0000f340: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+0000f350: 2053 7765 7074 2062 6c6f 636b 20e2 89a1 Swept block ...\n+0000f360: 2062 6c6f 636b 206f 6620 7377 6565 7073 block of sweeps\n+0000f370: 2e0a 2a20 5377 6565 7020 6661 6365 2073 ..* Sweep face s\n+0000f380: 706c 6974 7469 6e67 3a3a 2020 2020 2020 plitting:: \n+0000f390: 2020 4669 7869 6e67 2077 6172 7065 6420 Fixing warped \n+0000f3a0: 6661 6365 7320 7769 7468 2074 7269 616e faces with trian\n+0000f3b0: 676c 6573 2e0a 0a1f 0a46 696c 653a 2073 gles.....File: s\n+0000f3c0: 6b65 7463 682e 696e 666f 2c20 204e 6f64 ketch.info, Nod\n+0000f3d0: 653a 2053 7765 7074 2070 6f69 6e74 732c e: Swept points,\n+0000f3e0: 2020 4e65 7874 3a20 5377 6570 7420 6c69 Next: Swept li\n+0000f3f0: 6e65 732c 2020 5072 6576 3a20 5377 6565 nes, Prev: Swee\n+0000f400: 7073 2c20 2055 703a 2053 7765 6570 730a ps, Up: Sweeps.\n+0000f410: 0a33 2e32 2e36 2e31 2053 7765 7074 2070 .3.2.6.1 Swept p\n+0000f420: 6f69 6e74 730a 2e2e 2e2e 2e2e 2e2e 2e2e oints...........\n+0000f430: 2e2e 2e2e 2e2e 2e2e 2e2e 0a0a 4966 2053 ............If S\n+0000f440: 5745 5054 5f4f 424a 4543 5420 6973 2061 WEPT_OBJECT is a\n+0000f450: 2070 6f69 6e74 206c 6973 7420 616e 6420 point list and \n+0000f460: 7468 6572 6520 6973 206e 6f20 636c 6f73 there is no clos\n+0000f470: 7572 6520 7461 672c 2074 6865 6e0a e280 ure tag, then...\n+0000f480: 9873 7765 6570 e280 9920 636f 6e6e 6563 .sweep... connec\n+0000f490: 7473 204e 2b31 2073 7563 6365 7373 6976 ts N+1 successiv\n+0000f4a0: 6520 636f 7069 6573 206f 6620 6561 6368 e copies of each\n+0000f4b0: 2070 6f69 6e74 2028 696e 636c 7564 696e point (includin\n+0000f4c0: 6720 7468 650a 6f72 6967 696e 616c 2920 g the.original) \n+0000f4d0: 7769 7468 2073 7472 6169 6768 7420 6c69 with straight li\n+0000f4e0: 6e65 2073 6567 6d65 6e74 7320 746f 2066 ne segments to f\n+0000f4f0: 6f72 6d20 6120 706f 6c79 6c69 6e65 2e20 orm a polyline. \n+0000f500: 2049 6620 7468 6572 6520 6172 650a 6d20 If there are.m \n+0000f510: 706f 696e 7473 2069 6e20 7468 6520 6f72 points in the or\n+0000f520: 6967 696e 616c 2070 6f69 6e74 206c 6973 iginal point lis\n+0000f530: 742c 2074 6865 6e20 6d20 706f 6c79 6c69 t, then m polyli\n+0000f540: 6e65 7320 7769 7468 204e 2073 6567 6d65 nes with N segme\n+0000f550: 6e74 730a 6561 6368 2061 7265 2066 6f72 nts.each are for\n+0000f560: 6d65 6420 6279 2074 6865 2073 7765 6570 med by the sweep\n+0000f570: 2e20 2049 6e20 7468 6973 206d 616e 6e65 . In this manne\n+0000f580: 722c 20e2 8098 7377 6565 70e2 8099 2066 r, ...sweep... f\n+0000f590: 6f72 6d73 2061 2073 6574 206f 660a 6f6e orms a set of.on\n+0000f5a0: 652d 6469 6d65 6e73 696f 6e61 6c20 6f62 e-dimensional ob\n+0000f5b0: 6a65 6374 7320 2870 6f6c 796c 696e 6573 jects (polylines\n+0000f5c0: 2920 6672 6f6d 207a 6572 6f2d 6469 6d65 ) from zero-dime\n+0000f5d0: 6e73 696f 6e61 6c20 6f6e 6573 2028 706f nsional ones (po\n+0000f5e0: 696e 7473 292e 0a0a 2020 2057 6865 6e20 ints)... When \n+0000f5f0: 7468 6572 6520 5f69 735f 2061 2063 6c6f there _is_ a clo\n+0000f600: 7375 7265 2074 6167 2c20 e280 9873 7765 sure tag, ...swe\n+0000f610: 6570 e280 9920 636f 6e6e 6563 7473 204e ep... connects N\n+0000f620: 2073 7563 6365 7373 6976 6520 636f 7069 successive copi\n+0000f630: 6573 0a6f 6620 6561 6368 2070 6f69 6e74 es.of each point\n+0000f640: 2028 696e 636c 7564 696e 6720 7468 6520 (including the \n+0000f650: 6f72 6967 696e 616c 2920 7769 7468 2073 original) with s\n+0000f660: 7472 6169 6768 7420 6c69 6e65 2073 6567 traight line seg\n+0000f670: 6d65 6e74 7320 616e 640a 6669 6e61 6c6c ments and.finall\n+0000f680: 7920 636f 6e6e 6563 7473 2074 6865 206c y connects the l\n+0000f690: 6173 7420 636f 7079 2062 6163 6b20 746f ast copy back to\n+0000f6a0: 2074 6865 206f 7269 6769 6e61 6c20 746f the original to\n+0000f6b0: 2066 6f72 6d20 6120 706f 6c79 676f 6e0a form a polygon.\n+0000f6c0: 7769 7468 204e 2073 6964 6573 2e20 2049 with N sides. I\n+0000f6d0: 6620 7468 6572 6520 6172 6520 6d20 706f f there are m po\n+0000f6e0: 696e 7473 2069 6e20 7468 6520 6f72 6967 ints in the orig\n+0000f6f0: 696e 616c 2070 6f69 6e74 206c 6973 742c inal point list,\n+0000f700: 2074 6865 6e20 6d0a 706f 6c79 676f 6e73 then m.polygons\n+0000f710: 2077 6974 6820 4e20 7369 6465 7320 6561 with N sides ea\n+0000f720: 6368 2061 7265 2066 6f72 6d65 6420 6279 ch are formed by\n+0000f730: 2074 6865 2073 7765 6570 2e20 2049 6e20 the sweep. In \n+0000f740: 7468 6973 206d 616e 6e65 722c 0ae2 8098 this manner,....\n+0000f750: 7377 6565 70e2 8099 2066 6f72 6d73 2061 sweep... forms a\n+0000f760: 2073 6574 206f 6620 7477 6f2d 6469 6d65 set of two-dime\n+0000f770: 6e73 696f 6e61 6c20 6f62 6a65 6374 7320 nsional objects \n+0000f780: 2870 6f6c 7967 6f6e 7329 2066 726f 6d0a (polygons) from.\n+0000f790: 7a65 726f 2d64 696d 656e 7369 6f6e 616c zero-dimensional\n+0000f7a0: 206f 6e65 7320 2870 6f69 6e74 7329 2e0a ones (points)..\n+0000f7b0: 0a20 2020 4f70 7469 6f6e 7320 6f66 2074 . Options of t\n+0000f7c0: 6865 20e2 8098 7377 6565 70e2 8099 2061 he ...sweep... a\n+0000f7d0: 7265 2063 6f70 6965 6420 6469 7265 6374 re copied direct\n+0000f7e0: 6c79 2074 6f20 7468 6520 7265 7375 6c74 ly to the result\n+0000f7f0: 696e 670a 706f 6c79 6c69 6e65 2873 292e ing.polyline(s).\n+0000f800: 0a0a 1f0a 4669 6c65 3a20 736b 6574 6368 ....File: sketch\n+0000f810: 2e69 6e66 6f2c 2020 4e6f 6465 3a20 5377 .info, Node: Sw\n+0000f820: 6570 7420 6c69 6e65 732c 2020 4e65 7874 ept lines, Next\n+0000f830: 3a20 5377 6570 7420 706f 6c79 676f 6e73 : Swept polygons\n+0000f840: 2c20 2050 7265 763a 2053 7765 7074 2070 , Prev: Swept p\n+0000f850: 6f69 6e74 732c 2020 5570 3a20 5377 6565 oints, Up: Swee\n+0000f860: 7073 0a0a 332e 322e 362e 3220 5377 6570 ps..3.2.6.2 Swep\n+0000f870: 7420 6c69 6e65 730a 2e2e 2e2e 2e2e 2e2e t lines.........\n+0000f880: 2e2e 2e2e 2e2e 2e2e 2e2e 2e0a 0a49 6620 .............If \n+0000f890: 5357 4550 545f 4f42 4a45 4354 2069 7320 SWEPT_OBJECT is \n+0000f8a0: 6120 706f 6c79 6c69 6e65 2061 6e64 2074 a polyline and t\n+0000f8b0: 6865 7265 2069 7320 6e6f 2063 6c6f 7375 here is no closu\n+0000f8c0: 7265 2074 6167 2c20 7468 656e 20e2 8098 re tag, then ...\n+0000f8d0: 7377 6565 70e2 8099 0a63 6f6e 6e65 6374 sweep....connect\n+0000f8e0: 7320 4e2b 3120 7375 6363 6573 7369 7665 s N+1 successive\n+0000f8f0: 2063 6f70 6965 7320 6f66 2074 6865 2070 copies of the p\n+0000f900: 6f6c 796c 696e 6520 2869 6e63 6c75 6469 olyline (includi\n+0000f910: 6e67 2074 6865 206f 7269 6769 6e61 6c29 ng the original)\n+0000f920: 0a77 6974 6820 666f 7572 2d73 6964 6564 .with four-sided\n+0000f930: 2070 6f6c 7967 6f6e 732c 2065 6163 6820 polygons, each \n+0000f940: 7061 6972 206f 6620 636f 7069 6573 2067 pair of copies g\n+0000f950: 6976 696e 6720 7269 7365 2074 6f20 6120 iving rise to a \n+0000f960: 2270 6f6c 7967 6f6e 0a73 7472 6970 2e22 \"polygon.strip.\"\n+0000f970: 2020 4966 2074 6865 7265 2061 7265 206d If there are m\n+0000f980: 2070 6f69 6e74 7320 696e 2074 6865 206f points in the o\n+0000f990: 7269 6769 6e61 6c20 706f 6c79 6c69 6e65 riginal polyline\n+0000f9a0: 2c20 7468 656e 2028 6d2d 3129 4e0a 706f , then (m-1)N.po\n+0000f9b0: 6c79 676f 6e73 2061 7265 2066 6f72 6d65 lygons are forme\n+0000f9c0: 6420 6279 2074 6865 2073 7765 6570 2e20 d by the sweep. \n+0000f9d0: 2057 6520 6361 6c6c 2074 6865 7365 20e2 We call these .\n+0000f9e0: 809c 626f 6479 2070 6f6c 7967 6f6e 73e2 ..body polygons.\n+0000f9f0: 809d 2e20 2049 6e0a 7468 6973 206d 616e ... In.this man\n+0000fa00: 6e65 722c 20e2 8098 7377 6565 70e2 8099 ner, ...sweep...\n+0000fa10: 2066 6f72 6d73 2061 2074 776f 2d64 696d forms a two-dim\n+0000fa20: 656e 7369 6f6e 616c 2073 7572 6661 6365 ensional surface\n+0000fa30: 2066 726f 6d20 6672 6f6d 2061 0a6f 6e65 from from a.one\n+0000fa40: 2d64 696d 656e 7369 6f6e 616c 2070 6f6c -dimensional pol\n+0000fa50: 796c 696e 652e 0a0a 2020 2054 6865 206f yline... The o\n+0000fa60: 7264 6572 206f 6620 7665 7274 6963 6573 rder of vertices\n+0000fa70: 2070 726f 6475 6365 6420 6279 20e2 8098 produced by ...\n+0000fa80: 7377 6565 70e2 8099 2069 7320 696d 706f sweep... is impo\n+0000fa90: 7274 616e 742e 2020 4966 2061 0a70 6f6c rtant. If a.pol\n+0000faa0: 7967 6f6e 2773 2076 6572 7469 6365 7320 ygon's vertices \n+0000fab0: 646f 206e 6f74 2061 7070 6561 7220 696e do not appear in\n+0000fac0: 2063 6f75 6e74 6572 2d63 6c6f 636b 7769 counter-clockwi\n+0000fad0: 7365 206f 7264 6572 2069 6e20 7468 6520 se order in the \n+0000fae0: 6669 6e61 6c0a 696d 6167 652c 2074 6865 final.image, the\n+0000faf0: 2070 6f6c 7967 6f6e 2077 696c 6c20 6265 polygon will be\n+0000fb00: 2063 756c 6c65 6420 2875 6e6c 6573 7320 culled (unless \n+0000fb10: e280 9863 756c 6c3d 6661 6c73 65e2 8099 ...cull=false...\n+0000fb20: 2069 7320 7365 7429 2e20 2049 6620 7468 is set). If th\n+0000fb30: 650a 706f 696e 7473 2069 6e20 7468 6520 e.points in the \n+0000fb40: 6b27 7468 2063 6f70 7920 6f66 2074 6865 k'th copy of the\n+0000fb50: 2070 6f6c 796c 696e 6520 6172 6520 505f polyline are P_\n+0000fb60: 312c 2050 5f32 2c20 2e2e 2e2c 2050 5f6d 1, P_2, ..., P_m\n+0000fb70: 2c20 616e 6420 7468 650a 706f 696e 7473 , and the.points\n+0000fb80: 2069 6e20 7468 6520 6e65 7874 2063 6f70 in the next cop\n+0000fb90: 792c 2074 6865 2028 6b2b 3129 7374 2c20 y, the (k+1)st, \n+0000fba0: 6172 6520 505f 3127 2c20 505f 3227 2c20 are P_1', P_2', \n+0000fbb0: 2e2e 2e2c 2050 5f6d 272c 2074 6865 6e0a ..., P_m', then.\n+0000fbc0: 7468 6520 7665 7274 6578 206f 7264 6572 the vertex order\n+0000fbd0: 206f 6620 7468 6520 6765 6e65 7261 7465 of the generate\n+0000fbe0: 6420 706f 6c79 676f 6e73 2069 730a 2020 d polygons is. \n+0000fbf0: 2020 2042 6f64 7920 706f 6c79 676f 6e20 Body polygon \n+0000fc00: 313a 2050 5f32 2050 5f31 2050 5f31 2720 1: P_2 P_1 P_1' \n+0000fc10: 505f 3227 0a20 2020 2020 426f 6479 2070 P_2'. Body p\n+0000fc20: 6f6c 7967 6f6e 2032 3a20 505f 3320 505f olygon 2: P_3 P_\n+0000fc30: 3220 505f 3227 2050 5f33 270a 2020 2020 2 P_2' P_3'. \n+0000fc40: 202e 2e2e 0a20 2020 2020 426f 6479 2070 .... Body p\n+0000fc50: 6f6c 7967 6f6e 206d 2d31 3a20 505f 6d20 olygon m-1: P_m \n+0000fc60: 505f 7b6d 2d31 7d20 505f 7b6d 2d31 7d27 P_{m-1} P_{m-1}'\n+0000fc70: 2050 5f6d 270a 0a20 2020 4f70 7469 6f6e P_m'.. Option\n+0000fc80: 7320 6f66 2075 6e63 6c6f 7365 6420 6c69 s of unclosed li\n+0000fc90: 6e65 2073 7765 6570 7320 6172 6520 636f ne sweeps are co\n+0000fca0: 7069 6564 2074 6f20 6561 6368 206f 7574 pied to each out\n+0000fcb0: 7075 7420 706f 6c79 676f 6e2e 0a4f 7074 put polygon..Opt\n+0000fcc0: 696f 6e73 206f 6620 7468 6520 7377 6570 ions of the swep\n+0000fcd0: 7420 6c69 6e65 2061 7265 2069 676e 6f72 t line are ignor\n+0000fce0: 6564 2e0a 0a20 2020 5768 656e 2074 6865 ed... When the\n+0000fcf0: 7265 205f 6973 5f20 6120 636c 6f73 7572 re _is_ a closur\n+0000fd00: 6520 7461 672c 2074 6865 6e20 e280 9873 e tag, then ...s\n+0000fd10: 7765 6570 e280 9920 636f 6e6e 6563 7473 weep... connects\n+0000fd20: 204e 2073 7563 6365 7373 6976 650a 636f N successive.co\n+0000fd30: 7069 6573 206f 6620 7468 6520 706f 6c79 pies of the poly\n+0000fd40: 6c69 6e65 2028 696e 636c 7564 696e 6720 line (including \n+0000fd50: 7468 6520 6f72 6967 696e 616c 2920 7769 the original) wi\n+0000fd60: 7468 2066 6f75 722d 7369 6465 6420 626f th four-sided bo\n+0000fd70: 6479 0a70 6f6c 7967 6f6e 7320 6a75 7374 dy.polygons just\n+0000fd80: 2061 7320 7468 6520 6361 7365 2077 6974 as the case wit\n+0000fd90: 6820 6e6f 2063 6c6f 7375 7265 2074 6167 h no closure tag\n+0000fda0: 2e20 2049 7420 7468 656e 2063 6f6e 6e65 . It then conne\n+0000fdb0: 6374 7320 7468 650a 6c61 7374 2063 6f70 cts the.last cop\n+0000fdc0: 7920 6261 636b 2074 6f20 7468 6520 6f72 y back to the or\n+0000fdd0: 6967 696e 616c 2074 6f20 666f 726d 2061 iginal to form a\n+0000fde0: 2072 6962 626f 6e2d 7368 6170 6564 2073 ribbon-shaped s\n+0000fdf0: 7572 6661 6365 2074 6861 740a 636c 6f73 urface that.clos\n+0000fe00: 6573 206f 6e20 6974 7365 6c66 2077 6974 es on itself wit\n+0000fe10: 6820 7477 6f20 686f 6c65 7320 7265 6d61 h two holes rema\n+0000fe20: 696e 696e 672e 0a0a 2020 2046 696e 616c ining... Final\n+0000fe30: 6c79 2c20 7468 6520 7377 6565 7020 6164 ly, the sweep ad\n+0000fe40: 6473 2074 776f 206d 6f72 6520 706f 6c79 ds two more poly\n+0000fe50: 676f 6e73 2074 6f20 7365 616c 2074 6865 gons to seal the\n+0000fe60: 2068 6f6c 6573 2061 6e64 2066 6f72 6d0a holes and form.\n+0000fe70: 6120 636c 6f73 6564 2073 7572 6661 6365 a closed surface\n+0000fe80: 2074 6861 742c 2064 6570 656e 6469 6e67 that, depending\n+0000fe90: 206f 6e20 7468 6520 7377 6565 7020 7472 on the sweep tr\n+0000fea0: 616e 7366 6f72 6d73 2c20 6d61 7920 7265 ansforms, may re\n+0000feb0: 7072 6573 656e 740a 7468 6520 626f 756e present.the boun\n+0000fec0: 6461 7279 206f 6620 6120 736f 6c69 642e dary of a solid.\n+0000fed0: 2020 496e 2074 6869 7320 6d61 6e6e 6572 In this manner\n+0000fee0: 2c20 e280 9873 7765 6570 e280 9920 666f , ...sweep... fo\n+0000fef0: 726d 7320 7468 6520 626f 756e 6461 7279 rms the boundary\n+0000ff00: 206f 660a 6120 7468 7265 652d 6469 6d65 of.a three-dime\n+0000ff10: 6e73 696f 6e61 6c20 6f62 6a65 6374 2066 nsional object f\n+0000ff20: 726f 6d20 6120 6f6e 652d 6469 6d65 6e73 rom a one-dimens\n+0000ff30: 696f 6e61 6c20 706f 6c79 6c69 6e65 2e20 ional polyline. \n+0000ff40: 2057 6520 6361 6c6c 0a74 6865 7365 2068 We call.these h\n+0000ff50: 6f6c 652d 6669 6c6c 696e 6720 706f 6c79 ole-filling poly\n+0000ff60: 676f 6e73 20e2 809c 656e 6473 e280 9d2e gons ...ends....\n+0000ff70: 0a0a 2020 2054 6865 206f 7264 6572 206f .. The order o\n+0000ff80: 6620 7665 7274 6963 6573 206f 6620 656e f vertices of en\n+0000ff90: 6420 706f 6c79 676f 6e73 2069 7320 696d d polygons is im\n+0000ffa0: 706f 7274 616e 7420 666f 7220 636f 7272 portant for corr\n+0000ffb0: 6563 740a 6375 6c6c 696e 6720 6173 2064 ect.culling as d\n+0000ffc0: 6573 6372 6962 6564 2061 626f 7665 2e20 escribed above. \n+0000ffd0: 2049 6620 505f 315e 312c 2050 5f31 5e32 If P_1^1, P_1^2\n+0000ffe0: 2c20 2e2e 2e2c 2050 5f31 5e6e 2061 7265 , ..., P_1^n are\n+0000fff0: 2074 6865 204e 0a63 6f70 6965 7320 6f66 the N.copies of\n+00010000: 2074 6865 2066 6972 7374 2070 6f6c 796c the first polyl\n+00010010: 696e 6520 706f 696e 7420 616e 6420 505f ine point and P_\n+00010020: 6d5e 312c 2050 5f6d 5e32 2c20 2e2e 2e20 m^1, P_m^2, ... \n+00010030: 2c50 5f6d 5e6e 2061 7265 2074 6865 0a4e ,P_m^n are the.N\n+00010040: 2063 6f70 6965 7320 6f66 2074 6865 206c copies of the l\n+00010050: 6173 7420 706f 6c79 6c69 6e65 2070 6f69 ast polyline poi\n+00010060: 6e74 2c20 7468 656e 2074 6865 2065 6e64 nt, then the end\n+00010070: 2070 6f6c 7967 6f6e 2076 6572 7465 7820 polygon vertex \n+00010080: 6f72 6465 720a 6973 0a20 2020 2020 456e order.is. En\n+00010090: 6420 706f 6c79 676f 6e20 313a 2050 5f31 d polygon 1: P_1\n+000100a0: 5e6e 2c20 505f 315e 7b6e 2d31 7d2c 202e ^n, P_1^{n-1}, .\n+000100b0: 2e2e 202c 505f 315e 310a 2020 2020 2045 .. ,P_1^1. E\n+000100c0: 6e64 2070 6f6c 7967 6f6e 2032 3a20 505f nd polygon 2: P_\n+000100d0: 6d5e 312c 2050 5f6d 5e32 2c20 2e2e 2e20 m^1, P_m^2, ... \n+000100e0: 2c50 5f6d 5e6e 0a0a 2020 2049 6620 7468 ,P_m^n.. If th\n+000100f0: 6572 6520 6172 6520 6e6f 206f 7074 696f ere are no optio\n+00010100: 6e73 206f 6e20 7468 6520 7377 6570 7420 ns on the swept \n+00010110: 6c69 6e65 2c20 7468 656e 2074 6865 20e2 line, then the .\n+00010120: 8098 7377 6565 70e2 8099 206f 7074 696f ..sweep... optio\n+00010130: 6e73 0a61 7265 2063 6f70 6965 6420 746f ns.are copied to\n+00010140: 2065 6163 6820 6f75 7470 7574 2070 6f6c each output pol\n+00010150: 7967 6f6e 2e20 2049 6620 7468 6520 7377 ygon. If the sw\n+00010160: 6570 7420 6c69 6e65 2064 6f65 7320 6861 ept line does ha\n+00010170: 7665 206f 7074 696f 6e73 2c0a 7468 6573 ve options,.thes\n+00010180: 6520 6172 6520 636f 7069 6564 2074 6f20 e are copied to \n+00010190: 636f 7272 6573 706f 6e64 696e 6720 626f corresponding bo\n+000101a0: 6479 2070 6f6c 7967 6f6e 733b 2074 6865 dy polygons; the\n+000101b0: 2073 7765 6570 206f 7074 696f 6e73 2061 sweep options a\n+000101c0: 7265 0a63 6f70 6965 6420 746f 2074 6865 re.copied to the\n+000101d0: 2065 6e64 2070 6f6c 7967 6f6e 732e 2020 end polygons. \n+000101e0: 496e 2074 6869 7320 6d61 6e6e 6572 2c20 In this manner, \n+000101f0: 626f 6479 2061 6e64 2065 6e64 7320 6d61 body and ends ma\n+00010200: 7920 6265 2064 7261 776e 0a77 6974 6820 y be drawn.with \n+00010210: 6469 6666 6572 656e 7420 6368 6172 6163 different charac\n+00010220: 7465 7269 7374 6963 7320 7375 6368 2061 teristics such a\n+00010230: 7320 e280 9866 696c 6c63 6f6c 6f72 e280 s ...fillcolor..\n+00010240: 992e 0a0a 1f0a 4669 6c65 3a20 736b 6574 ......File: sket\n+00010250: 6368 2e69 6e66 6f2c 2020 4e6f 6465 3a20 ch.info, Node: \n+00010260: 5377 6570 7420 706f 6c79 676f 6e73 2c20 Swept polygons, \n+00010270: 204e 6578 743a 2053 7765 7074 2062 6c6f Next: Swept blo\n+00010280: 636b 732c 2020 5072 6576 3a20 5377 6570 cks, Prev: Swep\n+00010290: 7420 6c69 6e65 732c 2020 5570 3a20 5377 t lines, Up: Sw\n+000102a0: 6565 7073 0a0a 332e 322e 362e 3320 5377 eeps..3.2.6.3 Sw\n+000102b0: 6570 7420 706f 6c79 676f 6e73 0a2e 2e2e ept polygons....\n+000102c0: 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e ................\n+000102d0: 2e2e 2e0a 0a49 6620 5357 4550 545f 4f42 .....If SWEPT_OB\n+000102e0: 4a45 4354 2069 7320 6120 706f 6c79 676f JECT is a polygo\n+000102f0: 6e2c 2074 6865 20e2 8098 7377 6565 70e2 n, the ...sweep.\n+00010300: 8099 2063 6f6e 6e65 6374 7320 4e2b 3120 .. connects N+1 \n+00010310: 7375 6363 6573 7369 7665 2063 6f70 6965 successive copie\n+00010320: 730a 6f66 2074 6865 2063 6c6f 7365 6420 s.of the closed \n+00010330: 706f 6c79 6c69 6e65 2062 6f72 6465 7220 polyline border \n+00010340: 6f66 2074 6865 2070 6f6c 7967 6f6e 2074 of the polygon t\n+00010350: 6f20 666f 726d 2062 6f64 7920 706f 6c79 o form body poly\n+00010360: 676f 6e73 0a65 7861 6374 6c79 2061 7320 gons.exactly as \n+00010370: 7468 6f75 6768 2074 6865 2062 6f72 6465 though the borde\n+00010380: 7220 7765 7265 2061 2073 7765 7074 2070 r were a swept p\n+00010390: 6f6c 796c 696e 6520 6173 2064 6573 6372 olyline as descr\n+000103a0: 6962 6564 2069 6e20 2a6e 6f74 650a 5377 ibed in *note.Sw\n+000103b0: 6570 7420 6c69 6e65 733a 3a2e 2020 4966 ept lines::. If\n+000103c0: 2074 6865 7265 2061 7265 206d 2070 6f69 there are m poi\n+000103d0: 6e74 7320 696e 2074 6865 206f 7269 6769 nts in the origi\n+000103e0: 6e61 6c20 706f 6c79 676f 6e2c 2074 6865 nal polygon, the\n+000103f0: 6e20 6d4e 0a62 6f64 7920 706f 6c79 676f n mN.body polygo\n+00010400: 6e73 2061 7265 2066 6f72 6d65 6420 6279 ns are formed by\n+00010410: 2074 6869 7320 7377 6565 702e 2020 5468 this sweep. Th\n+00010420: 6520 626f 6479 2070 6f6c 7967 6f6e 7320 e body polygons \n+00010430: 666f 726d 2061 6e0a e280 9c65 7874 7275 form an....extru\n+00010440: 7369 6f6e e280 9d20 6f66 2074 6865 2062 sion... of the b\n+00010450: 6f75 6e64 6172 7920 6f66 2074 6865 206f oundary of the o\n+00010460: 7269 6769 6e61 6c20 706f 6c79 676f 6e20 riginal polygon \n+00010470: 7769 7468 2074 776f 2068 6f6c 6573 2061 with two holes a\n+00010480: 740a 7468 6520 6f70 656e 2065 6e64 732e t.the open ends.\n+00010490: 0a0a 2020 2046 696e 616c 6c79 2c20 7468 .. Finally, th\n+000104a0: 6520 7377 6565 7020 6164 6473 2074 776f e sweep adds two\n+000104b0: 2063 6f70 6965 7320 6f66 2074 6865 206f copies of the o\n+000104c0: 7269 6769 6e61 6c20 706f 6c79 676f 6e20 riginal polygon \n+000104d0: 746f 2063 6f76 6572 0a74 6865 2068 6f6c to cover.the hol\n+000104e0: 6573 2e20 2057 6520 6361 6c6c 2074 6865 es. We call the\n+000104f0: 7365 2068 6f6c 652d 6669 6c6c 696e 6720 se hole-filling \n+00010500: 706f 6c79 676f 6e73 20e2 809c 656e 6473 polygons ...ends\n+00010510: e280 9d2e 2020 496e 2074 6869 7320 6d61 .... In this ma\n+00010520: 6e6e 6572 2c0a e280 9873 7765 6570 e280 nner,....sweep..\n+00010530: 9920 666f 726d 7320 7468 6520 626f 756e . forms the boun\n+00010540: 6461 7279 206f 6620 6120 7468 7265 652d dary of a three-\n+00010550: 6469 6d65 6e73 696f 6e61 6c20 6f62 6a65 dimensional obje\n+00010560: 6374 2066 726f 6d20 610a 7477 6f2d 6469 ct from a.two-di\n+00010570: 6d65 6e73 696f 6e61 6c20 706f 6c79 676f mensional polygo\n+00010580: 6e2e 0a0a 2020 2054 6865 206f 7264 6572 n... The order\n+00010590: 206f 6620 7665 7274 6963 6573 206f 6620 of vertices of \n+000105a0: 656e 6420 706f 6c79 676f 6e73 2069 7320 end polygons is \n+000105b0: 696d 706f 7274 616e 7420 666f 7220 636f important for co\n+000105c0: 7272 6563 740a 6375 6c6c 696e 6720 6173 rrect.culling as\n+000105d0: 2064 6573 6372 6962 6564 2061 626f 7665 described above\n+000105e0: 2e20 2041 6e20 6578 6163 7420 636f 7079 . An exact copy\n+000105f0: 206f 6620 7468 6520 6f72 6967 696e 616c of the original\n+00010600: 2070 6f6c 7967 6f6e 2077 6974 680a 7665 polygon with.ve\n+00010610: 7274 6578 206f 7264 6572 2069 6e74 6163 rtex order intac\n+00010620: 7420 666f 726d 7320 7468 6520 6669 7273 t forms the firs\n+00010630: 7420 656e 6420 706f 6c79 676f 6e2e 2020 t end polygon. \n+00010640: 5468 6520 6f74 6865 7220 656e 6420 706f The other end po\n+00010650: 6c79 676f 6e0a 7265 7375 6c74 7320 6672 lygon.results fr\n+00010660: 6f6d 2074 7261 6e73 666f 726d 696e 6720 om transforming \n+00010670: 616e 6420 7468 6520 7265 7665 7273 696e and the reversin\n+00010680: 6720 7468 6520 6f72 6465 7220 6f66 2076 g the order of v\n+00010690: 6572 7469 6365 7320 696e 2074 6865 0a6f ertices in the.o\n+000106a0: 7269 6769 6e61 6c2e 2020 5468 6520 7472 riginal. The tr\n+000106b0: 616e 7366 6f72 6d20 706c 6163 6573 2074 ansform places t\n+000106c0: 6865 206f 7269 6769 6e61 6c20 706f 6c79 he original poly\n+000106d0: 676f 6e20 6174 2074 6865 2075 6e63 6f76 gon at the uncov\n+000106e0: 6572 6564 0a68 6f6c 653b 2069 7420 6973 ered.hole; it is\n+000106f0: 0a20 2020 2020 545f 315e 6e20 7468 656e . T_1^n then\n+00010700: 2054 5f32 5e6e 2074 6865 6e20 2e2e 2e20 T_2^n then ... \n+00010710: 7468 656e 2054 5f52 5e6e 2e0a 4966 2074 then T_R^n..If t\n+00010720: 6865 7265 2061 7265 206e 6f20 6f70 7469 here are no opti\n+00010730: 6f6e 7320 6f6e 2074 6865 2073 7765 7074 ons on the swept\n+00010740: 2070 6f6c 7967 6f6e 2c20 7468 656e 2074 polygon, then t\n+00010750: 6865 20e2 8098 7377 6565 70e2 8099 206f he ...sweep... o\n+00010760: 7074 696f 6e73 0a61 7265 2063 6f70 6965 ptions.are copie\n+00010770: 6420 746f 2065 6163 6820 6f75 7470 7574 d to each output\n+00010780: 2070 6f6c 7967 6f6e 2e20 2049 6620 7468 polygon. If th\n+00010790: 6520 7377 6570 7420 706f 6c79 676f 6e20 e swept polygon \n+000107a0: 646f 6573 2068 6176 650a 6f70 7469 6f6e does have.option\n+000107b0: 732c 2074 6865 7365 2061 7265 2063 6f70 s, these are cop\n+000107c0: 6965 6420 746f 2074 6865 2065 6e64 733b ied to the ends;\n+000107d0: 2074 6865 2073 7765 6570 206f 7074 696f the sweep optio\n+000107e0: 6e73 2061 7265 2063 6f70 6965 6420 746f ns are copied to\n+000107f0: 0a74 6865 2062 6f64 7920 706f 6c79 676f .the body polygo\n+00010800: 6e73 2e20 2049 6e20 7468 6973 206d 616e ns. In this man\n+00010810: 6e65 722c 2062 6f64 7920 616e 6420 656e ner, body and en\n+00010820: 6473 206d 6179 2062 6520 6472 6177 6e20 ds may be drawn \n+00010830: 7769 7468 0a64 6966 6665 7265 6e74 2063 with.different c\n+00010840: 6861 7261 6374 6572 6973 7469 6373 2073 haracteristics s\n+00010850: 7563 6820 6173 20e2 8098 6669 6c6c 636f uch as ...fillco\n+00010860: 6c6f 72e2 8099 2e0a 0a1f 0a46 696c 653a lor........File:\n+00010870: 2073 6b65 7463 682e 696e 666f 2c20 204e sketch.info, N\n+00010880: 6f64 653a 2053 7765 7074 2062 6c6f 636b ode: Swept block\n+00010890: 732c 2020 4e65 7874 3a20 5377 6565 7020 s, Next: Sweep \n+000108a0: 6661 6365 2073 706c 6974 7469 6e67 2c20 face splitting, \n+000108b0: 2050 7265 763a 2053 7765 7074 2070 6f6c Prev: Swept pol\n+000108c0: 7967 6f6e 732c 2020 5570 3a20 5377 6565 ygons, Up: Swee\n+000108d0: 7073 0a0a 332e 322e 362e 3420 5377 6570 ps..3.2.6.4 Swep\n+000108e0: 7420 626c 6f63 6b73 0a2e 2e2e 2e2e 2e2e t blocks........\n+000108f0: 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e0a 0a54 ...............T\n+00010900: 6865 2073 7765 7074 206f 626a 6563 7420 he swept object \n+00010910: 5357 4550 545f 4f42 4a45 4354 206d 6179 SWEPT_OBJECT may\n+00010920: 2061 6c73 6f20 6265 2061 6e79 2063 6f6c also be any col\n+00010930: 6c65 6374 696f 6e20 6f66 2070 6f6c 796c lection of polyl\n+00010940: 696e 6573 0a61 6e64 2070 6f6c 7967 6f6e ines.and polygon\n+00010950: 732e 2020 5468 6973 206d 6179 2062 6520 s. This may be \n+00010960: 6120 626c 6f63 6b20 636f 6d70 6f73 6564 a block composed\n+00010970: 206f 6620 e280 986c 696e 65e2 8099 2061 of ...line... a\n+00010980: 6e64 2f6f 7220 e280 9870 6f6c 7967 6f6e nd/or ...polygon\n+00010990: e280 990a 636f 6d6d 616e 6473 2069 6e20 ....commands in \n+000109a0: 6272 6163 6573 20e2 8098 7b20 7de2 8099 braces ...{ }...\n+000109b0: 2c20 6f72 2069 7420 6d61 7920 6265 2074 , or it may be t\n+000109c0: 6865 2072 6573 756c 7420 6f66 2061 20e2 he result of a .\n+000109d0: 8098 7265 7065 6174 e280 992c 2061 6e6f ..repeat..., ano\n+000109e0: 7468 6572 0ae2 8098 7377 6565 70e2 8099 ther....sweep...\n+000109f0: 2c20 6574 632e 2020 5468 6520 7377 6565 , etc. The swee\n+00010a00: 7020 6163 7473 2069 6e64 6570 656e 6465 p acts independe\n+00010a10: 6e74 6c79 206f 6e20 6561 6368 206f 626a ntly on each obj\n+00010a20: 6563 7420 696e 2074 6865 2062 6c6f 636b ect in the block\n+00010a30: 0a65 7861 6374 6c79 2061 7320 6966 2069 .exactly as if i\n+00010a40: 7420 7765 7265 2061 2073 696e 676c 6520 t were a single \n+00010a50: 7377 6570 7420 6f62 6a65 6374 2064 6573 swept object des\n+00010a60: 6372 6962 6564 2061 626f 7665 2069 6e20 cribed above in \n+00010a70: 2a6e 6f74 650a 5377 6570 7420 6c69 6e65 *note.Swept line\n+00010a80: 733a 3a20 616e 6420 2a6e 6f74 6520 5377 s:: and *note Sw\n+00010a90: 6570 7420 706f 6c79 676f 6e73 3a3a 2e0a ept polygons::..\n+00010aa0: 0a1f 0a46 696c 653a 2073 6b65 7463 682e ...File: sketch.\n+00010ab0: 696e 666f 2c20 204e 6f64 653a 2053 7765 info, Node: Swe\n+00010ac0: 6570 2066 6163 6520 7370 6c69 7474 696e ep face splittin\n+00010ad0: 672c 2020 5072 6576 3a20 5377 6570 7420 g, Prev: Swept \n+00010ae0: 626c 6f63 6b73 2c20 2055 703a 2053 7765 blocks, Up: Swe\n+00010af0: 6570 730a 0a33 2e32 2e36 2e35 2053 7765 eps..3.2.6.5 Swe\n+00010b00: 6570 2066 6163 6520 7370 6c69 7474 696e ep face splittin\n+00010b10: 670a 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e g...............\n+00010b20: 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 0a0a ................\n+00010b30: 4265 666f 7265 2073 656e 6469 6e67 2065 Before sending e\n+00010b40: 6163 6820 666f 7572 2d73 6964 6564 2062 ach four-sided b\n+00010b50: 6f64 7920 706f 6c79 676f 6e20 6f66 2061 ody polygon of a\n+00010b60: 20e2 8098 7377 6565 70e2 8099 2074 6f20 ...sweep... to \n+00010b70: 7468 6520 6f75 7470 7574 2c0a e280 9873 the output,....s\n+00010b80: 6b65 7463 68e2 8099 2074 6573 7473 2074 ketch... tests t\n+00010b90: 6f20 7365 6520 6966 2069 7420 6973 2072 o see if it is r\n+00010ba0: 6f75 6768 6c79 2070 6c61 6e61 722e 2020 oughly planar. \n+00010bb0: 5369 6e63 6520 706c 616e 6172 6974 7920 Since planarity \n+00010bc0: 6973 0a6e 6563 6573 7361 7279 2066 6f72 is.necessary for\n+00010bd0: 2070 726f 7065 7220 6675 6e63 7469 6f6e proper function\n+00010be0: 696e 6720 6f66 2074 6865 2068 6964 6465 ing of the hidde\n+00010bf0: 6e20 7375 7266 6163 6520 616c 676f 7269 n surface algori\n+00010c00: 7468 6d2c 0a22 7761 7270 6564 2220 706f thm,.\"warped\" po\n+00010c10: 6c79 676f 6e73 2061 7265 2061 7574 6f6d lygons are autom\n+00010c20: 6174 6963 616c 6c79 2073 706c 6974 2069 atically split i\n+00010c30: 6e74 6f20 7477 6f20 7472 6961 6e67 6c65 nto two triangle\n+00010c40: 732e 0a0a 2020 2048 6f6c 652d 6669 6c6c s... Hole-fill\n+00010c50: 696e 6720 706f 6c79 676f 6e73 2070 726f ing polygons pro\n+00010c60: 6475 6365 6420 6279 2063 6c6f 7375 7265 duced by closure\n+00010c70: 2d74 6167 6765 6420 6c69 6e65 2073 7765 -tagged line swe\n+00010c80: 6570 7320 6172 6520 6e6f 740a 7370 6c69 eps are not.spli\n+00010c90: 742e 2020 4e6f 7220 6172 6520 6f72 6967 t. Nor are orig\n+00010ca0: 696e 616c 2070 6f6c 7967 6f6e 7320 696e inal polygons in\n+00010cb0: 2070 6f6c 7967 6f6e 2073 7765 6570 732e polygon sweeps.\n+00010cc0: 2020 4974 2069 7320 7468 6520 7573 6572 It is the user\n+00010cd0: 2773 0a72 6573 706f 6e73 6962 696c 6974 's.responsibilit\n+00010ce0: 7920 746f 2065 6e73 7572 6520 7468 6573 y to ensure thes\n+00010cf0: 6520 6172 6520 706c 616e 6172 2e0a 0a1f e are planar....\n+00010d00: 0a46 696c 653a 2073 6b65 7463 682e 696e .File: sketch.in\n+00010d10: 666f 2c20 204e 6f64 653a 2042 6c6f 636b fo, Node: Block\n+00010d20: 732c 2020 4e65 7874 3a20 5265 7065 6174 s, Next: Repeat\n+00010d30: 732c 2020 5072 6576 3a20 5377 6565 7073 s, Prev: Sweeps\n+00010d40: 2c20 2055 703a 2044 7261 7761 626c 6573 , Up: Drawables\n+00010d50: 0a0a 332e 322e 3720 426c 6f63 6b73 0a2d ..3.2.7 Blocks.-\n+00010d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a41 6e79 -----------..Any\n+00010d70: 2073 6571 7565 6e63 6520 6f66 2064 7261 sequence of dra\n+00010d80: 7761 626c 6573 206d 6179 2062 6520 6772 wables may be gr\n+00010d90: 6f75 7065 6420 696e 2061 20e2 809c 626c ouped in a ...bl\n+00010da0: 6f63 6be2 809d 206d 6572 656c 7920 6279 ock... merely by\n+00010db0: 0a65 6e63 6c6f 7369 6e67 2074 6865 6d20 .enclosing them \n+00010dc0: 696e 2062 7261 6365 7320 e280 987b 207d in braces ...{ }\n+00010dd0: e280 992e 2020 4120 626c 6f63 6b20 6973 .... A block is\n+00010de0: 2069 7473 656c 6620 6472 6177 6162 6c65 itself drawable\n+00010df0: 2e20 2041 206b 6579 2075 7365 0a6f 6620 . A key use.of \n+00010e00: 626c 6f63 6b73 2069 7320 746f 2065 7874 blocks is to ext\n+00010e10: 656e 6420 7468 6520 6566 6665 6374 206f end the effect o\n+00010e20: 6620 6120 7369 6e67 6c65 20e2 8098 6465 f a single ...de\n+00010e30: 66e2 8099 2c20 2a6e 6f74 650a 4465 6669 f..., *note.Defi\n+00010e40: 6e69 7469 6f6e 733a 3a2c 20e2 8098 7075 nitions::, ...pu\n+00010e50: 74e2 8099 202a 6e6f 7465 2050 7574 733a t... *note Puts:\n+00010e60: 3a2c 20e2 8098 7377 6565 70e2 8099 202a :, ...sweep... *\n+00010e70: 6e6f 7465 2053 7765 6570 733a 3a2c 206f note Sweeps::, o\n+00010e80: 7220 e280 9872 6570 6561 74e2 8099 0a2a r ...repeat....*\n+00010e90: 6e6f 7465 2052 6570 6561 7473 3a3a 2074 note Repeats:: t\n+00010ea0: 6f20 696e 636c 7564 6520 7365 7665 7261 o include severa\n+00010eb0: 6c20 6f62 6a65 6374 7320 7261 7468 6572 l objects rather\n+00010ec0: 2074 6861 6e20 6f6e 652e 0a0a 2020 2044 than one... D\n+00010ed0: 6566 696e 6974 696f 6e73 2028 2a4e 6f74 efinitions (*Not\n+00010ee0: 6520 4465 6669 6e69 7469 6f6e 733a 3a2e e Definitions::.\n+00010ef0: 2920 2069 6e73 6964 6520 6120 626c 6f63 ) inside a bloc\n+00010f00: 6b20 6861 7665 20e2 809c 6c65 7869 6361 k have ...lexica\n+00010f10: 6c0a 7363 6f70 65e2 809d 2065 7874 656e l.scope... exten\n+00010f20: 6469 6e67 2066 726f 6d20 7468 6520 706c ding from the pl\n+00010f30: 6163 6520 6f66 2064 6566 696e 6974 696f ace of definitio\n+00010f40: 6e20 746f 2074 6865 2065 6e64 206f 6620 n to the end of \n+00010f50: 7468 6520 626c 6f63 6b2e 0a0a 1f0a 4669 the block.....Fi\n+00010f60: 6c65 3a20 736b 6574 6368 2e69 6e66 6f2c le: sketch.info,\n+00010f70: 2020 4e6f 6465 3a20 5265 7065 6174 732c Node: Repeats,\n+00010f80: 2020 4e65 7874 3a20 5075 7473 2c20 2050 Next: Puts, P\n+00010f90: 7265 763a 2042 6c6f 636b 732c 2020 5570 rev: Blocks, Up\n+00010fa0: 3a20 4472 6177 6162 6c65 730a 0a33 2e32 : Drawables..3.2\n+00010fb0: 2e38 2052 6570 6561 7473 0a2d 2d2d 2d2d .8 Repeats.-----\n+00010fc0: 2d2d 2d2d 2d2d 2d2d 0a0a 2a53 796e 7461 --------..*Synta\n+00010fd0: 783a 2a0a 2020 2020 2072 6570 6561 7420 x:*. repeat \n+00010fe0: 7b20 4e2c 2054 5f31 2c20 545f 322c 202e { N, T_1, T_2, .\n+00010ff0: 2e2e 2c20 545f 5220 7d20 5245 5045 4154 .., T_R } REPEAT\n+00011000: 4544 5f4f 424a 4543 540a 5468 6520 7265 ED_OBJECT.The re\n+00011010: 7065 6174 206d 616b 6573 204e 2074 7261 peat makes N tra\n+00011020: 6e73 666f 726d 6564 2063 6f70 6965 7320 nsformed copies \n+00011030: 6f66 2052 4550 4541 5445 445f 4f42 4a45 of REPEATED_OBJE\n+00011040: 4354 2028 696e 636c 7564 696e 6720 7468 CT (including th\n+00011050: 650a 6f72 6967 696e 616c 292e 2020 5468 e.original). Th\n+00011060: 6520 545f 4920 6172 6520 7472 616e 7366 e T_I are transf\n+00011070: 6f72 6d73 2e20 2054 6865 206b 2774 6820 orms. The k'th \n+00011080: 636f 7079 206f 6620 7468 650a 5245 5045 copy of the.REPE\n+00011090: 4154 4544 5f4f 424a 4543 5420 2866 6f72 ATED_OBJECT (for\n+000110a0: 206b 3d30 2c31 2c2e 2e2e 2c6e 2d31 2920 k=0,1,...,n-1) \n+000110b0: 6973 2070 726f 6475 6365 6420 696e 2074 is produced in t\n+000110c0: 6865 2073 616d 6520 6d61 6e6e 6572 2061 he same manner a\n+000110d0: 730a 666f 7220 e280 9873 7765 6570 e280 s.for ...sweep..\n+000110e0: 9973 2064 6573 6372 6962 6564 2069 6e20 .s described in \n+000110f0: 2a6e 6f74 6520 5377 6565 7073 3a3a 2e20 *note Sweeps::. \n+00011100: 2054 6869 7320 6973 2072 6570 6561 7465 This is repeate\n+00011110: 6420 6865 7265 2028 6e6f 2070 756e 0a69 d here (no pun.i\n+00011120: 6e74 656e 6465 6429 2066 6f72 2063 6f6e ntended) for con\n+00011130: 7665 6e69 656e 6365 2e20 2054 6f20 6d61 venience. To ma\n+00011140: 6b65 2074 6865 206b 2774 6820 636f 7079 ke the k'th copy\n+00011150: 2c20 7468 6520 666f 6c6c 6f77 696e 670a , the following.\n+00011160: 7472 616e 7366 6f72 6d20 6973 2061 7070 transform is app\n+00011170: 6c69 6564 2074 6f20 7468 6520 6f72 6967 lied to the orig\n+00011180: 696e 616c 206f 626a 6563 742e 0a20 2020 inal object.. \n+00011190: 2020 545f 315e 6b20 7468 656e 2054 5f32 T_1^k then T_2\n+000111a0: 5e6b 2074 6865 6e20 2e2e 2e20 7468 656e ^k then ... then\n+000111b0: 2054 5f52 5e6b 0a48 6572 6520 545e 6b20 T_R^k.Here T^k \n+000111c0: 6d65 616e 7320 2274 7261 6e73 666f 726d means \"transform\n+000111d0: 2054 2061 7070 6c69 6564 206b 2074 696d T applied k tim\n+000111e0: 6573 2e22 0a0a 1f0a 4669 6c65 3a20 736b es.\"....File: sk\n+000111f0: 6574 6368 2e69 6e66 6f2c 2020 4e6f 6465 etch.info, Node\n+00011200: 3a20 5075 7473 2c20 2050 7265 763a 2052 : Puts, Prev: R\n+00011210: 6570 6561 7473 2c20 2055 703a 2044 7261 epeats, Up: Dra\n+00011220: 7761 626c 6573 0a0a 332e 322e 3920 5075 wables..3.2.9 Pu\n+00011230: 7473 0a2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a2a ts.----------..*\n+00011240: 5379 6e74 6178 3a2a 0a20 2020 2020 7075 Syntax:*. pu\n+00011250: 7420 7b20 5420 7d20 5055 545f 4f42 4a45 t { T } PUT_OBJE\n+00011260: 4354 0a50 7574 206d 6572 656c 7920 6170 CT.Put merely ap\n+00011270: 706c 6965 7320 7472 616e 7366 6f72 6d20 plies transform \n+00011280: 5420 746f 2074 6865 2064 7261 7761 626c T to the drawabl\n+00011290: 6520 5055 545f 4f42 4a45 4354 2e0a 0a1f e PUT_OBJECT....\n+000112a0: 0a46 696c 653a 2073 6b65 7463 682e 696e .File: sketch.in\n+000112b0: 666f 2c20 204e 6f64 653a 2044 6566 696e fo, Node: Defin\n+000112c0: 6974 696f 6e73 2c20 204e 6578 743a 2047 itions, Next: G\n+000112d0: 6c6f 6261 6c20 656e 7669 726f 6e6d 656e lobal environmen\n+000112e0: 742c 2020 5072 6576 3a20 4472 6177 6162 t, Prev: Drawab\n+000112f0: 6c65 732c 2020 5570 3a20 496e 7075 7420 les, Up: Input \n+00011300: 6c61 6e67 7561 6765 0a0a 332e 3320 4465 language..3.3 De\n+00011310: 6669 6e69 7469 6f6e 730a 3d3d 3d3d 3d3d finitions.======\n+00011320: 3d3d 3d3d 3d3d 3d3d 3d0a 0a44 6566 696e =========..Defin\n+00011330: 6974 696f 6e73 2067 6976 6520 6e61 6d65 itions give name\n+00011340: 7320 746f 20e2 8098 736b 6574 6368 e280 s to ...sketch..\n+00011350: 9920 6f62 6a65 6374 732e 2020 4465 6669 . objects. Defi\n+00011360: 6e69 7469 6f6e 7320 616c 6f6e 6520 6172 nitions alone ar\n+00011370: 650a 6265 6e69 676e 2e20 2041 20e2 8098 e.benign. A ...\n+00011380: 736b 6574 6368 e280 9920 696e 7075 7420 sketch... input \n+00011390: 6669 6c65 2063 6f6e 7369 7374 696e 6720 file consisting \n+000113a0: 656e 7469 7265 6c79 206f 6620 6465 6669 entirely of defi\n+000113b0: 6e69 7469 6f6e 7320 7769 6c6c 0a67 656e nitions will.gen\n+000113c0: 6572 6174 6520 6e6f 2064 7261 7769 6e67 erate no drawing\n+000113d0: 2e20 204f 6e6c 7920 7768 656e 2064 6566 . Only when def\n+000113e0: 696e 6974 696f 6e73 2061 7265 20e2 809c initions are ...\n+000113f0: 7265 6665 7265 6e63 6564 e280 9d20 646f referenced... do\n+00011400: 2074 6865 790a 706f 7465 6e74 6961 6c6c they.potentiall\n+00011410: 7920 6c65 6164 2074 6f20 696e 6b20 6f6e y lead to ink on\n+00011420: 2074 6865 2064 7261 7769 6e67 2e0a 0a20 the drawing... \n+00011430: 2020 5468 6520 696e 7465 6e74 206f 6620 The intent of \n+00011440: 6465 6669 6e69 7469 6f6e 7320 6973 2074 definitions is t\n+00011450: 6f20 6d61 6b65 20e2 8098 736b 6574 6368 o make ...sketch\n+00011460: e280 9920 636f 6465 206d 6f72 6520 636f ... code more co\n+00011470: 6e63 6973 6520 616e 640a 7265 6164 6162 ncise and.readab\n+00011480: 6c65 2e20 2054 6865 7265 2069 7320 6e6f le. 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In\n+000115c0: 2061 6464 6974 696f 6e2c 20e2 809c 7461 addition, ...ta\n+000115d0: 6720 6465 6669 6e69 7469 6f6e 73e2 809d g definitions...\n+000115e0: 2c20 7768 6963 6820 6861 7665 206e 6f0a , which have no.\n+000115f0: 6173 736f 6369 6174 6564 206f 626a 6563 associated objec\n+00011600: 7420 6174 2061 6c6c 2c20 616c 6c6f 7720 t at all, allow \n+00011610: 7468 6520 6d65 616e 696e 6720 6f66 206f the meaning of o\n+00011620: 7468 6572 2064 6566 696e 6974 696f 6e73 ther definitions\n+00011630: 2074 6f20 6265 0a73 656c 6563 7465 6420 to be.selected \n+00011640: 6672 6f6d 2061 2073 6574 206f 6620 616c from a set of al\n+00011650: 7465 726e 6174 6976 6573 2e20 2053 696e ternatives. Sin\n+00011660: 6365 2074 6167 7320 6d61 7920 6265 2064 ce tags may be d\n+00011670: 6566 696e 6564 2028 616e 640a 756e 6465 efined (and.unde\n+00011680: 6669 6e65 6429 2069 6e20 7468 6520 636f fined) in the co\n+00011690: 6d6d 616e 6420 6c69 6e65 206f 6620 e280 mmand line of ..\n+000116a0: 9873 6b65 7463 68e2 8099 2c20 7468 6579 .sketch..., they\n+000116b0: 2063 616e 2062 6520 616e 2061 6964 2069 can be an aid i\n+000116c0: 6e20 7468 650a 7363 7269 7074 2d64 7269 n the.script-dri\n+000116d0: 7665 6e20 7072 6570 6172 6174 696f 6e20 ven preparation \n+000116e0: 6f66 2064 6f63 756d 656e 7473 2e0a 0a2a of documents...*\n+000116f0: 204d 656e 753a 0a0a 2a20 466f 726d 7320 Menu:..* Forms \n+00011700: 6f66 2064 6566 696e 6974 696f 6e73 3a3a of definitions::\n+00011710: 2020 2020 2020 2020 4469 6666 6572 656e Differen\n+00011720: 7420 6465 6673 2066 6f72 2064 6966 6665 t defs for diffe\n+00011730: 7265 6e74 2070 7572 706f 7365 732e 0a2a rent purposes..*\n+00011740: 2046 6f72 6d73 206f 6620 7265 6665 7265 Forms of refere\n+00011750: 6e63 6573 3a3a 2020 2020 2020 2020 2048 nces:: H\n+00011760: 6f77 2072 6566 6572 656e 6365 7320 6465 ow references de\n+00011770: 6e6f 7465 2074 7970 6573 2e0a 0a1f 0a46 note types.....F\n+00011780: 696c 653a 2073 6b65 7463 682e 696e 666f ile: sketch.info\n+00011790: 2c20 204e 6f64 653a 2046 6f72 6d73 206f , Node: Forms o\n+000117a0: 6620 6465 6669 6e69 7469 6f6e 732c 2020 f definitions, \n+000117b0: 4e65 7874 3a20 466f 726d 7320 6f66 2072 Next: Forms of r\n+000117c0: 6566 6572 656e 6365 732c 2020 5072 6576 eferences, Prev\n+000117d0: 3a20 4465 6669 6e69 7469 6f6e 732c 2020 : Definitions, \n+000117e0: 5570 3a20 4465 6669 6e69 7469 6f6e 730a Up: Definitions.\n+000117f0: 0a33 2e33 2e31 2046 6f72 6d73 206f 6620 .3.3.1 Forms of \n+00011800: 6465 6669 6e69 7469 6f6e 730a 2d2d 2d2d definitions.----\n 00011810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ----------------\n-00011820: 2d2d 0a0a 4465 6669 6e69 7469 6f6e 7320 --..Definitions \n-00011830: 6861 7665 2074 6872 6565 2070 6f73 7369 have three possi\n-00011840: 626c 6520 666f 726d 732c 20e2 809c 7369 ble forms, ...si\n-00011850: 6d70 6c65 e280 9d2c 20e2 809c 7769 7468 mple..., ...with\n-00011860: 2061 6c74 6572 6e61 7469 7665 73e2 809d alternatives...\n-00011870: 2c0a 616e 6420 e280 9c74 6167 e280 9d20 ,.and ...tag... \n-00011880: 6173 2073 686f 776e 2068 6572 6520 696e as shown here in\n-00011890: 206f 7264 6572 2e0a 0a2a 5379 6e74 6178 order...*Syntax\n-000118a0: 3a2a 0a20 2020 2020 6465 6620 4944 204f :*. def ID O\n-000118b0: 424a 4543 5420 2025 2073 696d 706c 6520 BJECT % simple \n-000118c0: 6465 660a 2020 2020 2064 6566 2049 4420 def. def ID \n-000118d0: 3c54 4147 5f31 3e20 4f42 4a45 4354 5f31 OBJECT_1\n-000118e0: 2020 2520 6465 6620 7769 7468 2061 6c74 % def with alt\n-000118f0: 6572 6e61 7469 7665 730a 2020 2020 2020 ernatives. \n-00011900: 2020 2020 2020 3c54 4147 5f32 3e20 4f42 OB\n-00011910: 4a45 4354 5f32 0a20 2020 2020 2020 2020 JECT_2. \n-00011920: 2020 202e 2e2e 0a20 2020 2020 2020 2020 .... \n-00011930: 2020 203c 3e20 4445 4641 554c 545f 4f42 <> DEFAULT_OB\n-00011940: 4a45 4354 0a20 2020 2020 6465 6620 4944 JECT. def ID\n-00011950: 203c 3e20 2025 2074 6167 2064 6566 0a54 <> % tag def.T\n-00011960: 6865 2073 696d 706c 6520 6465 6669 6e69 he simple defini\n-00011970: 7469 6f6e 206d 6572 656c 7920 6173 736f tion merely asso\n-00011980: 6369 6174 6573 204f 424a 4543 5420 7769 ciates OBJECT wi\n-00011990: 7468 2074 6865 2069 6465 6e74 6966 6965 th the identifie\n-000119a0: 7220 4944 2e0a 0a20 2020 5468 6520 6465 r ID... The de\n-000119b0: 6669 6e69 7469 6f6e 2077 6974 6820 616c finition with al\n-000119c0: 7465 726e 6174 6976 6573 2061 7373 6f63 ternatives assoc\n-000119d0: 6961 7465 7320 4f42 4a45 4354 5f49 2077 iates OBJECT_I w\n-000119e0: 6974 6820 4944 2c20 7768 6572 650a 5441 ith ID, where.TA\n-000119f0: 475f 4920 6973 2074 6865 2066 6972 7374 G_I is the first\n-00011a00: 2064 6566 696e 6564 2074 6167 2069 6e20 defined tag in \n-00011a10: 7468 6520 6c69 7374 206f 6620 616c 7465 the list of alte\n-00011a20: 726e 6174 6976 6520 7461 670a 7265 6665 rnative tag.refe\n-00011a30: 7265 6e63 6573 2e20 2049 6620 6e6f 2074 rences. If no t\n-00011a40: 6167 2069 6e20 7468 6520 6c69 7374 2069 ag in the list i\n-00011a50: 7320 6465 6669 6e65 642c 2074 6865 6e20 s defined, then \n-00011a60: 4445 4641 554c 545f 4f42 4a45 4354 2069 DEFAULT_OBJECT i\n-00011a70: 730a 6173 736f 6369 6174 6564 2077 6974 s.associated wit\n-00011a80: 6820 6964 656e 7469 6669 6572 2049 442e h identifier ID.\n-00011a90: 0a0a 2020 2054 6865 2066 696e 616c 2066 .. The final f\n-00011aa0: 6f72 6d20 6465 6669 6e65 7320 4944 2061 orm defines ID a\n-00011ab0: 7320 6120 7461 672e 2020 416e 6f74 6865 s a tag. Anothe\n-00011ac0: 7220 7761 7920 746f 2064 6566 696e 6520 r way to define \n-00011ad0: 6120 7461 6720 6973 0a77 6974 6820 7468 a tag is.with th\n-00011ae0: 6520 e280 982d 44e2 8099 2063 6f6d 6d61 e ...-D... comma\n-00011af0: 6e64 206c 696e 6520 6f70 7469 6f6e 2e20 nd line option. \n-00011b00: 202a 4e6f 7465 2043 6f6d 6d61 6e64 206c *Note Command l\n-00011b10: 696e 653a 3a2e 0a0a 1f0a 4669 6c65 3a20 ine::.....File: \n-00011b20: 736b 6574 6368 2e69 6e66 6f2c 2020 4e6f sketch.info, No\n-00011b30: 6465 3a20 466f 726d 7320 6f66 2072 6566 de: Forms of ref\n-00011b40: 6572 656e 6365 732c 2020 5072 6576 3a20 erences, Prev: \n-00011b50: 466f 726d 7320 6f66 2064 6566 696e 6974 Forms of definit\n-00011b60: 696f 6e73 2c20 2055 703a 2044 6566 696e ions, Up: Defin\n-00011b70: 6974 696f 6e73 0a0a 332e 332e 3220 466f itions..3.3.2 Fo\n-00011b80: 726d 7320 6f66 2072 6566 6572 656e 6365 rms of reference\n-00011b90: 730a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d s.--------------\n-00011ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a52 6566 -----------..Ref\n-00011bb0: 6572 656e 6365 7320 746f 2064 6566 696e erences to defin\n-00011bc0: 6564 206e 616d 6573 2061 7265 2065 6e63 ed names are enc\n-00011bd0: 6c6f 7365 6420 696e 2062 7261 636b 6574 losed in bracket\n-00011be0: 696e 6720 6465 6c69 6d69 7465 7273 2e20 ing delimiters. \n-00011bf0: 2054 6865 0a64 656c 696d 6974 6572 2063 The.delimiter c\n-00011c00: 6861 7261 6374 6572 7320 696d 706c 7920 haracters imply \n-00011c10: 7468 6520 7479 7065 206f 6620 7468 6520 the type of the \n-00011c20: 6173 736f 6369 6174 6564 2076 616c 7565 associated value\n-00011c30: 2061 7320 7368 6f77 6e20 696e 0a74 6865 as shown in.the\n-00011c40: 2074 6162 6c65 2062 656c 6f77 2e20 2041 table below. A\n-00011c50: 2074 7970 6520 6572 726f 7220 6973 2072 type error is r\n-00011c60: 6169 7365 6420 6966 2074 6865 2074 7970 aised if the typ\n-00011c70: 6520 6f66 2061 2072 6566 6572 656e 6365 e of a reference\n-00011c80: 2064 6f65 730a 6e6f 7420 6d61 7463 6820 does.not match \n-00011c90: 7468 6520 7479 7065 206f 6620 7468 6520 the type of the \n-00011ca0: 6465 6669 6e65 6420 7661 6c75 652e 2020 defined value. \n-00011cb0: 5468 6520 696e 7465 6e74 206f 6620 7468 The intent of th\n-00011cc0: 6973 206d 6563 6861 6e69 736d 0a69 732c is mechanism.is,\n-00011cd0: 2061 6761 696e 2c20 746f 206d 616b 6520 again, to make \n-00011ce0: e280 9873 6b65 7463 68e2 8099 2069 6e70 ...sketch... inp\n-00011cf0: 7574 2066 696c 6573 206d 6f72 6520 7265 ut files more re\n-00011d00: 6164 6162 6c65 2e0a 5479 7065 2020 2020 adable..Type \n-00011d10: 2020 2020 5265 6665 7265 6e63 650a 2d2d Reference.--\n-00011d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ----------------\n+00011820: 2d2d 2d2d 2d2d 0a0a 4465 6669 6e69 7469 ------..Definiti\n+00011830: 6f6e 7320 6861 7665 2074 6872 6565 2070 ons have three p\n+00011840: 6f73 7369 626c 6520 666f 726d 732c 20e2 ossible forms, .\n+00011850: 809c 7369 6d70 6c65 e280 9d2c 20e2 809c ..simple..., ...\n+00011860: 7769 7468 2061 6c74 6572 6e61 7469 7665 with alternative\n+00011870: 73e2 809d 2c0a 616e 6420 e280 9c74 6167 s...,.and ...tag\n+00011880: e280 9d20 6173 2073 686f 776e 2068 6572 ... as shown her\n+00011890: 6520 696e 206f 7264 6572 2e0a 0a2a 5379 e in order...*Sy\n+000118a0: 6e74 6178 3a2a 0a20 2020 2020 6465 6620 ntax:*. def \n+000118b0: 4944 204f 424a 4543 5420 2025 2073 696d ID OBJECT % sim\n+000118c0: 706c 6520 6465 660a 2020 2020 2064 6566 ple def. def\n+000118d0: 2049 4420 3c54 4147 5f31 3e20 4f42 4a45 ID OBJE\n+000118e0: 4354 5f31 2020 2520 6465 6620 7769 7468 CT_1 % def with\n+000118f0: 2061 6c74 6572 6e61 7469 7665 730a 2020 alternatives. \n+00011900: 2020 2020 2020 2020 2020 3c54 4147 5f32 OBJECT_2. \n+00011920: 2020 2020 2020 202e 2e2e 0a20 2020 2020 .... \n+00011930: 2020 2020 2020 203c 3e20 4445 4641 554c <> DEFAUL\n+00011940: 545f 4f42 4a45 4354 0a20 2020 2020 6465 T_OBJECT. de\n+00011950: 6620 4944 203c 3e20 2025 2074 6167 2064 f ID <> % tag d\n+00011960: 6566 0a54 6865 2073 696d 706c 6520 6465 ef.The simple de\n+00011970: 6669 6e69 7469 6f6e 206d 6572 656c 7920 finition merely \n+00011980: 6173 736f 6369 6174 6573 204f 424a 4543 associates OBJEC\n+00011990: 5420 7769 7468 2074 6865 2069 6465 6e74 T with the ident\n+000119a0: 6966 6965 7220 4944 2e0a 0a20 2020 5468 ifier ID... Th\n+000119b0: 6520 6465 6669 6e69 7469 6f6e 2077 6974 e definition wit\n+000119c0: 6820 616c 7465 726e 6174 6976 6573 2061 h alternatives a\n+000119d0: 7373 6f63 6961 7465 7320 4f42 4a45 4354 ssociates OBJECT\n+000119e0: 5f49 2077 6974 6820 4944 2c20 7768 6572 _I with ID, wher\n+000119f0: 650a 5441 475f 4920 6973 2074 6865 2066 e.TAG_I is the f\n+00011a00: 6972 7374 2064 6566 696e 6564 2074 6167 irst defined tag\n+00011a10: 2069 6e20 7468 6520 6c69 7374 206f 6620 in the list of \n+00011a20: 616c 7465 726e 6174 6976 6520 7461 670a alternative tag.\n+00011a30: 7265 6665 7265 6e63 6573 2e20 2049 6620 references. If \n+00011a40: 6e6f 2074 6167 2069 6e20 7468 6520 6c69 no tag in the li\n+00011a50: 7374 2069 7320 6465 6669 6e65 642c 2074 st is defined, t\n+00011a60: 6865 6e20 4445 4641 554c 545f 4f42 4a45 hen DEFAULT_OBJE\n+00011a70: 4354 2069 730a 6173 736f 6369 6174 6564 CT is.associated\n+00011a80: 2077 6974 6820 6964 656e 7469 6669 6572 with identifier\n+00011a90: 2049 442e 0a0a 2020 2054 6865 2066 696e ID... The fin\n+00011aa0: 616c 2066 6f72 6d20 6465 6669 6e65 7320 al form defines \n+00011ab0: 4944 2061 7320 6120 7461 672e 2020 416e ID as a tag. An\n+00011ac0: 6f74 6865 7220 7761 7920 746f 2064 6566 other way to def\n+00011ad0: 696e 6520 6120 7461 6720 6973 0a77 6974 ine a tag is.wit\n+00011ae0: 6820 7468 6520 e280 982d 44e2 8099 2063 h the ...-D... c\n+00011af0: 6f6d 6d61 6e64 206c 696e 6520 6f70 7469 ommand line opti\n+00011b00: 6f6e 2e20 202a 4e6f 7465 2043 6f6d 6d61 on. *Note Comma\n+00011b10: 6e64 206c 696e 653a 3a2e 0a0a 1f0a 4669 nd line::.....Fi\n+00011b20: 6c65 3a20 736b 6574 6368 2e69 6e66 6f2c le: sketch.info,\n+00011b30: 2020 4e6f 6465 3a20 466f 726d 7320 6f66 Node: Forms of\n+00011b40: 2072 6566 6572 656e 6365 732c 2020 5072 references, Pr\n+00011b50: 6576 3a20 466f 726d 7320 6f66 2064 6566 ev: Forms of def\n+00011b60: 696e 6974 696f 6e73 2c20 2055 703a 2044 initions, Up: D\n+00011b70: 6566 696e 6974 696f 6e73 0a0a 332e 332e efinitions..3.3.\n+00011b80: 3220 466f 726d 7320 6f66 2072 6566 6572 2 Forms of refer\n+00011b90: 656e 6365 730a 2d2d 2d2d 2d2d 2d2d 2d2d ences.----------\n+00011ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------.\n+00011bb0: 0a52 6566 6572 656e 6365 7320 746f 2064 .References to d\n+00011bc0: 6566 696e 6564 206e 616d 6573 2061 7265 efined names are\n+00011bd0: 2065 6e63 6c6f 7365 6420 696e 2062 7261 enclosed in bra\n+00011be0: 636b 6574 696e 6720 6465 6c69 6d69 7465 cketing delimite\n+00011bf0: 7273 2e20 2054 6865 0a64 656c 696d 6974 rs. The.delimit\n+00011c00: 6572 2063 6861 7261 6374 6572 7320 696d er characters im\n+00011c10: 706c 7920 7468 6520 7479 7065 206f 6620 ply the type of \n+00011c20: 7468 6520 6173 736f 6369 6174 6564 2076 the associated v\n+00011c30: 616c 7565 2061 7320 7368 6f77 6e20 696e alue as shown in\n+00011c40: 0a74 6865 2074 6162 6c65 2062 656c 6f77 .the table below\n+00011c50: 2e20 2041 2074 7970 6520 6572 726f 7220 . A type error \n+00011c60: 6973 2072 6169 7365 6420 6966 2074 6865 is raised if the\n+00011c70: 2074 7970 6520 6f66 2061 2072 6566 6572 type of a refer\n+00011c80: 656e 6365 2064 6f65 730a 6e6f 7420 6d61 ence does.not ma\n+00011c90: 7463 6820 7468 6520 7479 7065 206f 6620 tch the type of \n+00011ca0: 7468 6520 6465 6669 6e65 6420 7661 6c75 the defined valu\n+00011cb0: 652e 2020 5468 6520 696e 7465 6e74 206f e. The intent o\n+00011cc0: 6620 7468 6973 206d 6563 6861 6e69 736d f this mechanism\n+00011cd0: 0a69 732c 2061 6761 696e 2c20 746f 206d .is, again, to m\n+00011ce0: 616b 6520 e280 9873 6b65 7463 68e2 8099 ake ...sketch...\n+00011cf0: 2069 6e70 7574 2066 696c 6573 206d 6f72 input files mor\n+00011d00: 6520 7265 6164 6162 6c65 2e0a 5479 7065 e readable..Type\n+00011d10: 2020 2020 2020 2020 5265 6665 7265 6e63 Referenc\n+00011d20: 650a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d e.--------------\n 00011d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ----------------\n-00011d40: 2d2d 2d2d 2d2d 2d2d 0a73 6361 6c61 7220 --------.scalar \n-00011d50: 2020 2020 20e2 8098 4944 e280 990a 706f ...ID....po\n-00011d60: 696e 7420 2020 2020 2020 e280 9828 4944 int ...(ID\n-00011d70: 29e2 8099 0a76 6563 746f 7220 2020 2020 )....vector \n-00011d80: 20e2 8098 5b49 445d e280 990a 7472 616e ...[ID]....tran\n-00011d90: 7366 6f72 6d20 2020 e280 985b 5b49 445d sform ...[[ID]\n-00011da0: 5de2 8099 0a64 7261 7761 626c 6520 2020 ]....drawable \n-00011db0: 20e2 8098 7b49 447d e280 990a 6f70 7469 ...{ID}....opti\n-00011dc0: 6f6e 7320 2020 2020 e280 985b 4944 5de2 ons ...[ID].\n-00011dd0: 8099 206f 7220 e280 985b 4944 312c 2e2e .. or ...[ID1,..\n-00011de0: 2e2c 4944 4e5d e280 990a 7461 6720 2020 .,IDN]....tag \n-00011df0: 2020 2020 2020 e280 983c 4944 3ee2 8099 ......\n-00011e00: 0a0a 4e6f 7465 2074 6861 7420 7371 7561 ..Note that squa\n-00011e10: 7265 2062 7261 636b 6574 7320 e280 985b re brackets ...[\n-00011e20: 205d e280 9920 6172 6520 7573 6564 2062 ]... are used b\n-00011e30: 6f74 6820 666f 7220 7665 6374 6f72 2061 oth for vector a\n-00011e40: 6e64 2066 6f72 206f 7074 696f 6e73 0a72 nd for options.r\n-00011e50: 6566 6572 656e 6365 732e 2020 4465 7461 eferences. Deta\n-00011e60: 696c 7320 6f66 20e2 8098 736b 6574 6368 ils of ...sketch\n-00011e70: e280 9920 7379 6e74 6178 206d 616b 6520 ... syntax make \n-00011e80: 6974 2069 6d70 6f73 7369 626c 6520 666f it impossible fo\n-00011e90: 7220 7468 6573 6520 7477 6f0a 7265 6665 r these two.refe\n-00011ea0: 7265 6e63 6520 7479 7065 7320 746f 2062 rence types to b\n-00011eb0: 6520 636f 6e66 7573 6564 2e20 2054 6865 e confused. The\n-00011ec0: 2073 7065 6369 616c 206d 756c 7469 706c special multipl\n-00011ed0: 6520 7265 6665 7265 6e63 650a e280 985b e reference....[\n-00011ee0: 4944 312c 4944 322c 2e2e 2e2c 4944 4e5d ID1,ID2,...,IDN]\n-00011ef0: e280 9920 6163 7473 2061 7320 6966 2074 ... acts as if t\n-00011f00: 6865 2072 6573 7065 6374 6976 6520 6c69 he respective li\n-00011f10: 7374 7320 6f66 206f 7074 696f 6e73 2077 sts of options w\n-00011f20: 6572 650a 636f 6e63 6174 656e 6174 6564 ere.concatenated\n-00011f30: 2e0a 0a1f 0a46 696c 653a 2073 6b65 7463 .....File: sketc\n-00011f40: 682e 696e 666f 2c20 204e 6f64 653a 2047 h.info, Node: G\n-00011f50: 6c6f 6261 6c20 656e 7669 726f 6e6d 656e lobal environmen\n-00011f60: 742c 2020 5072 6576 3a20 4465 6669 6e69 t, Prev: Defini\n-00011f70: 7469 6f6e 732c 2020 5570 3a20 496e 7075 tions, Up: Inpu\n-00011f80: 7420 6c61 6e67 7561 6765 0a0a 332e 3420 t language..3.4 \n-00011f90: 476c 6f62 616c 2065 6e76 6972 6f6e 6d65 Global environme\n-00011fa0: 6e74 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d nt.=============\n-00011fb0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a41 6e20 6f70 =========..An op\n-00011fc0: 7469 6f6e 616c 2067 6c6f 6261 6c20 656e tional global en\n-00011fd0: 7669 726f 6e6d 656e 7420 626c 6f63 6b20 vironment block \n-00011fe0: 7072 6f76 6964 6573 2061 2066 6577 2077 provides a few w\n-00011ff0: 6179 7320 746f 2061 6666 6563 7420 7468 ays to affect th\n-00012000: 650a 656e 7469 7265 2073 6365 6e65 2e20 e.entire scene. \n-00012010: 2054 6865 2062 6c6f 636b 206d 7573 7420 The block must \n-00012020: 6170 7065 6172 2061 7320 7468 6520 6c61 appear as the la\n-00012030: 7374 2074 6578 7420 696e 2074 6865 20e2 st text in the .\n-00012040: 8098 736b 6574 6368 e280 990a 696e 7075 ..sketch....inpu\n-00012050: 7420 6669 6c65 2e20 2049 7420 6d61 7920 t file. It may \n-00012060: 696e 636c 7564 6520 6465 6669 6e69 7469 include definiti\n-00012070: 6f6e 732c 2062 7574 206e 6f74 6520 7468 ons, but note th\n-00012080: 6174 2070 7265 7669 6f75 730a 6465 6669 at previous.defi\n-00012090: 6e69 7469 6f6e 7320 6174 2074 6865 2074 nitions at the t\n-000120a0: 6f70 206c 6576 656c 2028 6e6f 7420 6e65 op level (not ne\n-000120b0: 7374 6564 2069 6e73 6964 6520 626c 6f63 sted inside bloc\n-000120c0: 6b73 2920 6172 6520 616c 736f 0a61 7661 ks) are also.ava\n-000120d0: 696c 6162 6c65 2e0a 0a2a 5379 6e74 6178 ilable...*Syntax\n-000120e0: 3a2a 0a20 2020 2020 676c 6f62 616c 207b :*. global {\n-000120f0: 2045 4e56 4952 4f4e 4d45 4e54 5f53 4554 ENVIRONMENT_SET\n-00012100: 5449 4e47 5320 7d0a 5468 6520 636f 6e74 TINGS }.The cont\n-00012110: 656e 7473 206f 6620 454e 5649 524f 4e4d ents of ENVIRONM\n-00012120: 454e 545f 5345 5454 494e 4753 2061 7265 ENT_SETTINGS are\n-00012130: 2064 6973 6375 7373 6564 2069 6e20 7468 discussed in th\n-00012140: 6520 7365 6374 696f 6e73 2074 6861 740a e sections that.\n-00012150: 666f 6c6c 6f77 2e0a 0a2a 204d 656e 753a follow...* Menu:\n-00012160: 0a0a 2a20 476c 6f62 616c 206f 7074 696f ..* Global optio\n-00012170: 6e73 3a3a 2020 2020 2020 2020 2020 2020 ns:: \n-00012180: 2020 4174 7472 6962 7574 6573 206f 6620 Attributes of \n-00012190: 7468 6520 656e 7469 7265 2064 7261 7769 the entire drawi\n-000121a0: 6e67 2e0a 2a20 4361 6d65 7261 3a3a 2020 ng..* Camera:: \n-000121b0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000121c0: 2020 2020 4120 6669 6e61 6c20 6361 6d65 A final came\n-000121d0: 7261 2074 7261 6e73 666f 726d 6174 696f ra transformatio\n-000121e0: 6e20 6f66 2074 6865 2073 6365 6e65 2e0a n of the scene..\n-000121f0: 2a20 5069 6374 7572 6520 626f 783a 3a20 * Picture box:: \n-00012200: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00012210: 5365 7474 696e 6720 7468 6520 626f 756e Setting the boun\n-00012220: 6469 6e67 2062 6f78 2061 6e64 2032 6420 ding box and 2d \n-00012230: 636c 6970 7069 6e67 2e0a 2a20 4672 616d clipping..* Fram\n-00012240: 653a 3a20 2020 2020 2020 2020 2020 2020 e:: \n-00012250: 2020 2020 2020 2020 2020 4164 6469 6e67 Adding\n-00012260: 2061 2062 6f78 2061 726f 756e 6420 7468 a box around th\n-00012270: 6520 6472 6177 696e 672e 0a2a 204c 616e e drawing..* Lan\n-00012280: 6775 6167 653a 3a20 2020 2020 2020 2020 guage:: \n-00012290: 2020 2020 2020 2020 2020 2053 6574 7469 Setti\n-000122a0: 6e67 2074 6865 206f 7574 7075 7420 6c61 ng the output la\n-000122b0: 6e67 7561 6765 2e0a 0a1f 0a46 696c 653a nguage.....File:\n-000122c0: 2073 6b65 7463 682e 696e 666f 2c20 204e sketch.info, N\n-000122d0: 6f64 653a 2047 6c6f 6261 6c20 6f70 7469 ode: Global opti\n-000122e0: 6f6e 732c 2020 4e65 7874 3a20 4361 6d65 ons, Next: Came\n-000122f0: 7261 2c20 2050 7265 763a 2047 6c6f 6261 ra, Prev: Globa\n-00012300: 6c20 656e 7669 726f 6e6d 656e 742c 2020 l environment, \n-00012310: 5570 3a20 476c 6f62 616c 2065 6e76 6972 Up: Global envir\n-00012320: 6f6e 6d65 6e74 0a0a 332e 342e 3120 476c onment..3.4.1 Gl\n-00012330: 6f62 616c 206f 7074 696f 6e73 0a2d 2d2d obal options.---\n-00012340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ----------------\n-00012350: 2d0a 0a2a 5379 6e74 6178 3a2a 0a20 2020 -..*Syntax:*. \n-00012360: 2020 7365 7420 5b20 4f50 5449 4f4e 5320 set [ OPTIONS \n-00012370: 5d0a 5468 6520 636f 6e74 656e 7473 206f ].The contents o\n-00012380: 6620 4f50 5449 4f4e 532c 2065 7863 6570 f OPTIONS, excep\n-00012390: 7420 666f 7220 e280 9873 6b65 7463 68e2 t for ...sketch.\n-000123a0: 8099 2069 6e74 6572 6e61 6c20 6f70 7469 .. internal opti\n-000123b0: 6f6e 732c 2061 7265 0a63 6f70 6965 6420 ons, are.copied \n-000123c0: 6173 2d69 7320 746f 2061 205c 7073 7365 as-is to a \\psse\n-000123d0: 7420 7468 6174 2061 7070 6561 7273 2062 t that appears b\n-000123e0: 6566 6f72 6520 616e 7974 6869 6e67 2065 efore anything e\n-000123f0: 6c73 6520 696e 2074 6865 206f 7574 7075 lse in the outpu\n-00012400: 740a 6669 6c65 2e20 2054 6869 7320 6973 t.file. This is\n-00012410: 2061 2067 6f6f 6420 706c 6163 6520 746f a good place to\n-00012420: 2073 6574 20e2 8098 756e 6974 e280 992c set ...unit...,\n-00012430: 2061 2064 6566 6175 6c74 20e2 8098 6c69 a default ...li\n-00012440: 6e65 7769 6474 68e2 8099 2c20 6574 632e newidth..., etc.\n-00012450: 0a0a 2020 2049 6e74 6572 6e61 6c20 6f70 .. Internal op\n-00012460: 7469 6f6e 7320 776f 726b 206f 6e20 616c tions work on al\n-00012470: 6c20 6f62 6a65 6374 7320 7768 6572 6520 l objects where \n-00012480: 7468 6579 206d 616b 6520 7365 6e73 652e they make sense.\n-00012490: 2020 5468 6973 0a69 6e63 6c75 6465 7320 This.includes \n-000124a0: e280 9863 756c 6ce2 8099 2061 6e64 20e2 ...cull... and .\n-000124b0: 8098 7370 6c69 74e2 8099 2028 6275 7420 ..split... (but \n-000124c0: 6e6f 7420 e280 986c 6179 e280 9929 2e20 not ...lay...). \n-000124d0: 202a 4e6f 7465 2049 6e74 6572 6e61 6c20 *Note Internal \n-000124e0: 6f70 7469 6f6e 733a 3a2e 0a0a 1f0a 4669 options::.....Fi\n-000124f0: 6c65 3a20 736b 6574 6368 2e69 6e66 6f2c le: sketch.info,\n-00012500: 2020 4e6f 6465 3a20 4361 6d65 7261 2c20 Node: Camera, \n-00012510: 204e 6578 743a 2050 6963 7475 7265 2062 Next: Picture b\n-00012520: 6f78 2c20 2050 7265 763a 2047 6c6f 6261 ox, Prev: Globa\n-00012530: 6c20 6f70 7469 6f6e 732c 2020 5570 3a20 l options, Up: \n-00012540: 476c 6f62 616c 2065 6e76 6972 6f6e 6d65 Global environme\n-00012550: 6e74 0a0a 332e 342e 3220 4361 6d65 7261 nt..3.4.2 Camera\n-00012560: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a2a .------------..*\n-00012570: 5379 6e74 6178 3a2a 0a20 2020 2020 6361 Syntax:*. ca\n-00012580: 6d65 7261 2054 5241 4e53 464f 524d 5f45 mera TRANSFORM_E\n-00012590: 5850 5245 5353 494f 4e0a 2020 2054 6865 XPRESSION. The\n-000125a0: 2054 5241 4e53 464f 524d 5f45 5850 5245 TRANSFORM_EXPRE\n-000125b0: 5353 494f 4e20 6973 2061 7070 6c69 6564 SSION is applied\n-000125c0: 2061 6674 6572 2061 6c6c 206f 7468 6572 after all other\n-000125d0: 2074 7261 6e73 666f 726d 6174 696f 6e73 transformations\n-000125e0: 0a6f 6620 7468 6520 7363 656e 652e 2020 .of the scene. \n-000125f0: 5468 6973 2069 7320 6375 7272 656e 746c This is currentl\n-00012600: 7920 6f6e 6c79 2075 7365 6675 6c20 666f y only useful fo\n-00012610: 7220 7472 616e 7366 6f72 6d69 6e67 2074 r transforming t\n-00012620: 6865 0a62 6f75 6e64 696e 6720 626f 782e he.bounding box.\n-00012630: 2020 2a4e 6f74 6520 5069 6374 7572 6520 *Note Picture \n-00012640: 626f 783a 3a2e 2020 4974 2077 696c 6c20 box::. It will \n-00012650: 706c 6179 2061 2072 6f6c 6520 696e 2061 play a role in a\n-00012660: 6e79 2066 7574 7572 650a 696d 706c 656d ny future.implem\n-00012670: 656e 7461 7469 6f6e 206f 6620 636c 6970 entation of clip\n-00012680: 7069 6e67 2e0a 0a1f 0a46 696c 653a 2073 ping.....File: s\n-00012690: 6b65 7463 682e 696e 666f 2c20 204e 6f64 ketch.info, Nod\n-000126a0: 653a 2050 6963 7475 7265 2062 6f78 2c20 e: Picture box, \n-000126b0: 204e 6578 743a 2046 7261 6d65 2c20 2050 Next: Frame, P\n-000126c0: 7265 763a 2043 616d 6572 612c 2020 5570 rev: Camera, Up\n-000126d0: 3a20 476c 6f62 616c 2065 6e76 6972 6f6e : Global environ\n-000126e0: 6d65 6e74 0a0a 332e 342e 3320 5069 6374 ment..3.4.3 Pict\n-000126f0: 7572 6520 626f 780a 2d2d 2d2d 2d2d 2d2d ure box.--------\n-00012700: 2d2d 2d2d 2d2d 2d2d 2d0a 0a2a 5379 6e74 ---------..*Synt\n-00012710: 6178 3a2a 0a20 2020 2020 7069 6374 7572 ax:*. pictur\n-00012720: 6562 6f78 5b42 4153 454c 494e 455d 0a20 ebox[BASELINE]. \n-00012730: 2020 2020 7069 6374 7572 6562 6f78 5b42 picturebox[B\n-00012740: 4153 454c 494e 455d 2028 5031 2928 5032 ASELINE] (P1)(P2\n-00012750: 290a 5468 6520 6669 7273 7420 666f 726d ).The first form\n-00012760: 206f 6620 e280 9870 6963 7475 7265 626f of ...picturebo\n-00012770: 78e2 8099 2063 6175 7365 7320 6120 7363 x... causes a sc\n-00012780: 616c 6172 2042 4153 454c 494e 4520 6672 alar BASELINE fr\n-00012790: 6163 7469 6f6e 2074 6f20 6265 0a65 6d69 action to be.emi\n-000127a0: 7474 6564 2069 6e20 7468 6520 e280 9870 tted in the ...p\n-000127b0: 7370 6963 7475 7265 e280 9920 656e 7669 spicture... envi\n-000127c0: 726f 6e6d 656e 7420 6f66 2074 6865 206f ronment of the o\n-000127d0: 7574 7075 742e 2020 5365 6520 e280 9850 utput. See ...P\n-000127e0: 5354 7269 636b 73e2 8099 0a64 6f63 756d STricks....docum\n-000127f0: 656e 7461 7469 6f6e 2066 6f72 20e2 8098 entation for ...\n-00012800: 7073 7069 6374 7572 65e2 8099 2e0a 0a20 pspicture...... \n-00012810: 2020 496e 2074 6865 2073 6563 6f6e 6420 In the second \n-00012820: 666f 726d 2c20 7468 6520 4241 5345 4c49 form, the BASELI\n-00012830: 4e45 2066 7261 6374 696f 6e20 6973 206f NE fraction is o\n-00012840: 7074 696f 6e61 6c2c 2061 6e64 2074 6865 ptional, and the\n-00012850: 2074 776f 0a70 6f69 6e74 7320 7468 6174 two.points that\n-00012860: 2066 6f6c 6c6f 7720 6465 6669 6e65 2074 follow define t\n-00012870: 6865 2064 6961 676f 6e61 6c20 6f66 2061 he diagonal of a\n-00012880: 2074 6872 6565 2d64 696d 656e 7369 6f6e three-dimension\n-00012890: 616c 2062 6f75 6e64 696e 670a 626f 7820 al bounding.box \n-000128a0: 666f 7220 7468 6520 636f 6d70 6c65 7465 for the complete\n-000128b0: 6420 7363 656e 652e 2020 5468 6520 7061 d scene. The pa\n-000128c0: 7261 6c6c 656c 2070 726f 6a65 6374 696f rallel projectio\n-000128d0: 6e20 6f66 2074 6865 2062 6f75 6e64 696e n of the boundin\n-000128e0: 670a 626f 7820 6465 7465 726d 696e 6573 g.box determines\n-000128f0: 2074 6865 2063 6f72 6e65 7273 206f 6620 the corners of \n-00012900: 7468 6520 6472 6177 696e 6727 7320 e280 the drawing's ..\n-00012910: 9870 7370 6963 7475 7265 2ae2 8099 2065 .pspicture*... e\n-00012920: 6e76 6972 6f6e 6d65 6e74 2c0a 7768 6963 nvironment,.whic\n-00012930: 6820 6973 2075 7365 6420 696e 2070 6c61 h is used in pla\n-00012940: 6365 206f 6620 e280 9870 7370 6963 7475 ce of ...pspictu\n-00012950: 7265 e280 992e 2020 5468 6973 2063 6175 re.... This cau\n-00012960: 7365 7320 506f 7374 5363 7269 7074 2074 ses PostScript t\n-00012970: 6f20 636c 6970 0a74 6865 2066 696e 616c o clip.the final\n-00012980: 2064 7261 7769 6e67 2074 6f20 7468 6520 drawing to the \n-00012990: 626f 756e 6469 6e67 2062 6f78 2069 6e20 bounding box in \n-000129a0: 3264 2e20 2049 6620 7468 6572 6520 6973 2d. If there is\n-000129b0: 2061 20e2 8098 6361 6d65 7261 e280 990a a ...camera....\n-000129c0: 7370 6563 6966 6965 642c 2074 6865 2063 specified, the c\n-000129d0: 616d 6572 6120 7472 616e 666f 726d 6174 amera tranformat\n-000129e0: 696f 6e20 6973 2061 7070 6c69 6564 2074 ion is applied t\n-000129f0: 6f20 7468 6520 626f 756e 6469 6e67 2062 o the bounding b\n-00012a00: 6f78 2c20 616e 640a 7468 6520 e280 9870 ox, and.the ...p\n-00012a10: 7370 6963 7475 7265 e280 9920 6973 2073 spicture... is s\n-00012a20: 6574 206a 7573 7420 6c61 7267 6520 656e et just large en\n-00012a30: 6f75 6768 2074 6f20 696e 636c 7564 6520 ough to include \n-00012a40: 7468 6520 7472 616e 7366 6f72 6d65 6420 the transformed \n-00012a50: 626f 782e 0a0a 2020 2057 6865 6e20 6e6f box... When no\n-00012a60: 2062 6f75 6e64 696e 6720 626f 7820 6973 bounding box is\n-00012a70: 2067 6976 656e 2c20 e280 9873 6b65 7463 given, ...sketc\n-00012a80: 68e2 8099 2063 6f6d 7075 7465 7320 6f6e h... computes on\n-00012a90: 6520 6175 746f 6d61 7469 6361 6c6c 792e e automatically.\n-00012aa0: 0a0a 1f0a 4669 6c65 3a20 736b 6574 6368 ....File: sketch\n-00012ab0: 2e69 6e66 6f2c 2020 4e6f 6465 3a20 4672 .info, Node: Fr\n-00012ac0: 616d 652c 2020 4e65 7874 3a20 4c61 6e67 ame, Next: Lang\n-00012ad0: 7561 6765 2c20 2050 7265 763a 2050 6963 uage, Prev: Pic\n-00012ae0: 7475 7265 2062 6f78 2c20 2055 703a 2047 ture box, Up: G\n-00012af0: 6c6f 6261 6c20 656e 7669 726f 6e6d 656e lobal environmen\n-00012b00: 740a 0a33 2e34 2e34 2046 7261 6d65 0a2d t..3.4.4 Frame.-\n-00012b10: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 2a53 796e ----------..*Syn\n-00012b20: 7461 783a 2a0a 2020 2020 2066 7261 6d65 tax:*. frame\n-00012b30: 205b 4f50 5449 4f4e 535d 0a43 6175 7365 [OPTIONS].Cause\n-00012b40: 7320 6120 5c70 7366 7261 6d65 626f 7820 s a \\psframebox \n-00012b50: 746f 2073 7572 726f 756e 6420 7468 6520 to surround the \n-00012b60: e280 9870 7370 6963 7475 7265 e280 9920 ...pspicture... \n-00012b70: 656e 7669 726f 6e6d 656e 7420 696e 2074 environment in t\n-00012b80: 6865 0a6f 7574 7075 742e 2020 4966 204f he.output. If O\n-00012b90: 5054 494f 4e53 2061 7265 2070 7265 7365 PTIONS are prese\n-00012ba0: 6e74 2c20 7468 6579 2061 7265 2063 6f70 nt, they are cop\n-00012bb0: 6965 6420 6173 2d69 732e 2020 4e6f 726d ied as-is. Norm\n-00012bc0: 616c 6c79 206f 6e65 0a77 6f75 6c64 2077 ally one.would w\n-00012bd0: 616e 7420 746f 2073 6574 20e2 8098 6c69 ant to set ...li\n-00012be0: 6e65 7769 6474 68e2 8099 2c20 e280 986c newidth..., ...l\n-00012bf0: 696e 6573 7479 6c65 e280 992c 20e2 8098 inestyle..., ...\n-00012c00: 6c69 6e65 636f 6c6f 72e2 8099 2c20 6574 linecolor..., et\n-00012c10: 632e 2020 4966 0a6f 6d69 7474 6564 2c20 c. If.omitted, \n-00012c20: 7468 656e 20e2 8098 6672 616d 6573 6570 then ...framesep\n-00012c30: 3d30 7074 e280 9920 6973 2061 6464 6564 =0pt... is added\n-00012c40: 2073 6f20 7468 6174 2074 6865 2066 7261 so that the fra\n-00012c50: 6d65 2074 6967 6874 6c79 2068 7567 7320 me tightly hugs \n-00012c60: 7468 650a e280 9870 7370 6963 7475 7265 the....pspicture\n-00012c70: e280 992e 0a0a 1f0a 4669 6c65 3a20 736b ........File: sk\n-00012c80: 6574 6368 2e69 6e66 6f2c 2020 4e6f 6465 etch.info, Node\n-00012c90: 3a20 4c61 6e67 7561 6765 2c20 2050 7265 : Language, Pre\n-00012ca0: 763a 2046 7261 6d65 2c20 2055 703a 2047 v: Frame, Up: G\n-00012cb0: 6c6f 6261 6c20 656e 7669 726f 6e6d 656e lobal environmen\n-00012cc0: 740a 0a33 2e34 2e35 204c 616e 6775 6167 t..3.4.5 Languag\n-00012cd0: 650a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d e.--------------\n-00012ce0: 0a0a 2020 2020 206c 616e 6775 6167 6520 .. language \n-00012cf0: 7469 6b7a 0a20 2020 2020 6c61 6e67 7561 tikz. langua\n-00012d00: 6765 2074 696b 7a2c 2063 6f6e 7465 7874 ge tikz, context\n-00012d10: 0a20 2020 2020 6c61 6e67 7561 6765 2070 . language p\n-00012d20: 7374 7269 636b 730a 2020 2020 206c 616e stricks. lan\n-00012d30: 6775 6167 6520 7073 7472 6963 6b73 2c20 guage pstricks, \n-00012d40: 6c61 7465 780a 5365 7473 2074 6865 206f latex.Sets the o\n-00012d50: 7574 7075 7420 6c61 6e67 7561 6765 2067 utput language g\n-00012d60: 656e 6572 6174 6564 2062 7920 e280 9873 enerated by ...s\n-00012d70: 6b65 7463 68e2 8099 2e20 2054 6865 2073 ketch.... 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If no\n-00012ec0: 206c 616e 6775 6167 6520 6973 0a73 7065 language is.spe\n-00012ed0: 6369 6669 6564 2c20 7468 6520 6465 6661 cified, the defa\n-00012ee0: 756c 7420 e280 9870 7374 7269 636b 73e2 ult ...pstricks.\n-00012ef0: 8099 2069 7320 7573 6564 2e0a 0a20 2020 .. is used... \n-00012f00: 416e 206f 7074 696f 6e61 6c20 636f 6d6d An optional comm\n-00012f10: 6120 666f 6c6c 6f77 6564 2062 7920 e280 a followed by ..\n-00012f20: 986c 6174 6578 e280 9920 6f72 20e2 8098 .latex... or ...\n-00012f30: 636f 6e74 6578 74e2 8099 2073 7065 6369 context... speci\n-00012f40: 6669 6573 2074 6865 0a6d 6163 726f 2070 fies the.macro p\n-00012f50: 6163 6b61 6765 2074 6861 7420 7468 6520 ackage that the \n-00012f60: 6f75 7470 7574 2073 686f 756c 6420 6173 output should as\n-00012f70: 7375 6d65 2e20 2054 6869 7320 6166 6665 sume. This affe\n-00012f80: 6374 7320 7468 6520 e280 9870 6963 7475 cts the ...pictu\n-00012f90: 7265 e280 990a 656e 7669 726f 6e6d 656e re....environmen\n-00012fa0: 7420 636f 6d6d 616e 6473 2065 6d69 7474 t commands emitt\n-00012fb0: 6564 2061 6e64 2074 6865 2064 6f63 756d ed and the docum\n-00012fc0: 656e 7420 7465 6d70 6c61 7465 2075 7365 ent template use\n-00012fd0: 6420 7769 7468 2074 6865 0ae2 8098 2d54 d with the....-T\n-00012fe0: e280 9920 6f70 7469 6f6e 2e20 202a 4e6f ... option. *No\n-00012ff0: 7465 2043 6f6d 6d61 6e64 206c 696e 653a te Command line:\n-00013000: 3a2e 2020 4e6f 7465 2074 6861 7420 6174 :. Note that at\n-00013010: 2074 6865 2074 696d 6520 7468 6973 206d the time this m\n-00013020: 616e 7561 6c0a 7761 7320 6765 6e65 7261 anual.was genera\n-00013030: 7465 642c 20e2 8098 5053 5472 6963 6b73 ted, ...PSTricks\n-00013040: e280 9920 7761 7320 6e6f 7420 7375 7070 ... was not supp\n-00013050: 6f72 7465 6420 6279 204c 6154 6558 206f orted by LaTeX o\n-00013060: 7220 6279 20e2 8098 436f 6e54 6558 74e2 r by ...ConTeXt.\n-00013070: 8099 2e0a 0a1f 0a46 696c 653a 2073 6b65 .......File: ske\n-00013080: 7463 682e 696e 666f 2c20 204e 6f64 653a tch.info, Node:\n-00013090: 2042 7569 6c64 696e 6720 6120 6472 6177 Building a draw\n-000130a0: 696e 672c 2020 4e65 7874 3a20 436f 6d6d ing, Next: Comm\n-000130b0: 616e 6420 6c69 6e65 2c20 2050 7265 763a and line, Prev:\n-000130c0: 2049 6e70 7574 206c 616e 6775 6167 652c Input language,\n-000130d0: 2020 5570 3a20 546f 700a 0a34 2042 7569 Up: Top..4 Bui\n-000130e0: 6c64 696e 6720 6120 6472 6177 696e 670a lding a drawing.\n-000130f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ****************\n-00013100: 2a2a 2a2a 0a0a 5375 6363 6573 7366 756c ****..Successful\n-00013110: 2064 7261 7769 6e67 7320 7769 7468 20e2 drawings with .\n-00013120: 8098 736b 6574 6368 e280 9920 616e 6420 ..sketch... and \n-00013130: 7769 7468 2061 6e79 2073 6365 6e65 2064 with any scene d\n-00013140: 6573 6372 6970 7469 6f6e 0a6c 616e 6775 escription.langu\n-00013150: 6167 6520 7265 7175 6972 6520 7468 6174 age require that\n-00013160: 2074 6865 2075 7365 7220 6465 7665 6c6f the user develo\n-00013170: 7020 616e 2061 6363 7572 6174 6520 6d65 p an accurate me\n-00013180: 6e74 616c 2070 6963 7475 7265 206f 6620 ntal picture of \n-00013190: 6865 720a 636f 6465 2061 6e64 2069 7473 her.code and its\n-000131a0: 206d 6561 6e69 6e67 2e20 2054 6869 7320 meaning. This \n-000131b0: 696d 6167 6520 6973 2062 6573 7420 6275 image is best bu\n-000131c0: 696c 7420 696e 2073 6d61 6c6c 2070 6965 ilt in small pie\n-000131d0: 6365 732e 0a54 6865 7265 666f 7265 2c20 ces..Therefore, \n-000131e0: e280 9873 6b65 7463 68e2 8099 2069 6e70 ...sketch... inp\n-000131f0: 7574 7320 6172 6520 6265 7374 2063 7265 uts are best cre\n-00013200: 6174 6564 2069 6e20 736d 616c 6c20 696e ated in small in\n-00013210: 6372 656d 656e 7473 2077 6974 680a 6672 crements with.fr\n-00013220: 6571 7565 6e74 2070 6175 7365 7320 746f equent pauses to\n-00013230: 2063 6f6d 7069 6c65 2061 6e64 2076 6965 compile and vie\n-00013240: 7720 7468 6520 7265 7375 6c74 732e 2020 w the results. \n-00013250: 4361 7265 6675 6c20 636f 6d6d 656e 7473 Careful comments\n-00013260: 2069 6e0a 7468 6520 696e 7075 7420 6f66 in.the input of\n-00013270: 7465 6e20 6865 6c70 2061 7320 6120 7363 ten help as a sc\n-00013280: 656e 6520 6772 6f77 7320 696e 2063 6f6d ene grows in com\n-00013290: 706c 6578 6974 792e 0a0a 2a20 4d65 6e75 plexity...* Menu\n-000132a0: 3a0a 0a2a 204f 7665 7276 6965 773a 3a20 :..* Overview:: \n-000132b0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000132c0: 2020 2042 7569 6c64 696e 6720 6120 7375 Building a su\n-000132d0: 6273 7461 6e74 6961 6c20 6472 6177 696e bstantial drawin\n-000132e0: 672e 0a2a 2041 2074 6563 686e 6963 616c g..* A technical\n-000132f0: 2064 7261 7769 6e67 3a3a 2020 2020 2020 drawing:: \n-00013300: 2020 2041 6e20 6578 616d 706c 6520 7769 An example wi\n-00013310: 7468 2066 696e 6520 706c 6163 656d 656e th fine placemen\n-00013320: 742e 0a2a 2041 2068 6965 7261 7263 6869 t..* A hierarchi\n-00013330: 6361 6c20 6d6f 6465 6c3a 3a20 2020 2020 cal model:: \n-00013340: 2020 2041 6e20 6578 616d 706c 6520 7769 An example wi\n-00013350: 7468 2073 7765 6570 7320 616e 6420 7075 th sweeps and pu\n-00013360: 7473 2e0a 2a20 4361 7665 6174 733a 3a20 ts..* Caveats:: \n-00013370: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00013380: 2020 2020 5768 6572 6520 7472 6f75 626c Where troubl\n-00013390: 6520 6361 6e20 6f63 6375 722e 0a0a 1f0a e can occur.....\n-000133a0: 4669 6c65 3a20 736b 6574 6368 2e69 6e66 File: sketch.inf\n-000133b0: 6f2c 2020 4e6f 6465 3a20 4f76 6572 7669 o, Node: Overvi\n-000133c0: 6577 2c20 204e 6578 743a 2041 2074 6563 ew, Next: A tec\n-000133d0: 686e 6963 616c 2064 7261 7769 6e67 2c20 hnical drawing, \n-000133e0: 2050 7265 763a 2042 7569 6c64 696e 6720 Prev: Building \n-000133f0: 6120 6472 6177 696e 672c 2020 5570 3a20 a drawing, Up: \n-00013400: 4275 696c 6469 6e67 2061 2064 7261 7769 Building a drawi\n-00013410: 6e67 0a0a 342e 3120 4f76 6572 7669 6577 ng..4.1 Overview\n-00013420: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a41 .============..A\n-00013430: 7320 616e 206f 7665 7276 6965 772c 206c s an overview, l\n-00013440: 6574 2773 2064 6576 656c 6f70 2061 2064 et's develop a d\n-00013450: 6961 6772 616d 2074 6861 7420 7368 6f77 iagram that show\n-00013460: 7320 686f 7720 6120 7065 7273 7065 6374 s how a perspect\n-00013470: 6976 650a 7072 6f6a 6563 7469 6f6e 2074 ive.projection t\n-00013480: 7261 6e73 666f 726d 2077 6f72 6b73 2e20 ransform works. \n-00013490: 2057 6527 6c6c 2073 7461 7274 2077 6974 We'll start wit\n-000134a0: 6820 7468 6520 7472 6164 6974 696f 6e61 h the traditiona\n-000134b0: 6c20 7265 6665 7265 6e63 650a 6f62 6a65 l reference.obje\n-000134c0: 6374 2075 7365 6420 696e 2063 6f6d 7075 ct used in compu\n-000134d0: 7465 7220 6772 6170 6869 6373 2074 6578 ter graphics tex\n-000134e0: 7462 6f6f 6b73 2c20 6120 686f 7573 652d tbooks, a house-\n-000134f0: 7368 6170 6564 2070 7269 736d 2e20 2042 shaped prism. B\n-00013500: 6567 696e 0a62 7920 6465 6669 6e69 6e67 egin.by defining\n-00013510: 2074 6865 2070 6f69 6e74 7320 6f66 2074 the points of t\n-00013520: 6865 2068 6f75 7365 2e20 2052 6174 6865 he house. Rathe\n-00013530: 7220 7468 616e 2064 6566 696e 696e 6720 r than defining \n-00013540: 7468 6520 6661 6365 7320 6f66 0a74 6865 the faces of.the\n-00013550: 2068 6f75 7365 2061 7320 706f 6c79 676f house as polygo\n-00013560: 6e73 2061 6e64 2074 7261 6e73 666f 726d ns and transform\n-00013570: 696e 6720 7468 6f73 652c 2077 6520 6172 ing those, we ar\n-00013580: 6520 676f 696e 6720 746f 2074 7261 6e73 e going to trans\n-00013590: 666f 726d 0a74 6865 2070 6f69 6e74 7320 form.the points \n-000135a0: 7468 656d 7365 6c76 6573 2077 6974 6820 themselves with \n-000135b0: e280 9873 6b65 7463 68e2 8099 2061 7269 ...sketch... ari\n-000135c0: 7468 6d65 7469 6320 736f 2074 6861 7420 thmetic so that \n-000135d0: 7765 2068 6176 6520 6e61 6d65 7320 666f we have names fo\n-000135e0: 720a 7468 6520 7472 616e 7366 6f72 6d65 r.the transforme\n-000135f0: 6420 706f 696e 7473 206c 6174 6572 2e0a d points later..\n-00013600: 2020 2520 7269 6768 7420 7369 6465 2028 % right side (\n-00013610: 6f75 7473 6964 6520 746f 2072 6967 6874 outside to right\n-00013620: 290a 2020 6465 6620 5231 2028 312c 312c ). def R1 (1,1,\n-00013630: 3129 2064 6566 2052 3220 2831 2c2d 312c 1) def R2 (1,-1,\n-00013640: 3129 2064 6566 2052 3320 2831 2c2d 312c 1) def R3 (1,-1,\n-00013650: 2d31 2920 6465 6620 5234 2028 312c 312c -1) def R4 (1,1,\n-00013660: 2d31 290a 2020 6465 6620 5235 2028 312c -1). def R5 (1,\n-00013670: 312e 352c 3029 0a0a 2020 2520 6c65 6674 1.5,0).. % left\n-00013680: 2073 6964 6520 286f 7574 7369 6465 2074 side (outside t\n-00013690: 6f20 7269 6768 742d 2d62 6163 6b77 6172 o right--backwar\n-000136a0: 6429 0a20 2064 6566 2057 2020 5b32 2c30 d). def W [2,0\n-000136b0: 2c30 5d0a 2020 6465 6620 4c31 2028 5231 ,0]. def L1 (R1\n-000136c0: 292d 5b57 5d20 6465 6620 4c32 2028 5232 )-[W] def L2 (R2\n-000136d0: 292d 5b57 5d20 6465 6620 4c33 2028 5233 )-[W] def L3 (R3\n-000136e0: 292d 5b57 5d20 6465 6620 4c34 2028 5234 )-[W] def L4 (R4\n-000136f0: 292d 5b57 5d0a 2020 6465 6620 4c35 2028 )-[W]. def L5 (\n-00013700: 5235 292d 5b57 5d0a 546f 2061 6464 2061 R5)-[W].To add a\n-00013710: 2064 6f6f 7220 746f 2074 6865 2068 6f75 door to the hou\n-00013720: 7365 2c20 7765 2075 7365 2061 2070 6f6c se, we use a pol\n-00013730: 7967 6f6e 2073 6c69 6768 746c 7920 696e ygon slightly in\n-00013740: 2066 726f 6e74 206f 6620 7468 650a 666f front of the.fo\n-00013750: 7265 6d6f 7374 2066 6163 6520 6f66 2074 remost face of t\n-00013760: 6865 2068 6f75 7365 2e0a 2020 2520 646f he house.. % do\n-00013770: 6f72 0a20 2064 6566 2065 202e 3031 0a20 or. def e .01. \n-00013780: 2064 6566 2044 3120 2830 2c2d 312c 312b def D1 (0,-1,1+\n-00013790: 6529 2064 6566 2044 3220 282e 352c 2d31 e) def D2 (.5,-1\n-000137a0: 2c31 2b65 2920 6465 6620 4433 2028 2e35 ,1+e) def D3 (.5\n-000137b0: 2c30 2c31 2b65 2920 6465 6620 4434 2028 ,0,1+e) def D4 (\n-000137c0: 302c 302c 312b 6529 0a4e 6f77 206c 6574 0,0,1+e).Now let\n-000137d0: 2773 2063 7265 6174 6520 6120 6e65 7720 's create a new \n-000137e0: 7365 7420 6f66 2070 6f69 6e74 7320 7468 set of points th\n-000137f0: 6174 2061 7265 2061 2074 6f2d 6265 2d64 at are a to-be-d\n-00013800: 6574 6572 6d69 6e65 640a 7472 616e 7366 etermined.transf\n-00013810: 6f72 6d20 6f66 2074 6865 206f 7269 6769 orm of the origi\n-00013820: 6e61 6c73 2e0a 2020 6465 6620 6870 2073 nals.. def hp s\n-00013830: 6361 6c65 2831 2920 2520 686f 7573 6520 cale(1) % house \n-00013840: 706f 7369 7469 6f6e 6572 0a20 2064 6566 positioner. def\n-00013850: 2070 5231 205b 5b68 705d 5d2a 2852 3129 pR1 [[hp]]*(R1)\n-00013860: 2064 6566 2070 5232 205b 5b68 705d 5d2a def pR2 [[hp]]*\n-00013870: 2852 3229 2064 6566 2070 5233 205b 5b68 (R2) def pR3 [[h\n-00013880: 705d 5d2a 2852 3329 0a20 2064 6566 2070 p]]*(R3). def p\n-00013890: 5234 205b 5b68 705d 5d2a 2852 3429 2064 R4 [[hp]]*(R4) d\n-000138a0: 6566 2070 5235 205b 5b68 705d 5d2a 2852 ef pR5 [[hp]]*(R\n-000138b0: 3529 0a20 2064 6566 2070 4c31 205b 5b68 5). def pL1 [[h\n-000138c0: 705d 5d2a 284c 3129 2064 6566 2070 4c32 p]]*(L1) def pL2\n-000138d0: 205b 5b68 705d 5d2a 284c 3229 2064 6566 [[hp]]*(L2) def\n-000138e0: 2070 4c33 205b 5b68 705d 5d2a 284c 3329 pL3 [[hp]]*(L3)\n-000138f0: 0a20 2064 6566 2070 4c34 205b 5b68 705d . def pL4 [[hp]\n-00013900: 5d2a 284c 3429 2064 6566 2070 4c35 205b ]*(L4) def pL5 [\n-00013910: 5b68 705d 5d2a 284c 3529 0a20 2064 6566 [hp]]*(L5). def\n-00013920: 2070 4431 205b 5b68 705d 5d2a 2844 3129 pD1 [[hp]]*(D1)\n-00013930: 2064 6566 2070 4432 205b 5b68 705d 5d2a def pD2 [[hp]]*\n-00013940: 2844 3229 2064 6566 2070 4433 205b 5b68 (D2) def pD3 [[h\n-00013950: 705d 5d2a 2844 3329 0a20 2064 6566 2070 p]]*(D3). def p\n-00013960: 4434 205b 5b68 705d 5d2a 2844 3429 200a D4 [[hp]]*(D4) .\n-00013970: 4e6f 7465 2074 6865 2075 7365 206f 6620 Note the use of \n-00013980: 6120 e280 9c74 7261 6e73 666f 726d 2064 a ...transform d\n-00013990: 6566 696e 6974 696f 6ee2 809d 2061 6e64 efinition... and\n-000139a0: 20e2 809c 7472 616e 7366 6f72 6d20 7265 ...transform re\n-000139b0: 6665 7265 6e63 6573 e280 9d2e 0a4e 6f77 ferences.....Now\n-000139c0: 2064 6566 696e 6520 7468 6520 7365 7665 define the seve\n-000139d0: 6e20 706f 6c79 676f 6e61 6c20 6661 6365 n polygonal face\n-000139e0: 7320 6f66 2074 6865 2068 6f75 7365 2061 s of the house a\n-000139f0: 6e64 2074 6865 2064 6f6f 7220 7573 696e nd the door usin\n-00013a00: 6720 7468 650a 7472 616e 7366 6f72 6d65 g the.transforme\n-00013a10: 6420 706f 696e 7473 2061 7320 7665 7274 d points as vert\n-00013a20: 6963 6573 2e20 2042 6520 6361 7265 6675 ices. Be carefu\n-00013a30: 6c20 7769 7468 2076 6572 7465 7820 6f72 l with vertex or\n-00013a40: 6465 7221 0a20 2064 6566 2072 6774 2070 der!. def rgt p\n-00013a50: 6f6c 7967 6f6e 2028 7052 3129 2870 5232 olygon (pR1)(pR2\n-00013a60: 2928 7052 3329 2870 5234 2928 7052 3529 )(pR3)(pR4)(pR5)\n-00013a70: 0a20 2064 6566 206c 6674 2070 6f6c 7967 . def lft polyg\n-00013a80: 6f6e 2028 704c 3529 2870 4c34 2928 704c on (pL5)(pL4)(pL\n-00013a90: 3329 2870 4c32 2928 704c 3129 0a20 2064 3)(pL2)(pL1). d\n-00013aa0: 6566 2066 7274 2070 6f6c 7967 6f6e 2028 ef frt polygon (\n-00013ab0: 7052 3229 2870 5231 2928 704c 3129 2870 pR2)(pR1)(pL1)(p\n-00013ac0: 4c32 290a 2020 6465 6620 6263 6b20 706f L2). def bck po\n-00013ad0: 6c79 676f 6e20 2870 5234 2928 7052 3329 lygon (pR4)(pR3)\n-00013ae0: 2870 4c33 2928 704c 3429 0a20 2064 6566 (pL3)(pL4). def\n-00013af0: 2074 6672 2070 6f6c 7967 6f6e 2028 7052 tfr polygon (pR\n-00013b00: 3129 2870 5235 2928 704c 3529 2870 4c31 1)(pR5)(pL5)(pL1\n-00013b10: 290a 2020 6465 6620 7462 6b20 706f 6c79 ). def tbk poly\n-00013b20: 676f 6e20 2870 5235 2928 7052 3429 2870 gon (pR5)(pR4)(p\n-00013b30: 4c34 2928 704c 3529 0a20 2064 6566 2062 L4)(pL5). def b\n-00013b40: 6f74 2070 6f6c 7967 6f6e 2028 7052 3229 ot polygon (pR2)\n-00013b50: 2870 4c32 2928 704c 3329 2870 5233 290a (pL2)(pL3)(pR3).\n-00013b60: 2020 6465 6620 646f 6f72 2070 6f6c 7967 def door polyg\n-00013b70: 6f6e 5b66 696c 6c63 6f6c 6f72 3d62 726f on[fillcolor=bro\n-00013b80: 776e 5d20 2870 4431 2928 7044 3229 2870 wn] (pD1)(pD2)(p\n-00013b90: 4433 2928 7044 3429 0a20 2064 6566 2068 D3)(pD4). def h\n-00013ba0: 6f75 7365 207b 207b 7267 747d 7b6c 6674 ouse { {rgt}{lft\n-00013bb0: 7d7b 6672 747d 7b62 636b 7d7b 7466 727d }{frt}{bck}{tfr}\n-00013bc0: 7b74 626b 7d7b 626f 747d 7b64 6f6f 727d {tbk}{bot}{door}\n-00013bd0: 207d 0a20 2020 5469 6d65 2066 6f72 2061 }. Time for a\n-00013be0: 2073 616e 6974 7920 6368 6563 6b2e 2020 sanity check. \n-00013bf0: 4164 6420 7468 6520 6c69 6e65 0a20 207b Add the line. {\n-00013c00: 686f 7573 657d 0a61 6e64 2074 6869 7320 house}.and this \n-00013c10: 6973 2077 6861 7420 7765 2067 6574 2e0a is what we get..\n-00013c20: 0a00 085b 696d 6167 6520 7372 633d 2265 ...[image src=\"e\n-00013c30: 7831 3330 2e70 6e67 2220 7465 7874 3d22 x130.png\" text=\"\n-00013c40: 496d 6167 6520 6578 3133 3020 6f6d 6974 Image ex130 omit\n-00013c50: 7465 6420 696e 2074 6578 7420 7665 7273 ted in text vers\n-00013c60: 696f 6e20 6f66 2074 6869 7320 646f 6375 ion of this docu\n-00013c70: 6d65 6e74 2e22 0008 5d0a 0a54 6869 7320 ment.\"..]..This \n-00013c80: 6973 2063 6f72 7265 6374 2c20 6275 7420 is correct, but \n-00013c90: 646f 6573 206e 6f74 2072 6576 6561 6c20 does not reveal \n-00013ca0: 7665 7279 206d 7563 682e 2020 436f 6d6d very much. Comm\n-00013cb0: 6f6e 2065 7272 6f72 7320 6172 650a 6d69 on errors are.mi\n-00013cc0: 7370 6c61 6365 6420 7665 7274 6963 6573 splaced vertices\n-00013cd0: 2061 6e64 2070 6f6c 7967 6f6e 7320 6d69 and polygons mi\n-00013ce0: 7373 696e 6720 656e 7469 7265 6c79 2064 ssing entirely d\n-00013cf0: 7565 2074 6f20 696e 636f 7272 6563 7420 ue to incorrect \n-00013d00: 7665 7274 6578 0a6f 7264 6572 2e20 2054 vertex.order. T\n-00013d10: 6f20 7275 6c65 2074 6865 7365 206f 7574 o rule these out\n-00013d20: 2c20 6c65 7427 7320 696e 7370 6563 7420 , let's inspect \n-00013d30: 616c 6c20 7369 6465 7320 6f66 2074 6865 all sides of the\n-00013d40: 2068 6f75 7365 2e20 2054 6869 730a 6973 house. This.is\n-00013d50: 206e 6f74 2068 6172 642e 2020 4d65 7265 not hard. Mere\n-00013d60: 6c79 2072 6570 6c61 6365 2074 6865 2072 ly replace the r\n-00013d70: 6566 6572 656e 6365 207b 686f 7573 657d eference {house}\n-00013d80: 2077 6974 6820 6120 e280 9872 6570 6561 with a ...repea\n-00013d90: 74e2 8099 2e0a 2a4e 6f74 6520 5265 7065 t.....*Note Repe\n-00013da0: 6174 733a 3a2e 0a20 2072 6570 6561 7420 ats::.. repeat \n-00013db0: 7b20 3133 2c20 726f 7461 7465 2833 302c { 13, rotate(30,\n-00013dc0: 205b 312c 322c 335d 292c 2074 7261 6e73 [1,2,3]), trans\n-00013dd0: 6c61 7465 285b 332c 302c 305d 2920 7d20 late([3,0,0]) } \n-00013de0: 7b68 6f75 7365 7d0a 0008 5b69 6d61 6765 {house}...[image\n-00013df0: 2073 7263 3d22 6578 3134 302e 706e 6722 src=\"ex140.png\"\n-00013e00: 2074 6578 743d 2249 6d61 6765 2065 7831 text=\"Image ex1\n-00013e10: 3430 206f 6d69 7474 6564 2069 6e20 7465 40 omitted in te\n-00013e20: 7874 2076 6572 7369 6f6e 206f 6620 7468 xt version of th\n-00013e30: 6973 2064 6f63 756d 656e 742e 2200 085d is document.\"..]\n-00013e40: 0a0a 4167 6169 6e20 7468 696e 6773 206c ..Again things l\n-00013e50: 6f6f 6b20 636f 7272 6563 742e 2020 4e6f ook correct. No\n-00013e60: 7465 2074 6861 7420 7468 6520 6869 6464 te that the hidd\n-00013e70: 656e 2073 7572 6661 6365 2061 6c67 6f72 en surface algor\n-00013e80: 6974 686d 0a68 616e 646c 6573 2069 6e74 ithm.handles int\n-00013e90: 6572 7365 6374 696e 6720 706f 6c79 676f ersecting polygo\n-00013ea0: 6e73 2063 6f72 7265 6374 6c79 2077 6865 ns correctly whe\n-00013eb0: 7265 2073 6f6d 6520 636f 7069 6573 206f re some copies o\n-00013ec0: 6620 7468 6520 686f 7573 650a 6f76 6572 f the house.over\n-00013ed0: 6c61 702e 0a0a 2020 204c 6574 2773 206c lap... Let's l\n-00013ee0: 6179 206f 7574 2074 6865 2067 656f 6d65 ay out the geome\n-00013ef0: 7472 7920 6f66 2070 6572 7370 6563 7469 try of perspecti\n-00013f00: 7665 2070 726f 6a65 6374 696f 6e20 6f66 ve projection of\n-00013f10: 2074 6865 2068 6f75 7365 0a6f 6e74 6f20 the house.onto \n-00013f20: 6120 706c 616e 6520 7769 7468 2072 6179 a plane with ray\n-00013f30: 7320 7061 7373 696e 6720 7468 726f 7567 s passing throug\n-00013f40: 6820 7468 6520 6f72 6967 696e 2e20 2042 h the origin. B\n-00013f50: 6567 696e 2062 7920 706f 7369 7469 6f6e egin by position\n-00013f60: 696e 670a 7468 6520 686f 7573 6520 7477 ing.the house tw\n-00013f70: 656c 7665 2075 6e69 7473 2062 6163 6b20 elve units back \n-00013f80: 6f6e 2074 6865 206e 6567 6174 6976 6520 on the negative \n-00013f90: 7a2d 6178 6973 2061 6e64 2061 6464 696e z-axis and addin\n-00013fa0: 6720 6120 7365 7420 6f66 0a63 6f6f 7264 g a set of.coord\n-00013fb0: 696e 6174 6520 6178 6573 2e20 2054 6f20 inate axes. To \n-00013fc0: 6d6f 7665 2074 6865 2068 6f75 7365 2077 move the house w\n-00013fd0: 6520 6e65 6564 206f 6e6c 7920 6368 616e e need only chan\n-00013fe0: 6765 2074 6865 2022 686f 7573 650a 706f ge the \"house.po\n-00013ff0: 7369 7469 6f6e 696e 6722 2074 7261 6e73 sitioning\" trans\n-00014000: 666f 726d 2064 6566 696e 6564 2065 6172 form defined ear\n-00014010: 6c69 6572 2e0a 2020 6465 6620 6870 2072 lier.. def hp r\n-00014020: 6f74 6174 6528 2d34 302c 205b 302c 312c otate(-40, [0,1,\n-00014030: 305d 2920 7468 656e 2074 7261 6e73 6c61 0]) then transla\n-00014040: 7465 285b 302c 302c 2d31 325d 290a 2020 te([0,0,-12]). \n-00014050: 6465 6620 6178 6573 207b 0a20 2020 2064 def axes {. d\n-00014060: 6566 2073 7a20 310a 2020 2020 6c69 6e65 ef sz 1. line\n-00014070: 205b 6172 726f 7773 3d3c 2d3e 5d20 2873 [arrows=<->] (s\n-00014080: 7a2c 302c 3029 284f 2928 302c 737a 2c30 z,0,0)(O)(0,sz,0\n-00014090: 290a 2020 2020 6c69 6e65 205b 6172 726f ). line [arro\n-000140a0: 7773 3d2d 3e5d 2020 284f 2928 302c 302c ws=->] (O)(0,0,\n-000140b0: 737a 290a 2020 2020 6c69 6e65 205b 6c69 sz). line [li\n-000140c0: 6e65 7769 6474 683d 2e32 7074 2c6c 696e newidth=.2pt,lin\n-000140d0: 6563 6f6c 6f72 3d62 6c75 652c 6c69 6e65 ecolor=blue,line\n-000140e0: 7374 796c 653d 6461 7368 6564 5d20 284f style=dashed] (O\n-000140f0: 2928 302c 302c 2d31 3029 0a20 2020 2073 )(0,0,-10). s\n-00014100: 7065 6369 616c 207c 5c75 7075 745b 725d pecial |\\uput[r]\n-00014110: 2331 7b24 7824 7d5c 7570 7574 5b75 5d23 #1{$x$}\\uput[u]#\n-00014120: 327b 2479 247d 5c75 7075 745b 6c5d 2333 2{$y$}\\uput[l]#3\n-00014130: 7b24 7a24 7d7c 0a20 2020 2020 2028 737a {$z$}|. (sz\n-00014140: 2c30 2c30 2928 302c 737a 2c30 2928 302c ,0,0)(0,sz,0)(0,\n-00014150: 302c 737a 290a 2020 7d0a 0a20 2020 5469 0,sz). }.. Ti\n-00014160: 6d65 2066 6f72 2061 6e6f 7468 6572 2074 me for another t\n-00014170: 6573 742e 2020 4c65 7427 7320 6275 696c est. Let's buil\n-00014180: 6420 6120 7265 616c 2076 6965 7720 7472 d a real view tr\n-00014190: 616e 7366 6f72 6d2c 2063 7265 6174 696e ansform, creatin\n-000141a0: 6720 610a e280 9c76 6972 7475 616c 2063 g a....virtual c\n-000141b0: 616d 6572 61e2 809d 2074 6f20 6c6f 6f6b amera... to look\n-000141c0: 2061 7420 7468 6520 7363 656e 6520 7765 at the scene we\n-000141d0: 2061 7265 2063 6f6e 7374 7275 6374 696e are constructin\n-000141e0: 672e 2020 5265 706c 6163 6520 7468 650a g. Replace the.\n-000141f0: e280 9872 6570 6561 74e2 8099 2077 6974 ...repeat... wit\n-00014200: 680a 2020 6465 6620 6579 6520 2831 302c h. def eye (10,\n-00014210: 342c 3130 290a 2020 6465 6620 6c6f 6f6b 4,10). def look\n-00014220: 5f61 7420 2830 2c30 2c2d 3529 0a20 2070 _at (0,0,-5). p\n-00014230: 7574 207b 2076 6965 7728 2865 7965 292c ut { view((eye),\n-00014240: 2028 6c6f 6f6b 5f61 7429 2920 7d20 7b20 (look_at)) } { \n-00014250: 7b68 6f75 7365 7d7b 6178 6573 7d20 7d0a {house}{axes} }.\n-00014260: 2020 2054 6865 20e2 809c 7669 6577 2074 The ...view t\n-00014270: 7261 6e73 666f 726d e280 9d20 7265 706f ransform... repo\n-00014280: 7369 7469 6f6e 7320 7468 6520 7363 656e sitions the scen\n-00014290: 6520 736f 2074 6861 7420 7468 6520 706f e so that the po\n-000142a0: 696e 7420 e280 9865 7965 e280 9920 6973 int ...eye... is\n-000142b0: 0a61 7420 7468 6520 6f72 6967 696e 2061 .at the origin a\n-000142c0: 6e64 2074 6865 2064 6972 6563 7469 6f6e nd the direction\n-000142d0: 2066 726f 6d20 e280 9865 7965 e280 9920 from ...eye... \n-000142e0: 746f 20e2 8098 6c6f 6f6b 5f61 74e2 8099 to ...look_at...\n-000142f0: 2069 7320 7468 6520 6e65 6761 7469 7665 is the negative\n-00014300: 0a7a 2d61 7869 732e 2020 5468 6973 2072 .z-axis. This r\n-00014310: 6571 7569 7265 7320 6120 726f 7461 7469 equires a rotati\n-00014320: 6f6e 2061 6e64 2061 2074 7261 6e73 6c61 on and a transla\n-00014330: 7469 6f6e 2074 6861 7420 6172 6520 616c tion that are al\n-00014340: 6c20 7061 636b 6564 0a69 6e74 6f20 7468 l packed.into th\n-00014350: 6520 636f 6e73 7472 7563 746f 7220 e280 e constructor ..\n-00014360: 9876 6965 77e2 8099 2e0a 0a00 085b 696d .view........[im\n-00014370: 6167 6520 7372 633d 2265 7831 3530 2e70 age src=\"ex150.p\n-00014380: 6e67 2220 7465 7874 3d22 496d 6167 6520 ng\" text=\"Image \n-00014390: 6578 3135 3020 6f6d 6974 7465 6420 696e ex150 omitted in\n-000143a0: 2074 6578 7420 7665 7273 696f 6e20 6f66 text version of\n-000143b0: 2074 6869 7320 646f 6375 6d65 6e74 2e22 this document.\"\n-000143c0: 0008 5d0a 0a54 6869 7320 6973 2073 7461 ..]..This is sta\n-000143d0: 7274 696e 6720 746f 206c 6f6f 6b20 676f rting to look go\n-000143e0: 6f64 2120 2041 6464 2074 6865 2070 726f od! Add the pro\n-000143f0: 6a65 6374 696f 6e20 706c 616e 6520 6861 jection plane ha\n-00014400: 6c66 2077 6179 0a62 6574 7765 656e 2074 lf way.between t\n-00014410: 6865 206f 7269 6769 6e20 616e 6420 7468 he origin and th\n-00014420: 6520 686f 7573 6520 6174 207a 3d2d 352e e house at z=-5.\n-00014430: 2020 5765 276c 6c20 7472 7920 7468 6520 We'll try the \n-00014440: 616e 676c 6520 6172 6775 6d65 6e74 0a66 angle argument.f\n-00014450: 6561 7475 7265 206f 6620 e280 9873 7065 eature of ...spe\n-00014460: 6369 616c e280 9920 746f 2070 6f73 6974 cial... to posit\n-00014470: 696f 6e20 6120 6c61 6265 6c2e 0a20 2064 ion a label.. d\n-00014480: 6566 2070 2035 2025 2070 726f 6a65 6374 ef p 5 % project\n-00014490: 696f 6e20 6469 7374 616e 6365 2028 706c ion distance (pl\n-000144a0: 616e 6520 6174 207a 3d2d 7029 0a20 2064 ane at z=-p). d\n-000144b0: 6566 2070 726f 6a65 6374 696f 6e5f 706c ef projection_pl\n-000144c0: 616e 6520 7b0a 2020 2020 6465 6620 737a ane {. def sz\n-000144d0: 2031 2e35 0a20 2020 2070 6f6c 7967 6f6e 1.5. polygon\n-000144e0: 2028 2d73 7a2c 2d73 7a2c 2d70 2928 737a (-sz,-sz,-p)(sz\n-000144f0: 2c2d 737a 2c2d 7029 2873 7a2c 737a 2c2d ,-sz,-p)(sz,sz,-\n-00014500: 7029 282d 737a 2c73 7a2c 2d70 290a 2020 p)(-sz,sz,-p). \n-00014510: 2020 7370 6563 6961 6c20 7c5c 7270 7574 special |\\rput\n-00014520: 5b62 5d23 312d 3223 337b 5c66 6f6f 746e [b]#1-2#3{\\footn\n-00014530: 6f74 6573 697a 655c 7366 2070 726f 6a65 otesize\\sf proje\n-00014540: 6374 696f 6e20 706c 616e 657d 7c20 0a20 ction plane}| . \n-00014550: 2020 2020 2028 2d73 7a2c 2d73 7a2c 2d70 (-sz,-sz,-p\n-00014560: 2928 737a 2c2d 737a 2c2d 7029 2830 2c2d )(sz,-sz,-p)(0,-\n-00014570: 737a 2b2e 312c 2d70 290a 2020 7d0a 4164 sz+.1,-p). }.Ad\n-00014580: 6420 7b70 726f 6a65 6374 696f 6e5f 706c d {projection_pl\n-00014590: 616e 657d 2074 6f20 7468 6520 6c69 7374 ane} to the list\n-000145a0: 206f 6620 6f62 6a65 6374 7320 696e 2074 of objects in t\n-000145b0: 6865 20e2 8098 7075 74e2 8099 2061 626f he ...put... abo\n-000145c0: 7665 2e0a 0a00 085b 696d 6167 6520 7372 ve.....[image sr\n-000145d0: 633d 2265 7831 3630 2e70 6e67 2220 7465 c=\"ex160.png\" te\n-000145e0: 7874 3d22 496d 6167 6520 6578 3136 3020 xt=\"Image ex160 \n-000145f0: 6f6d 6974 7465 6420 696e 2074 6578 7420 omitted in text \n-00014600: 7665 7273 696f 6e20 6f66 2074 6869 7320 version of this \n-00014610: 646f 6375 6d65 6e74 2e22 0008 5d0a 0a20 document.\"..].. \n-00014620: 2020 5468 6520 7761 7920 7765 2063 6f6e The way we con\n-00014630: 7374 7275 6374 6564 2074 6865 2070 6f69 structed the poi\n-00014640: 6e74 7320 6f66 2074 6865 2068 6f75 7365 nts of the house\n-00014650: 206e 6f77 206d 616b 6573 2069 7420 6561 now makes it ea\n-00014660: 7379 2074 6f0a 6472 6177 2072 6179 7320 sy to.draw rays \n-00014670: 6f66 2070 726f 6a65 6374 696f 6e2e 2020 of projection. \n-00014680: 5765 276c 6c20 6361 7374 206f 6e65 2072 We'll cast one r\n-00014690: 6179 2066 726f 6d20 6576 6572 7920 7669 ay from every vi\n-000146a0: 7369 626c 6520 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line[ray\n-00014790: 6f70 745d 284f 2928 7052 3329 0a20 2020 opt](O)(pR3). \n-000147a0: 206c 696e 6520 5b72 6179 6f70 745d 284f line [rayopt](O\n-000147b0: 2928 7052 3429 206c 696e 6520 5b72 6179 )(pR4) line [ray\n-000147c0: 6f70 745d 284f 2928 7052 3529 0a20 2020 opt](O)(pR5). \n-000147d0: 206c 696e 6520 5b72 6179 6f70 745d 284f line [rayopt](O\n-000147e0: 2928 704c 3129 206c 696e 6520 5b72 6179 )(pL1) line [ray\n-000147f0: 6f70 745d 284f 2928 704c 3229 206c 696e opt](O)(pL2) lin\n-00014800: 655b 7261 796f 7074 5d28 4f29 2870 4c35 e[rayopt](O)(pL5\n-00014810: 290a 2020 2020 6c69 6e65 205b 7261 796f ). line [rayo\n-00014820: 7074 5d28 4f29 2870 4431 2920 6c69 6e65 pt](O)(pD1) line\n-00014830: 205b 7261 796f 7074 5d28 4f29 2870 4432 [rayopt](O)(pD2\n-00014840: 2920 0a20 2020 206c 696e 6520 5b72 6179 ) . line [ray\n-00014850: 6f70 745d 284f 2928 7044 3329 206c 696e opt](O)(pD3) lin\n-00014860: 6520 5b72 6179 6f70 745d 284f 2928 7044 e [rayopt](O)(pD\n-00014870: 3429 200a 2020 7d0a 5468 6520 7265 7375 4) . }.The resu\n-00014880: 6c74 2069 7320 7368 6f77 6e20 6865 7265 lt is shown here\n-00014890: 2e0a 0a00 085b 696d 6167 6520 7372 633d .....[image src=\n-000148a0: 2265 7831 3730 2e70 6e67 2220 7465 7874 \"ex170.png\" text\n-000148b0: 3d22 496d 6167 6520 6578 3137 3020 6f6d =\"Image ex170 om\n-000148c0: 6974 7465 6420 696e 2074 6578 7420 7665 itted in text ve\n-000148d0: 7273 696f 6e20 6f66 2074 6869 7320 646f rsion of this do\n-000148e0: 6375 6d65 6e74 2e22 0008 5d0a 0a54 6865 cument.\"..]..The\n-000148f0: 2072 6179 7320 7069 6572 6365 2074 6865 rays pierce the\n-00014900: 2070 726f 6a65 6374 696f 6e20 706c 616e projection plan\n-00014910: 6520 6174 2074 6865 2063 6f72 7265 7370 e at the corresp\n-00014920: 6f6e 6469 6e67 2070 6f69 6e74 7320 6f6e onding points on\n-00014930: 2074 6865 0a70 6572 7370 6563 7469 7665 the.perspective\n-00014940: 2069 6d61 6765 2077 6520 6172 6520 7472 image we are tr\n-00014950: 7969 6e67 2074 6f20 6472 6177 2e20 2041 ying to draw. A\n-00014960: 6c62 7265 6368 7420 44c3 bc72 6572 2061 lbrecht D..rer a\n-00014970: 6e64 2068 6973 0a52 656e 6169 7373 616e nd his.Renaissan\n-00014980: 6365 2063 6f6e 7465 6d70 6f72 6172 6965 ce contemporarie\n-00014990: 7320 6861 6420 7468 6520 7361 6d65 2069 s had the same i\n-000149a0: 6465 6120 696e 2074 6865 2065 6172 6c79 dea in the early\n-000149b0: 2031 3530 3027 732e 0a0a 0008 5b69 6d61 1500's.....[ima\n-000149c0: 6765 2073 7263 3d22 6475 6572 6572 2e70 ge src=\"duerer.p\n-000149d0: 6e67 2220 7465 7874 3d22 496d 6167 6520 ng\" text=\"Image \n-000149e0: 6f66 2044 7565 7265 7227 7320 7065 7273 of Duerer's pers\n-000149f0: 7065 6374 6976 6520 636f 6d70 7574 6572 pective computer\n-00014a00: 206f 6d69 7474 6564 2069 6e20 7465 7874 omitted in text\n-00014a10: 2076 6572 7369 6f6e 206f 6620 7468 6973 version of this\n-00014a20: 2064 6f63 756d 656e 742e 2200 085d 0a0a document.\"..]..\n-00014a30: 2020 2041 6c6c 2074 6861 7427 7320 6c65 All that's le\n-00014a40: 6674 2069 7320 746f 2066 696e 6420 6120 ft is to find a \n-00014a50: 7761 7920 746f 2063 6f6e 6e65 6374 2074 way to connect t\n-00014a60: 6865 2070 6f69 6e74 7320 6f66 2074 6865 he points of the\n-00014a70: 2068 6f75 7365 0a6f 6e20 7468 6520 7072 house.on the pr\n-00014a80: 6f6a 6563 7469 6f6e 2070 6c61 6e65 2e20 ojection plane. \n-00014a90: 2057 6520 636f 756c 6420 7075 6c6c 206f We could pull o\n-00014aa0: 7574 2061 2067 6f6f 6420 636f 6d70 7574 ut a good comput\n-00014ab0: 6572 2067 7261 7068 6963 730a 7465 7874 er graphics.text\n-00014ac0: 2c20 6669 6e64 2074 6865 206e 6563 6573 , find the neces\n-00014ad0: 7361 7279 206d 6174 7269 782c 2061 6e64 sary matrix, and\n-00014ae0: 2065 6e74 6572 2069 7420 6f75 7273 656c enter it oursel\n-00014af0: 7665 7320 6173 2061 2074 7261 6e73 666f ves as a transfo\n-00014b00: 726d 0a6c 6974 6572 616c 2e20 202a 4e6f rm.literal. *No\n-00014b10: 7465 2054 7261 6e73 666f 726d 206c 6974 te Transform lit\n-00014b20: 6572 616c 733a 3a2e 2020 5468 6174 2077 erals::. That w\n-00014b30: 6f72 6b20 6973 2061 6c72 6561 6479 2064 ork is already d\n-00014b40: 6f6e 652c 0a68 6f77 6576 6572 2e20 2057 one,.however. W\n-00014b50: 6520 6361 6e20 7573 6520 7468 6520 e280 e can use the ..\n-00014b60: 9870 726f 6a65 6374 2870 29e2 8099 2063 .project(p)... c\n-00014b70: 6f6e 7374 7275 6374 6f72 2e0a 0a20 2020 onstructor... \n-00014b80: 5468 6572 6520 6172 6520 7374 696c 6c20 There are still \n-00014b90: 736f 6d65 2064 6574 6169 6c73 2074 6861 some details tha\n-00014ba0: 7420 7265 7175 6972 6520 6361 7265 2e20 t require care. \n-00014bb0: 2050 726f 6a65 6374 696f 6e20 7769 6c6c Projection will\n-00014bc0: 0a66 6c61 7474 656e 2077 6861 7465 7665 .flatten whateve\n-00014bd0: 7220 6973 2074 7261 6e73 666f 726d 6564 r is transformed\n-00014be0: 206f 6e74 6f20 7468 6520 706c 616e 6520 onto the plane \n-00014bf0: 7a3d 2d70 2e20 2054 6865 7265 666f 7265 z=-p. Therefore\n-00014c00: 2061 6e79 2070 6172 740a 6f66 2074 6865 any part.of the\n-00014c10: 2068 6f75 7365 2063 6f75 6c64 2064 6973 house could dis\n-00014c20: 6170 7065 6172 2062 6568 696e 6420 7468 appear behind th\n-00014c30: 6520 7072 6f6a 6563 7469 6f6e 2070 6c61 e projection pla\n-00014c40: 6e65 2028 7468 6520 6869 6464 656e 0a73 ne (the hidden.s\n-00014c50: 7572 6661 6365 2061 6c67 6f72 6974 686d urface algorithm\n-00014c60: 206f 7264 6572 7320 6f62 6a65 6374 7320 orders objects \n-00014c70: 6174 2074 6865 2073 616d 6520 6465 7074 at the same dept\n-00014c80: 6820 6172 6269 7472 6172 696c 7929 2e20 h arbitrarily). \n-00014c90: 2054 6865 0a64 6f6f 7220 6d61 7920 616c The.door may al\n-00014ca0: 736f 2064 6973 6170 7065 6172 2062 6568 so disappear beh\n-00014cb0: 696e 6420 7468 6520 6672 6f6e 7420 6f66 ind the front of\n-00014cc0: 2074 6865 2068 6f75 7365 2e20 2054 6f20 the house. To \n-00014cd0: 6d61 6b65 2073 7572 650a 6576 6572 7974 make sure.everyt\n-00014ce0: 6869 6e67 2072 656d 6169 6e73 2076 6973 hing remains vis\n-00014cf0: 6962 6c65 2c20 7765 276c 6c20 706c 6163 ible, we'll plac\n-00014d00: 6520 7468 6520 686f 7573 6520 6120 7469 e the house a ti\n-00014d10: 6e79 2062 6974 2069 6e20 6672 6f6e 7420 ny bit in front \n-00014d20: 6f66 0a74 6865 2070 726f 6a65 6374 696f of.the projectio\n-00014d30: 6e20 706c 616e 6520 616e 6420 6120 7365 n plane and a se\n-00014d40: 636f 6e64 2063 6f70 7920 6f66 2074 6865 cond copy of the\n-00014d50: 2064 6f6f 7220 696e 2066 726f 6e74 206f door in front o\n-00014d60: 6620 7468 650a 686f 7573 652e 0a20 2064 f the.house.. d\n-00014d70: 6566 2070 726f 6a65 6374 696f 6e20 7b0a ef projection {.\n-00014d80: 2020 2020 2520 6520 6973 2061 2073 6d61 % e is a sma\n-00014d90: 6c6c 206e 756d 6265 7220 6465 6669 6e65 ll number define\n-00014da0: 6420 6162 6f76 650a 2020 2020 7075 7420 d above. put \n-00014db0: 7b20 7072 6f6a 6563 7428 7029 2074 6865 { project(p) the\n-00014dc0: 6e20 7472 616e 736c 6174 6528 5b30 2c30 n translate([0,0\n-00014dd0: 2c31 2a65 5d29 207d 207b 686f 7573 657d ,1*e]) } {house}\n-00014de0: 0a20 2020 2070 7574 207b 2070 726f 6a65 . put { proje\n-00014df0: 6374 2870 2920 7468 656e 2074 7261 6e73 ct(p) then trans\n-00014e00: 6c61 7465 285b 302c 302c 322a 655d 2920 late([0,0,2*e]) \n-00014e10: 7d20 7b64 6f6f 727d 0a20 207d 0a0a 0008 } {door}. }....\n-00014e20: 5b69 6d61 6765 2073 7263 3d22 6578 3138 [image src=\"ex18\n-00014e30: 302e 706e 6722 2074 6578 743d 2249 6d61 0.png\" text=\"Ima\n-00014e40: 6765 2065 7831 3830 206f 6d69 7474 6564 ge ex180 omitted\n-00014e50: 2069 6e20 7465 7874 2076 6572 7369 6f6e in text version\n-00014e60: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen\n-00014e70: 742e 2200 085d 0a0a 2020 2049 6620 796f t.\"..].. If yo\n-00014e80: 7520 6861 7665 2073 7475 6469 6564 2061 u have studied a\n-00014e90: 6e64 2075 6e64 6572 7374 616e 6420 616c nd understand al\n-00014ea0: 6c20 7468 6973 2c20 796f 7520 6172 6520 l this, you are \n-00014eb0: 7765 6c6c 206f 6e20 7468 6520 7761 790a well on the way.\n-00014ec0: 746f 2073 7563 6365 7373 2077 6974 6820 to success with \n-00014ed0: e280 9873 6b65 7463 68e2 8099 2e20 204e ...sketch.... N\n-00014ee0: 6f74 2073 686f 776e 2061 7265 2074 6865 ot shown are the\n-00014ef0: 2032 3020 6f72 2073 6f20 6974 6572 6174 20 or so iterat\n-00014f00: 696f 6e73 2074 6861 740a 7765 7265 2072 ions that.were r\n-00014f10: 6571 7569 7265 6420 746f 2066 696e 6420 equired to find \n-00014f20: 6120 7265 6173 6f6e 6162 6c65 2076 6965 a reasonable vie\n-00014f30: 7769 6e67 2061 6e67 6c65 2061 6e64 2068 wing angle and h\n-00014f40: 6f75 7365 2070 6f73 6974 696f 6e2c 0a65 ouse position,.e\n-00014f50: 7463 2e20 204e 6f6e 6574 6865 6c65 7373 tc. 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2063 6861 6e67 6520 7468 Just change th\n-00015050: 6520 7669 6577 2074 7261 6e73 666f 726d e view transform\n-00015060: 0a74 6f0a 2020 7075 7420 7b20 7669 6577 .to. put { view\n-00015070: 2828 6579 6529 2c20 286c 6f6f 6b5f 6174 ((eye), (look_at\n-00015080: 2929 2074 6865 6e20 7065 7273 7065 6374 )) then perspect\n-00015090: 6976 6528 3929 207d 207b 202e 2e2e 0a61 ive(9) } { ....a\n-000150a0: 6e64 2070 726f 6475 6365 2074 6869 732e nd produce this.\n-000150b0: 0a0a 0008 5b69 6d61 6765 2073 7263 3d22 ....[image src=\"\n-000150c0: 6578 3139 302e 706e 6722 2074 6578 743d ex190.png\" text=\n-000150d0: 2249 6d61 6765 2065 7831 3930 206f 6d69 \"Image ex190 omi\n-000150e0: 7474 6564 2069 6e20 7465 7874 2076 6572 tted in text ver\n-000150f0: 7369 6f6e 206f 6620 7468 6973 2064 6f63 sion of this doc\n-00015100: 756d 656e 742e 2200 085d 0a0a 1f0a 4669 ument.\"..]....Fi\n-00015110: 6c65 3a20 736b 6574 6368 2e69 6e66 6f2c le: sketch.info,\n-00015120: 2020 4e6f 6465 3a20 4120 7465 6368 6e69 Node: A 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2062 6520 636f 6e66 7573 6564 2e20 to be confused. \n+00011ec0: 2054 6865 2073 7065 6369 616c 206d 756c The special mul\n+00011ed0: 7469 706c 6520 7265 6665 7265 6e63 650a tiple reference.\n+00011ee0: e280 985b 4944 312c 4944 322c 2e2e 2e2c ...[ID1,ID2,...,\n+00011ef0: 4944 4e5d e280 9920 6163 7473 2061 7320 IDN]... acts as \n+00011f00: 6966 2074 6865 2072 6573 7065 6374 6976 if the respectiv\n+00011f10: 6520 6c69 7374 7320 6f66 206f 7074 696f e lists of optio\n+00011f20: 6e73 2077 6572 650a 636f 6e63 6174 656e ns were.concaten\n+00011f30: 6174 6564 2e0a 0a1f 0a46 696c 653a 2073 ated.....File: s\n+00011f40: 6b65 7463 682e 696e 666f 2c20 204e 6f64 ketch.info, Nod\n+00011f50: 653a 2047 6c6f 6261 6c20 656e 7669 726f e: Global enviro\n+00011f60: 6e6d 656e 742c 2020 5072 6576 3a20 4465 nment, Prev: De\n+00011f70: 6669 6e69 7469 6f6e 732c 2020 5570 3a20 finitions, Up: \n+00011f80: 496e 7075 7420 6c61 6e67 7561 6765 0a0a Input language..\n+00011f90: 332e 3420 476c 6f62 616c 2065 6e76 6972 3.4 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The block m\n+00012020: 7573 7420 6170 7065 6172 2061 7320 7468 ust appear as th\n+00012030: 6520 6c61 7374 2074 6578 7420 696e 2074 e last text in t\n+00012040: 6865 20e2 8098 736b 6574 6368 e280 990a he ...sketch....\n+00012050: 696e 7075 7420 6669 6c65 2e20 2049 7420 input file. It \n+00012060: 6d61 7920 696e 636c 7564 6520 6465 6669 may include defi\n+00012070: 6e69 7469 6f6e 732c 2062 7574 206e 6f74 nitions, but not\n+00012080: 6520 7468 6174 2070 7265 7669 6f75 730a e that previous.\n+00012090: 6465 6669 6e69 7469 6f6e 7320 6174 2074 definitions at t\n+000120a0: 6865 2074 6f70 206c 6576 656c 2028 6e6f he top level (no\n+000120b0: 7420 6e65 7374 6564 2069 6e73 6964 6520 t nested inside \n+000120c0: 626c 6f63 6b73 2920 6172 6520 616c 736f blocks) are also\n+000120d0: 0a61 7661 696c 6162 6c65 2e0a 0a2a 5379 .available...*Sy\n+000120e0: 6e74 6178 3a2a 0a20 2020 2020 676c 6f62 ntax:*. glob\n+000120f0: 616c 207b 2045 4e56 4952 4f4e 4d45 4e54 al { ENVIRONMENT\n+00012100: 5f53 4554 5449 4e47 5320 7d0a 5468 6520 _SETTINGS }.The \n+00012110: 636f 6e74 656e 7473 206f 6620 454e 5649 contents of ENVI\n+00012120: 524f 4e4d 454e 545f 5345 5454 494e 4753 RONMENT_SETTINGS\n+00012130: 2061 7265 2064 6973 6375 7373 6564 2069 are discussed i\n+00012140: 6e20 7468 6520 7365 6374 696f 6e73 2074 n the sections t\n+00012150: 6861 740a 666f 6c6c 6f77 2e0a 0a2a 204d hat.follow...* M\n+00012160: 656e 753a 0a0a 2a20 476c 6f62 616c 206f enu:..* Global o\n+00012170: 7074 696f 6e73 3a3a 2020 2020 2020 2020 ptions:: \n+00012180: 2020 2020 2020 4174 7472 6962 7574 6573 Attributes\n+00012190: 206f 6620 7468 6520 656e 7469 7265 2064 of the entire d\n+000121a0: 7261 7769 6e67 2e0a 2a20 4361 6d65 7261 rawing..* Camera\n+000121b0: 3a3a 2020 2020 2020 2020 2020 2020 2020 :: \n+000121c0: 2020 2020 2020 2020 4120 6669 6e61 6c20 A final \n+000121d0: 6361 6d65 7261 2074 7261 6e73 666f 726d camera transform\n+000121e0: 6174 696f 6e20 6f66 2074 6865 2073 6365 ation of the sce\n+000121f0: 6e65 2e0a 2a20 5069 6374 7572 6520 626f ne..* Picture bo\n+00012200: 783a 3a20 2020 2020 2020 2020 2020 2020 x:: \n+00012210: 2020 2020 5365 7474 696e 6720 7468 6520 Setting the \n+00012220: 626f 756e 6469 6e67 2062 6f78 2061 6e64 bounding box and\n+00012230: 2032 6420 636c 6970 7069 6e67 2e0a 2a20 2d clipping..* \n+00012240: 4672 616d 653a 3a20 2020 2020 2020 2020 Frame:: \n+00012250: 2020 2020 2020 2020 2020 2020 2020 4164 Ad\n+00012260: 6469 6e67 2061 2062 6f78 2061 726f 756e ding a box aroun\n+00012270: 6420 7468 6520 6472 6177 696e 672e 0a2a d the drawing..*\n+00012280: 204c 616e 6775 6167 653a 3a20 2020 2020 Language:: \n+00012290: 2020 2020 2020 2020 2020 2020 2020 2053 S\n+000122a0: 6574 7469 6e67 2074 6865 206f 7574 7075 etting the outpu\n+000122b0: 7420 6c61 6e67 7561 6765 2e0a 0a1f 0a46 t language.....F\n+000122c0: 696c 653a 2073 6b65 7463 682e 696e 666f ile: sketch.info\n+000122d0: 2c20 204e 6f64 653a 2047 6c6f 6261 6c20 , Node: Global \n+000122e0: 6f70 7469 6f6e 732c 2020 4e65 7874 3a20 options, Next: \n+000122f0: 4361 6d65 7261 2c20 2050 7265 763a 2047 Camera, Prev: G\n+00012300: 6c6f 6261 6c20 656e 7669 726f 6e6d 656e lobal environmen\n+00012310: 742c 2020 5570 3a20 476c 6f62 616c 2065 t, Up: Global e\n+00012320: 6e76 6972 6f6e 6d65 6e74 0a0a 332e 342e nvironment..3.4.\n+00012330: 3120 476c 6f62 616c 206f 7074 696f 6e73 1 Global options\n+00012340: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .---------------\n+00012350: 2d2d 2d2d 2d0a 0a2a 5379 6e74 6178 3a2a -----..*Syntax:*\n+00012360: 0a20 2020 2020 7365 7420 5b20 4f50 5449 . set [ OPTI\n+00012370: 4f4e 5320 5d0a 5468 6520 636f 6e74 656e ONS ].The conten\n+00012380: 7473 206f 6620 4f50 5449 4f4e 532c 2065 ts of OPTIONS, e\n+00012390: 7863 6570 7420 666f 7220 e280 9873 6b65 xcept for ...ske\n+000123a0: 7463 68e2 8099 2069 6e74 6572 6e61 6c20 tch... internal \n+000123b0: 6f70 7469 6f6e 732c 2061 7265 0a63 6f70 options, are.cop\n+000123c0: 6965 6420 6173 2d69 7320 746f 2061 205c ied as-is to a \\\n+000123d0: 7073 7365 7420 7468 6174 2061 7070 6561 psset that appea\n+000123e0: 7273 2062 6566 6f72 6520 616e 7974 6869 rs before anythi\n+000123f0: 6e67 2065 6c73 6520 696e 2074 6865 206f ng else in the o\n+00012400: 7574 7075 740a 6669 6c65 2e20 2054 6869 utput.file. Thi\n+00012410: 7320 6973 2061 2067 6f6f 6420 706c 6163 s is a good plac\n+00012420: 6520 746f 2073 6574 20e2 8098 756e 6974 e to set ...unit\n+00012430: e280 992c 2061 2064 6566 6175 6c74 20e2 ..., a default .\n+00012440: 8098 6c69 6e65 7769 6474 68e2 8099 2c20 ..linewidth..., \n+00012450: 6574 632e 0a0a 2020 2049 6e74 6572 6e61 etc... Interna\n+00012460: 6c20 6f70 7469 6f6e 7320 776f 726b 206f l options work o\n+00012470: 6e20 616c 6c20 6f62 6a65 6374 7320 7768 n all objects wh\n+00012480: 6572 6520 7468 6579 206d 616b 6520 7365 ere they make se\n+00012490: 6e73 652e 2020 5468 6973 0a69 6e63 6c75 nse. This.inclu\n+000124a0: 6465 7320 e280 9863 756c 6ce2 8099 2061 des ...cull... a\n+000124b0: 6e64 20e2 8098 7370 6c69 74e2 8099 2028 nd ...split... (\n+000124c0: 6275 7420 6e6f 7420 e280 986c 6179 e280 but not ...lay..\n+000124d0: 9929 2e20 202a 4e6f 7465 2049 6e74 6572 .). *Note Inter\n+000124e0: 6e61 6c20 6f70 7469 6f6e 733a 3a2e 0a0a nal options::...\n+000124f0: 1f0a 4669 6c65 3a20 736b 6574 6368 2e69 ..File: sketch.i\n+00012500: 6e66 6f2c 2020 4e6f 6465 3a20 4361 6d65 nfo, Node: Came\n+00012510: 7261 2c20 204e 6578 743a 2050 6963 7475 ra, Next: Pictu\n+00012520: 7265 2062 6f78 2c20 2050 7265 763a 2047 re box, Prev: G\n+00012530: 6c6f 6261 6c20 6f70 7469 6f6e 732c 2020 lobal options, \n+00012540: 5570 3a20 476c 6f62 616c 2065 6e76 6972 Up: Global envir\n+00012550: 6f6e 6d65 6e74 0a0a 332e 342e 3220 4361 onment..3.4.2 Ca\n+00012560: 6d65 7261 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d mera.-----------\n+00012570: 2d0a 0a2a 5379 6e74 6178 3a2a 0a20 2020 -..*Syntax:*. \n+00012580: 2020 6361 6d65 7261 2054 5241 4e53 464f camera TRANSFO\n+00012590: 524d 5f45 5850 5245 5353 494f 4e0a 2020 RM_EXPRESSION. \n+000125a0: 2054 6865 2054 5241 4e53 464f 524d 5f45 The TRANSFORM_E\n+000125b0: 5850 5245 5353 494f 4e20 6973 2061 7070 XPRESSION is app\n+000125c0: 6c69 6564 2061 6674 6572 2061 6c6c 206f lied after all o\n+000125d0: 7468 6572 2074 7261 6e73 666f 726d 6174 ther transformat\n+000125e0: 696f 6e73 0a6f 6620 7468 6520 7363 656e ions.of the scen\n+000125f0: 652e 2020 5468 6973 2069 7320 6375 7272 e. This is curr\n+00012600: 656e 746c 7920 6f6e 6c79 2075 7365 6675 ently only usefu\n+00012610: 6c20 666f 7220 7472 616e 7366 6f72 6d69 l for transformi\n+00012620: 6e67 2074 6865 0a62 6f75 6e64 696e 6720 ng the.bounding \n+00012630: 626f 782e 2020 2a4e 6f74 6520 5069 6374 box. *Note Pict\n+00012640: 7572 6520 626f 783a 3a2e 2020 4974 2077 ure box::. It w\n+00012650: 696c 6c20 706c 6179 2061 2072 6f6c 6520 ill play a role \n+00012660: 696e 2061 6e79 2066 7574 7572 650a 696d in any future.im\n+00012670: 706c 656d 656e 7461 7469 6f6e 206f 6620 plementation of \n+00012680: 636c 6970 7069 6e67 2e0a 0a1f 0a46 696c clipping.....Fil\n+00012690: 653a 2073 6b65 7463 682e 696e 666f 2c20 e: sketch.info, \n+000126a0: 204e 6f64 653a 2050 6963 7475 7265 2062 Node: Picture b\n+000126b0: 6f78 2c20 204e 6578 743a 2046 7261 6d65 ox, Next: Frame\n+000126c0: 2c20 2050 7265 763a 2043 616d 6572 612c , Prev: Camera,\n+000126d0: 2020 5570 3a20 476c 6f62 616c 2065 6e76 Up: Global env\n+000126e0: 6972 6f6e 6d65 6e74 0a0a 332e 342e 3320 ironment..3.4.3 \n+000126f0: 5069 6374 7572 6520 626f 780a 2d2d 2d2d Picture box.----\n+00012700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a2a -------------..*\n+00012710: 5379 6e74 6178 3a2a 0a20 2020 2020 7069 Syntax:*. pi\n+00012720: 6374 7572 6562 6f78 5b42 4153 454c 494e cturebox[BASELIN\n+00012730: 455d 0a20 2020 2020 7069 6374 7572 6562 E]. pictureb\n+00012740: 6f78 5b42 4153 454c 494e 455d 2028 5031 ox[BASELINE] (P1\n+00012750: 2928 5032 290a 5468 6520 6669 7273 7420 )(P2).The first \n+00012760: 666f 726d 206f 6620 e280 9870 6963 7475 form of ...pictu\n+00012770: 7265 626f 78e2 8099 2063 6175 7365 7320 rebox... causes \n+00012780: 6120 7363 616c 6172 2042 4153 454c 494e a scalar BASELIN\n+00012790: 4520 6672 6163 7469 6f6e 2074 6f20 6265 E fraction to be\n+000127a0: 0a65 6d69 7474 6564 2069 6e20 7468 6520 .emitted in the \n+000127b0: e280 9870 7370 6963 7475 7265 e280 9920 ...pspicture... \n+000127c0: 656e 7669 726f 6e6d 656e 7420 6f66 2074 environment of t\n+000127d0: 6865 206f 7574 7075 742e 2020 5365 6520 he output. See \n+000127e0: e280 9850 5354 7269 636b 73e2 8099 0a64 ...PSTricks....d\n+000127f0: 6f63 756d 656e 7461 7469 6f6e 2066 6f72 ocumentation for\n+00012800: 20e2 8098 7073 7069 6374 7572 65e2 8099 ...pspicture...\n+00012810: 2e0a 0a20 2020 496e 2074 6865 2073 6563 ... In the sec\n+00012820: 6f6e 6420 666f 726d 2c20 7468 6520 4241 ond form, the BA\n+00012830: 5345 4c49 4e45 2066 7261 6374 696f 6e20 SELINE fraction \n+00012840: 6973 206f 7074 696f 6e61 6c2c 2061 6e64 is optional, and\n+00012850: 2074 6865 2074 776f 0a70 6f69 6e74 7320 the two.points \n+00012860: 7468 6174 2066 6f6c 6c6f 7720 6465 6669 that follow defi\n+00012870: 6e65 2074 6865 2064 6961 676f 6e61 6c20 ne the diagonal \n+00012880: 6f66 2061 2074 6872 6565 2d64 696d 656e of a three-dimen\n+00012890: 7369 6f6e 616c 2062 6f75 6e64 696e 670a sional bounding.\n+000128a0: 626f 7820 666f 7220 7468 6520 636f 6d70 box for the comp\n+000128b0: 6c65 7465 6420 7363 656e 652e 2020 5468 leted scene. Th\n+000128c0: 6520 7061 7261 6c6c 656c 2070 726f 6a65 e parallel proje\n+000128d0: 6374 696f 6e20 6f66 2074 6865 2062 6f75 ction of the bou\n+000128e0: 6e64 696e 670a 626f 7820 6465 7465 726d nding.box determ\n+000128f0: 696e 6573 2074 6865 2063 6f72 6e65 7273 ines the corners\n+00012900: 206f 6620 7468 6520 6472 6177 696e 6727 of the drawing'\n+00012910: 7320 e280 9870 7370 6963 7475 7265 2ae2 s ...pspicture*.\n+00012920: 8099 2065 6e76 6972 6f6e 6d65 6e74 2c0a .. environment,.\n+00012930: 7768 6963 6820 6973 2075 7365 6420 696e which is used in\n+00012940: 2070 6c61 6365 206f 6620 e280 9870 7370 place of ...psp\n+00012950: 6963 7475 7265 e280 992e 2020 5468 6973 icture.... This\n+00012960: 2063 6175 7365 7320 506f 7374 5363 7269 causes PostScri\n+00012970: 7074 2074 6f20 636c 6970 0a74 6865 2066 pt to clip.the f\n+00012980: 696e 616c 2064 7261 7769 6e67 2074 6f20 inal drawing to \n+00012990: 7468 6520 626f 756e 6469 6e67 2062 6f78 the bounding box\n+000129a0: 2069 6e20 3264 2e20 2049 6620 7468 6572 in 2d. If ther\n+000129b0: 6520 6973 2061 20e2 8098 6361 6d65 7261 e is a ...camera\n+000129c0: e280 990a 7370 6563 6966 6965 642c 2074 ....specified, t\n+000129d0: 6865 2063 616d 6572 6120 7472 616e 666f he camera tranfo\n+000129e0: 726d 6174 696f 6e20 6973 2061 7070 6c69 rmation is appli\n+000129f0: 6564 2074 6f20 7468 6520 626f 756e 6469 ed to the boundi\n+00012a00: 6e67 2062 6f78 2c20 616e 640a 7468 6520 ng box, and.the \n+00012a10: e280 9870 7370 6963 7475 7265 e280 9920 ...pspicture... \n+00012a20: 6973 2073 6574 206a 7573 7420 6c61 7267 is set just larg\n+00012a30: 6520 656e 6f75 6768 2074 6f20 696e 636c e enough to incl\n+00012a40: 7564 6520 7468 6520 7472 616e 7366 6f72 ude the transfor\n+00012a50: 6d65 6420 626f 782e 0a0a 2020 2057 6865 med box... Whe\n+00012a60: 6e20 6e6f 2062 6f75 6e64 696e 6720 626f n no bounding bo\n+00012a70: 7820 6973 2067 6976 656e 2c20 e280 9873 x is given, ...s\n+00012a80: 6b65 7463 68e2 8099 2063 6f6d 7075 7465 ketch... compute\n+00012a90: 7320 6f6e 6520 6175 746f 6d61 7469 6361 s one automatica\n+00012aa0: 6c6c 792e 0a0a 1f0a 4669 6c65 3a20 736b lly.....File: sk\n+00012ab0: 6574 6368 2e69 6e66 6f2c 2020 4e6f 6465 etch.info, Node\n+00012ac0: 3a20 4672 616d 652c 2020 4e65 7874 3a20 : Frame, Next: \n+00012ad0: 4c61 6e67 7561 6765 2c20 2050 7265 763a Language, Prev:\n+00012ae0: 2050 6963 7475 7265 2062 6f78 2c20 2055 Picture box, U\n+00012af0: 703a 2047 6c6f 6261 6c20 656e 7669 726f p: Global enviro\n+00012b00: 6e6d 656e 740a 0a33 2e34 2e34 2046 7261 nment..3.4.4 Fra\n+00012b10: 6d65 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a me.-----------..\n+00012b20: 2a53 796e 7461 783a 2a0a 2020 2020 2066 *Syntax:*. f\n+00012b30: 7261 6d65 205b 4f50 5449 4f4e 535d 0a43 rame [OPTIONS].C\n+00012b40: 6175 7365 7320 6120 5c70 7366 7261 6d65 auses a \\psframe\n+00012b50: 626f 7820 746f 2073 7572 726f 756e 6420 box to surround \n+00012b60: 7468 6520 e280 9870 7370 6963 7475 7265 the ...pspicture\n+00012b70: e280 9920 656e 7669 726f 6e6d 656e 7420 ... environment \n+00012b80: 696e 2074 6865 0a6f 7574 7075 742e 2020 in the.output. \n+00012b90: 4966 204f 5054 494f 4e53 2061 7265 2070 If OPTIONS are p\n+00012ba0: 7265 7365 6e74 2c20 7468 6579 2061 7265 resent, they are\n+00012bb0: 2063 6f70 6965 6420 6173 2d69 732e 2020 copied as-is. \n+00012bc0: 4e6f 726d 616c 6c79 206f 6e65 0a77 6f75 Normally one.wou\n+00012bd0: 6c64 2077 616e 7420 746f 2073 6574 20e2 ld want to set .\n+00012be0: 8098 6c69 6e65 7769 6474 68e2 8099 2c20 ..linewidth..., \n+00012bf0: e280 986c 696e 6573 7479 6c65 e280 992c ...linestyle...,\n+00012c00: 20e2 8098 6c69 6e65 636f 6c6f 72e2 8099 ...linecolor...\n+00012c10: 2c20 6574 632e 2020 4966 0a6f 6d69 7474 , etc. If.omitt\n+00012c20: 6564 2c20 7468 656e 20e2 8098 6672 616d ed, then ...fram\n+00012c30: 6573 6570 3d30 7074 e280 9920 6973 2061 esep=0pt... is a\n+00012c40: 6464 6564 2073 6f20 7468 6174 2074 6865 dded so that the\n+00012c50: 2066 7261 6d65 2074 6967 6874 6c79 2068 frame tightly h\n+00012c60: 7567 7320 7468 650a e280 9870 7370 6963 ugs the....pspic\n+00012c70: 7475 7265 e280 992e 0a0a 1f0a 4669 6c65 ture........File\n+00012c80: 3a20 736b 6574 6368 2e69 6e66 6f2c 2020 : sketch.info, \n+00012c90: 4e6f 6465 3a20 4c61 6e67 7561 6765 2c20 Node: Language, \n+00012ca0: 2050 7265 763a 2046 7261 6d65 2c20 2055 Prev: Frame, U\n+00012cb0: 703a 2047 6c6f 6261 6c20 656e 7669 726f p: Global enviro\n+00012cc0: 6e6d 656e 740a 0a33 2e34 2e35 204c 616e nment..3.4.5 Lan\n+00012cd0: 6775 6167 650a 2d2d 2d2d 2d2d 2d2d 2d2d guage.----------\n+00012ce0: 2d2d 2d2d 0a0a 2020 2020 206c 616e 6775 ----.. langu\n+00012cf0: 6167 6520 7469 6b7a 0a20 2020 2020 6c61 age tikz. la\n+00012d00: 6e67 7561 6765 2074 696b 7a2c 2063 6f6e nguage tikz, con\n+00012d10: 7465 7874 0a20 2020 2020 6c61 6e67 7561 text. langua\n+00012d20: 6765 2070 7374 7269 636b 730a 2020 2020 ge pstricks. \n+00012d30: 206c 616e 6775 6167 6520 7073 7472 6963 language pstric\n+00012d40: 6b73 2c20 6c61 7465 780a 5365 7473 2074 ks, latex.Sets t\n+00012d50: 6865 206f 7574 7075 7420 6c61 6e67 7561 he output langua\n+00012d60: 6765 2067 656e 6572 6174 6564 2062 7920 ge generated by \n+00012d70: e280 9873 6b65 7463 68e2 8099 2e20 2054 ...sketch.... T\n+00012d80: 6865 2073 6574 206f 6620 6f70 7469 6f6e he set of option\n+00012d90: 730a 756e 6465 7273 746f 6f64 2062 7920 s.understood by \n+00012da0: 736b 6574 6368 2061 6c73 6f20 6368 616e sketch also chan\n+00012db0: 6765 732e 2020 466f 7220 6578 616d 706c ges. For exampl\n+00012dc0: 652c 2074 6865 20e2 8098 5053 5472 6963 e, the ...PSTric\n+00012dd0: 6b73 e280 9920 6f70 7469 6f6e 0ae2 8098 ks... option....\n+00012de0: 6c69 6e65 7769 6474 68e2 8099 2077 696c linewidth... wil\n+00012df0: 6c20 6e6f 7420 6265 2070 726f 7065 726c l not be properl\n+00012e00: 7920 6861 6e64 6c65 6420 6966 20e2 8098 y handled if ...\n+00012e10: 6c61 6e67 7561 6765 e280 9920 6973 2073 language... is s\n+00012e20: 6574 2074 6f20 e280 9874 696b 7ae2 8099 et to ...tikz...\n+00012e30: 2e0a 5369 6d69 6c61 726c 792c 2074 6865 ..Similarly, the\n+00012e40: 20e2 8098 5469 6b5a e280 9920 6f70 7469 ...TikZ... opti\n+00012e50: 6f6e 20e2 8098 6c69 6e65 2073 7479 6c65 on ...line style\n+00012e60: e280 9920 286e 6f74 6520 7468 6520 7370 ... (note the sp\n+00012e70: 6163 6529 2077 696c 6c20 6e6f 7420 6265 ace) will not be\n+00012e80: 0a70 726f 7065 726c 7920 6861 6e64 6c65 .properly handle\n+00012e90: 6420 6966 20e2 8098 6c61 6e67 7561 6765 d if ...language\n+00012ea0: e280 9920 6973 2073 6574 2074 6f20 e280 ... is set to ..\n+00012eb0: 9870 7374 7269 636b 73e2 8099 2e20 2049 .pstricks.... I\n+00012ec0: 6620 6e6f 206c 616e 6775 6167 6520 6973 f no language is\n+00012ed0: 0a73 7065 6369 6669 6564 2c20 7468 6520 .specified, the \n+00012ee0: 6465 6661 756c 7420 e280 9870 7374 7269 default ...pstri\n+00012ef0: 636b 73e2 8099 2069 7320 7573 6564 2e0a cks... is used..\n+00012f00: 0a20 2020 416e 206f 7074 696f 6e61 6c20 . An optional \n+00012f10: 636f 6d6d 6120 666f 6c6c 6f77 6564 2062 comma followed b\n+00012f20: 7920 e280 986c 6174 6578 e280 9920 6f72 y ...latex... or\n+00012f30: 20e2 8098 636f 6e74 6578 74e2 8099 2073 ...context... s\n+00012f40: 7065 6369 6669 6573 2074 6865 0a6d 6163 pecifies the.mac\n+00012f50: 726f 2070 6163 6b61 6765 2074 6861 7420 ro package that \n+00012f60: 7468 6520 6f75 7470 7574 2073 686f 756c the output shoul\n+00012f70: 6420 6173 7375 6d65 2e20 2054 6869 7320 d assume. This \n+00012f80: 6166 6665 6374 7320 7468 6520 e280 9870 affects the ...p\n+00012f90: 6963 7475 7265 e280 990a 656e 7669 726f icture....enviro\n+00012fa0: 6e6d 656e 7420 636f 6d6d 616e 6473 2065 nment commands e\n+00012fb0: 6d69 7474 6564 2061 6e64 2074 6865 2064 mitted and the d\n+00012fc0: 6f63 756d 656e 7420 7465 6d70 6c61 7465 ocument template\n+00012fd0: 2075 7365 6420 7769 7468 2074 6865 0ae2 used with the..\n+00012fe0: 8098 2d54 e280 9920 6f70 7469 6f6e 2e20 ..-T... option. \n+00012ff0: 202a 4e6f 7465 2043 6f6d 6d61 6e64 206c *Note Command l\n+00013000: 696e 653a 3a2e 2020 4e6f 7465 2074 6861 ine::. Note tha\n+00013010: 7420 6174 2074 6865 2074 696d 6520 7468 t at the time th\n+00013020: 6973 206d 616e 7561 6c0a 7761 7320 6765 is manual.was ge\n+00013030: 6e65 7261 7465 642c 20e2 8098 5053 5472 nerated, ...PSTr\n+00013040: 6963 6b73 e280 9920 7761 7320 6e6f 7420 icks... was not \n+00013050: 7375 7070 6f72 7465 6420 6279 204c 6154 supported by LaT\n+00013060: 6558 206f 7220 6279 20e2 8098 436f 6e54 eX or by ...ConT\n+00013070: 6558 74e2 8099 2e0a 0a1f 0a46 696c 653a eXt........File:\n+00013080: 2073 6b65 7463 682e 696e 666f 2c20 204e sketch.info, N\n+00013090: 6f64 653a 2042 7569 6c64 696e 6720 6120 ode: Building a \n+000130a0: 6472 6177 696e 672c 2020 4e65 7874 3a20 drawing, Next: \n+000130b0: 436f 6d6d 616e 6420 6c69 6e65 2c20 2050 Command line, P\n+000130c0: 7265 763a 2049 6e70 7574 206c 616e 6775 rev: Input langu\n+000130d0: 6167 652c 2020 5570 3a20 546f 700a 0a34 age, Up: Top..4\n+000130e0: 2042 7569 6c64 696e 6720 6120 6472 6177 Building a draw\n+000130f0: 696e 670a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ing.************\n+00013100: 2a2a 2a2a 2a2a 2a2a 0a0a 5375 6363 6573 ********..Succes\n+00013110: 7366 756c 2064 7261 7769 6e67 7320 7769 sful drawings wi\n+00013120: 7468 20e2 8098 736b 6574 6368 e280 9920 th ...sketch... \n+00013130: 616e 6420 7769 7468 2061 6e79 2073 6365 and with any sce\n+00013140: 6e65 2064 6573 6372 6970 7469 6f6e 0a6c ne description.l\n+00013150: 616e 6775 6167 6520 7265 7175 6972 6520 anguage require \n+00013160: 7468 6174 2074 6865 2075 7365 7220 6465 that the user de\n+00013170: 7665 6c6f 7020 616e 2061 6363 7572 6174 velop an accurat\n+00013180: 6520 6d65 6e74 616c 2070 6963 7475 7265 e mental picture\n+00013190: 206f 6620 6865 720a 636f 6465 2061 6e64 of her.code and\n+000131a0: 2069 7473 206d 6561 6e69 6e67 2e20 2054 its meaning. T\n+000131b0: 6869 7320 696d 6167 6520 6973 2062 6573 his image is bes\n+000131c0: 7420 6275 696c 7420 696e 2073 6d61 6c6c t built in small\n+000131d0: 2070 6965 6365 732e 0a54 6865 7265 666f pieces..Therefo\n+000131e0: 7265 2c20 e280 9873 6b65 7463 68e2 8099 re, ...sketch...\n+000131f0: 2069 6e70 7574 7320 6172 6520 6265 7374 inputs are best\n+00013200: 2063 7265 6174 6564 2069 6e20 736d 616c created in smal\n+00013210: 6c20 696e 6372 656d 656e 7473 2077 6974 l increments wit\n+00013220: 680a 6672 6571 7565 6e74 2070 6175 7365 h.frequent pause\n+00013230: 7320 746f 2063 6f6d 7069 6c65 2061 6e64 s to compile and\n+00013240: 2076 6965 7720 7468 6520 7265 7375 6c74 view the result\n+00013250: 732e 2020 4361 7265 6675 6c20 636f 6d6d s. Careful comm\n+00013260: 656e 7473 2069 6e0a 7468 6520 696e 7075 ents in.the inpu\n+00013270: 7420 6f66 7465 6e20 6865 6c70 2061 7320 t often help as \n+00013280: 6120 7363 656e 6520 6772 6f77 7320 696e a scene grows in\n+00013290: 2063 6f6d 706c 6578 6974 792e 0a0a 2a20 complexity...* \n+000132a0: 4d65 6e75 3a0a 0a2a 204f 7665 7276 6965 Menu:..* Overvie\n+000132b0: 773a 3a20 2020 2020 2020 2020 2020 2020 w:: \n+000132c0: 2020 2020 2020 2042 7569 6c64 696e 6720 Building \n+000132d0: 6120 7375 6273 7461 6e74 6961 6c20 6472 a substantial dr\n+000132e0: 6177 696e 672e 0a2a 2041 2074 6563 686e awing..* A techn\n+000132f0: 6963 616c 2064 7261 7769 6e67 3a3a 2020 ical drawing:: \n+00013300: 2020 2020 2020 2041 6e20 6578 616d 706c An exampl\n+00013310: 6520 7769 7468 2066 696e 6520 706c 6163 e with fine plac\n+00013320: 656d 656e 742e 0a2a 2041 2068 6965 7261 ement..* A hiera\n+00013330: 7263 6869 6361 6c20 6d6f 6465 6c3a 3a20 rchical model:: \n+00013340: 2020 2020 2020 2041 6e20 6578 616d 706c An exampl\n+00013350: 6520 7769 7468 2073 7765 6570 7320 616e e with sweeps an\n+00013360: 6420 7075 7473 2e0a 2a20 4361 7665 6174 d puts..* Caveat\n+00013370: 733a 3a20 2020 2020 2020 2020 2020 2020 s:: \n+00013380: 2020 2020 2020 2020 5768 6572 6520 7472 Where tr\n+00013390: 6f75 626c 6520 6361 6e20 6f63 6375 722e ouble can occur.\n+000133a0: 0a0a 1f0a 4669 6c65 3a20 736b 6574 6368 ....File: sketch\n+000133b0: 2e69 6e66 6f2c 2020 4e6f 6465 3a20 4f76 .info, Node: Ov\n+000133c0: 6572 7669 6577 2c20 204e 6578 743a 2041 erview, Next: A\n+000133d0: 2074 6563 686e 6963 616c 2064 7261 7769 technical drawi\n+000133e0: 6e67 2c20 2050 7265 763a 2042 7569 6c64 ng, Prev: Build\n+000133f0: 696e 6720 6120 6472 6177 696e 672c 2020 ing a drawing, \n+00013400: 5570 3a20 4275 696c 6469 6e67 2061 2064 Up: Building a d\n+00013410: 7261 7769 6e67 0a0a 342e 3120 4f76 6572 rawing..4.1 Over\n+00013420: 7669 6577 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d view.===========\n+00013430: 3d0a 0a41 7320 616e 206f 7665 7276 6965 =..As an overvie\n+00013440: 772c 206c 6574 2773 2064 6576 656c 6f70 w, let's develop\n+00013450: 2061 2064 6961 6772 616d 2074 6861 7420 a diagram that \n+00013460: 7368 6f77 7320 686f 7720 6120 7065 7273 shows how a pers\n+00013470: 7065 6374 6976 650a 7072 6f6a 6563 7469 pective.projecti\n+00013480: 6f6e 2074 7261 6e73 666f 726d 2077 6f72 on transform wor\n+00013490: 6b73 2e20 2057 6527 6c6c 2073 7461 7274 ks. We'll start\n+000134a0: 2077 6974 6820 7468 6520 7472 6164 6974 with the tradit\n+000134b0: 696f 6e61 6c20 7265 6665 7265 6e63 650a ional reference.\n+000134c0: 6f62 6a65 6374 2075 7365 6420 696e 2063 object used in c\n+000134d0: 6f6d 7075 7465 7220 6772 6170 6869 6373 omputer graphics\n+000134e0: 2074 6578 7462 6f6f 6b73 2c20 6120 686f textbooks, a ho\n+000134f0: 7573 652d 7368 6170 6564 2070 7269 736d use-shaped prism\n+00013500: 2e20 2042 6567 696e 0a62 7920 6465 6669 . Begin.by defi\n+00013510: 6e69 6e67 2074 6865 2070 6f69 6e74 7320 ning the points \n+00013520: 6f66 2074 6865 2068 6f75 7365 2e20 2052 of the house. R\n+00013530: 6174 6865 7220 7468 616e 2064 6566 696e ather than defin\n+00013540: 696e 6720 7468 6520 6661 6365 7320 6f66 ing the faces of\n+00013550: 0a74 6865 2068 6f75 7365 2061 7320 706f .the house as po\n+00013560: 6c79 676f 6e73 2061 6e64 2074 7261 6e73 lygons and trans\n+00013570: 666f 726d 696e 6720 7468 6f73 652c 2077 forming those, w\n+00013580: 6520 6172 6520 676f 696e 6720 746f 2074 e are going to t\n+00013590: 7261 6e73 666f 726d 0a74 6865 2070 6f69 ransform.the poi\n+000135a0: 6e74 7320 7468 656d 7365 6c76 6573 2077 nts themselves w\n+000135b0: 6974 6820 e280 9873 6b65 7463 68e2 8099 ith ...sketch...\n+000135c0: 2061 7269 7468 6d65 7469 6320 736f 2074 arithmetic so t\n+000135d0: 6861 7420 7765 2068 6176 6520 6e61 6d65 hat we have name\n+000135e0: 7320 666f 720a 7468 6520 7472 616e 7366 s for.the transf\n+000135f0: 6f72 6d65 6420 706f 696e 7473 206c 6174 ormed points lat\n+00013600: 6572 2e0a 2020 2520 7269 6768 7420 7369 er.. % right si\n+00013610: 6465 2028 6f75 7473 6964 6520 746f 2072 de (outside to r\n+00013620: 6967 6874 290a 2020 6465 6620 5231 2028 ight). def R1 (\n+00013630: 312c 312c 3129 2064 6566 2052 3220 2831 1,1,1) def R2 (1\n+00013640: 2c2d 312c 3129 2064 6566 2052 3320 2831 ,-1,1) def R3 (1\n+00013650: 2c2d 312c 2d31 2920 6465 6620 5234 2028 ,-1,-1) def R4 (\n+00013660: 312c 312c 2d31 290a 2020 6465 6620 5235 1,1,-1). def R5\n+00013670: 2028 312c 312e 352c 3029 0a0a 2020 2520 (1,1.5,0).. % \n+00013680: 6c65 6674 2073 6964 6520 286f 7574 7369 left side (outsi\n+00013690: 6465 2074 6f20 7269 6768 742d 2d62 6163 de to right--bac\n+000136a0: 6b77 6172 6429 0a20 2064 6566 2057 2020 kward). def W \n+000136b0: 5b32 2c30 2c30 5d0a 2020 6465 6620 4c31 [2,0,0]. def L1\n+000136c0: 2028 5231 292d 5b57 5d20 6465 6620 4c32 (R1)-[W] def L2\n+000136d0: 2028 5232 292d 5b57 5d20 6465 6620 4c33 (R2)-[W] def L3\n+000136e0: 2028 5233 292d 5b57 5d20 6465 6620 4c34 (R3)-[W] def L4\n+000136f0: 2028 5234 292d 5b57 5d0a 2020 6465 6620 (R4)-[W]. def \n+00013700: 4c35 2028 5235 292d 5b57 5d0a 546f 2061 L5 (R5)-[W].To a\n+00013710: 6464 2061 2064 6f6f 7220 746f 2074 6865 dd a door to the\n+00013720: 2068 6f75 7365 2c20 7765 2075 7365 2061 house, we use a\n+00013730: 2070 6f6c 7967 6f6e 2073 6c69 6768 746c polygon slightl\n+00013740: 7920 696e 2066 726f 6e74 206f 6620 7468 y in front of th\n+00013750: 650a 666f 7265 6d6f 7374 2066 6163 6520 e.foremost face \n+00013760: 6f66 2074 6865 2068 6f75 7365 2e0a 2020 of the house.. \n+00013770: 2520 646f 6f72 0a20 2064 6566 2065 202e % door. def e .\n+00013780: 3031 0a20 2064 6566 2044 3120 2830 2c2d 01. def D1 (0,-\n+00013790: 312c 312b 6529 2064 6566 2044 3220 282e 1,1+e) def D2 (.\n+000137a0: 352c 2d31 2c31 2b65 2920 6465 6620 4433 5,-1,1+e) def D3\n+000137b0: 2028 2e35 2c30 2c31 2b65 2920 6465 6620 (.5,0,1+e) def \n+000137c0: 4434 2028 302c 302c 312b 6529 0a4e 6f77 D4 (0,0,1+e).Now\n+000137d0: 206c 6574 2773 2063 7265 6174 6520 6120 let's create a \n+000137e0: 6e65 7720 7365 7420 6f66 2070 6f69 6e74 new set of point\n+000137f0: 7320 7468 6174 2061 7265 2061 2074 6f2d s that are a to-\n+00013800: 6265 2d64 6574 6572 6d69 6e65 640a 7472 be-determined.tr\n+00013810: 616e 7366 6f72 6d20 6f66 2074 6865 206f ansform of the o\n+00013820: 7269 6769 6e61 6c73 2e0a 2020 6465 6620 riginals.. def \n+00013830: 6870 2073 6361 6c65 2831 2920 2520 686f hp scale(1) % ho\n+00013840: 7573 6520 706f 7369 7469 6f6e 6572 0a20 use positioner. \n+00013850: 2064 6566 2070 5231 205b 5b68 705d 5d2a def pR1 [[hp]]*\n+00013860: 2852 3129 2064 6566 2070 5232 205b 5b68 (R1) def pR2 [[h\n+00013870: 705d 5d2a 2852 3229 2064 6566 2070 5233 p]]*(R2) def pR3\n+00013880: 205b 5b68 705d 5d2a 2852 3329 0a20 2064 [[hp]]*(R3). d\n+00013890: 6566 2070 5234 205b 5b68 705d 5d2a 2852 ef pR4 [[hp]]*(R\n+000138a0: 3429 2064 6566 2070 5235 205b 5b68 705d 4) def pR5 [[hp]\n+000138b0: 5d2a 2852 3529 0a20 2064 6566 2070 4c31 ]*(R5). def pL1\n+000138c0: 205b 5b68 705d 5d2a 284c 3129 2064 6566 [[hp]]*(L1) def\n+000138d0: 2070 4c32 205b 5b68 705d 5d2a 284c 3229 pL2 [[hp]]*(L2)\n+000138e0: 2064 6566 2070 4c33 205b 5b68 705d 5d2a def pL3 [[hp]]*\n+000138f0: 284c 3329 0a20 2064 6566 2070 4c34 205b (L3). def pL4 [\n+00013900: 5b68 705d 5d2a 284c 3429 2064 6566 2070 [hp]]*(L4) def p\n+00013910: 4c35 205b 5b68 705d 5d2a 284c 3529 0a20 L5 [[hp]]*(L5). \n+00013920: 2064 6566 2070 4431 205b 5b68 705d 5d2a def pD1 [[hp]]*\n+00013930: 2844 3129 2064 6566 2070 4432 205b 5b68 (D1) def pD2 [[h\n+00013940: 705d 5d2a 2844 3229 2064 6566 2070 4433 p]]*(D2) def pD3\n+00013950: 205b 5b68 705d 5d2a 2844 3329 0a20 2064 [[hp]]*(D3). d\n+00013960: 6566 2070 4434 205b 5b68 705d 5d2a 2844 ef pD4 [[hp]]*(D\n+00013970: 3429 200a 4e6f 7465 2074 6865 2075 7365 4) .Note the use\n+00013980: 206f 6620 6120 e280 9c74 7261 6e73 666f of a ...transfo\n+00013990: 726d 2064 6566 696e 6974 696f 6ee2 809d rm definition...\n+000139a0: 2061 6e64 20e2 809c 7472 616e 7366 6f72 and ...transfor\n+000139b0: 6d20 7265 6665 7265 6e63 6573 e280 9d2e m references....\n+000139c0: 0a4e 6f77 2064 6566 696e 6520 7468 6520 .Now define the \n+000139d0: 7365 7665 6e20 706f 6c79 676f 6e61 6c20 seven polygonal \n+000139e0: 6661 6365 7320 6f66 2074 6865 2068 6f75 faces of the hou\n+000139f0: 7365 2061 6e64 2074 6865 2064 6f6f 7220 se and the door \n+00013a00: 7573 696e 6720 7468 650a 7472 616e 7366 using the.transf\n+00013a10: 6f72 6d65 6420 706f 696e 7473 2061 7320 ormed points as \n+00013a20: 7665 7274 6963 6573 2e20 2042 6520 6361 vertices. Be ca\n+00013a30: 7265 6675 6c20 7769 7468 2076 6572 7465 reful with verte\n+00013a40: 7820 6f72 6465 7221 0a20 2064 6566 2072 x order!. def r\n+00013a50: 6774 2070 6f6c 7967 6f6e 2028 7052 3129 gt polygon (pR1)\n+00013a60: 2870 5232 2928 7052 3329 2870 5234 2928 (pR2)(pR3)(pR4)(\n+00013a70: 7052 3529 0a20 2064 6566 206c 6674 2070 pR5). def lft p\n+00013a80: 6f6c 7967 6f6e 2028 704c 3529 2870 4c34 olygon (pL5)(pL4\n+00013a90: 2928 704c 3329 2870 4c32 2928 704c 3129 )(pL3)(pL2)(pL1)\n+00013aa0: 0a20 2064 6566 2066 7274 2070 6f6c 7967 . def frt polyg\n+00013ab0: 6f6e 2028 7052 3229 2870 5231 2928 704c on (pR2)(pR1)(pL\n+00013ac0: 3129 2870 4c32 290a 2020 6465 6620 6263 1)(pL2). def bc\n+00013ad0: 6b20 706f 6c79 676f 6e20 2870 5234 2928 k polygon (pR4)(\n+00013ae0: 7052 3329 2870 4c33 2928 704c 3429 0a20 pR3)(pL3)(pL4). \n+00013af0: 2064 6566 2074 6672 2070 6f6c 7967 6f6e def tfr polygon\n+00013b00: 2028 7052 3129 2870 5235 2928 704c 3529 (pR1)(pR5)(pL5)\n+00013b10: 2870 4c31 290a 2020 6465 6620 7462 6b20 (pL1). def tbk \n+00013b20: 706f 6c79 676f 6e20 2870 5235 2928 7052 polygon (pR5)(pR\n+00013b30: 3429 2870 4c34 2928 704c 3529 0a20 2064 4)(pL4)(pL5). d\n+00013b40: 6566 2062 6f74 2070 6f6c 7967 6f6e 2028 ef bot polygon (\n+00013b50: 7052 3229 2870 4c32 2928 704c 3329 2870 pR2)(pL2)(pL3)(p\n+00013b60: 5233 290a 2020 6465 6620 646f 6f72 2070 R3). def door p\n+00013b70: 6f6c 7967 6f6e 5b66 696c 6c63 6f6c 6f72 olygon[fillcolor\n+00013b80: 3d62 726f 776e 5d20 2870 4431 2928 7044 =brown] (pD1)(pD\n+00013b90: 3229 2870 4433 2928 7044 3429 0a20 2064 2)(pD3)(pD4). d\n+00013ba0: 6566 2068 6f75 7365 207b 207b 7267 747d ef house { {rgt}\n+00013bb0: 7b6c 6674 7d7b 6672 747d 7b62 636b 7d7b {lft}{frt}{bck}{\n+00013bc0: 7466 727d 7b74 626b 7d7b 626f 747d 7b64 tfr}{tbk}{bot}{d\n+00013bd0: 6f6f 727d 207d 0a20 2020 5469 6d65 2066 oor} }. Time f\n+00013be0: 6f72 2061 2073 616e 6974 7920 6368 6563 or a sanity chec\n+00013bf0: 6b2e 2020 4164 6420 7468 6520 6c69 6e65 k. Add the line\n+00013c00: 0a20 207b 686f 7573 657d 0a61 6e64 2074 . {house}.and t\n+00013c10: 6869 7320 6973 2077 6861 7420 7765 2067 his is what we g\n+00013c20: 6574 2e0a 0a00 085b 696d 6167 6520 7372 et.....[image sr\n+00013c30: 633d 2265 7831 3330 2e70 6e67 2220 7465 c=\"ex130.png\" te\n+00013c40: 7874 3d22 496d 6167 6520 6578 3133 3020 xt=\"Image ex130 \n+00013c50: 6f6d 6974 7465 6420 696e 2074 6578 7420 omitted in text \n+00013c60: 7665 7273 696f 6e20 6f66 2074 6869 7320 version of this \n+00013c70: 646f 6375 6d65 6e74 2e22 0008 5d0a 0a54 document.\"..]..T\n+00013c80: 6869 7320 6973 2063 6f72 7265 6374 2c20 his is correct, \n+00013c90: 6275 7420 646f 6573 206e 6f74 2072 6576 but does not rev\n+00013ca0: 6561 6c20 7665 7279 206d 7563 682e 2020 eal very much. \n+00013cb0: 436f 6d6d 6f6e 2065 7272 6f72 7320 6172 Common errors ar\n+00013cc0: 650a 6d69 7370 6c61 6365 6420 7665 7274 e.misplaced vert\n+00013cd0: 6963 6573 2061 6e64 2070 6f6c 7967 6f6e ices and polygon\n+00013ce0: 7320 6d69 7373 696e 6720 656e 7469 7265 s missing entire\n+00013cf0: 6c79 2064 7565 2074 6f20 696e 636f 7272 ly due to incorr\n+00013d00: 6563 7420 7665 7274 6578 0a6f 7264 6572 ect vertex.order\n+00013d10: 2e20 2054 6f20 7275 6c65 2074 6865 7365 . To rule these\n+00013d20: 206f 7574 2c20 6c65 7427 7320 696e 7370 out, let's insp\n+00013d30: 6563 7420 616c 6c20 7369 6465 7320 6f66 ect all sides of\n+00013d40: 2074 6865 2068 6f75 7365 2e20 2054 6869 the house. Thi\n+00013d50: 730a 6973 206e 6f74 2068 6172 642e 2020 s.is not hard. \n+00013d60: 4d65 7265 6c79 2072 6570 6c61 6365 2074 Merely replace t\n+00013d70: 6865 2072 6566 6572 656e 6365 207b 686f he reference {ho\n+00013d80: 7573 657d 2077 6974 6820 6120 e280 9872 use} with a ...r\n+00013d90: 6570 6561 74e2 8099 2e0a 2a4e 6f74 6520 epeat.....*Note \n+00013da0: 5265 7065 6174 733a 3a2e 0a20 2072 6570 Repeats::.. rep\n+00013db0: 6561 7420 7b20 3133 2c20 726f 7461 7465 eat { 13, rotate\n+00013dc0: 2833 302c 205b 312c 322c 335d 292c 2074 (30, [1,2,3]), t\n+00013dd0: 7261 6e73 6c61 7465 285b 332c 302c 305d ranslate([3,0,0]\n+00013de0: 2920 7d20 7b68 6f75 7365 7d0a 0008 5b69 ) } {house}...[i\n+00013df0: 6d61 6765 2073 7263 3d22 6578 3134 302e mage src=\"ex140.\n+00013e00: 706e 6722 2074 6578 743d 2249 6d61 6765 png\" text=\"Image\n+00013e10: 2065 7831 3430 206f 6d69 7474 6564 2069 ex140 omitted i\n+00013e20: 6e20 7465 7874 2076 6572 7369 6f6e 206f n text version o\n+00013e30: 6620 7468 6973 2064 6f63 756d 656e 742e f this document.\n+00013e40: 2200 085d 0a0a 4167 6169 6e20 7468 696e \"..]..Again thin\n+00013e50: 6773 206c 6f6f 6b20 636f 7272 6563 742e gs look correct.\n+00013e60: 2020 4e6f 7465 2074 6861 7420 7468 6520 Note that the \n+00013e70: 6869 6464 656e 2073 7572 6661 6365 2061 hidden surface a\n+00013e80: 6c67 6f72 6974 686d 0a68 616e 646c 6573 lgorithm.handles\n+00013e90: 2069 6e74 6572 7365 6374 696e 6720 706f intersecting po\n+00013ea0: 6c79 676f 6e73 2063 6f72 7265 6374 6c79 lygons correctly\n+00013eb0: 2077 6865 7265 2073 6f6d 6520 636f 7069 where some copi\n+00013ec0: 6573 206f 6620 7468 6520 686f 7573 650a es of the house.\n+00013ed0: 6f76 6572 6c61 702e 0a0a 2020 204c 6574 overlap... Let\n+00013ee0: 2773 206c 6179 206f 7574 2074 6865 2067 's lay out the g\n+00013ef0: 656f 6d65 7472 7920 6f66 2070 6572 7370 eometry of persp\n+00013f00: 6563 7469 7665 2070 726f 6a65 6374 696f ective projectio\n+00013f10: 6e20 6f66 2074 6865 2068 6f75 7365 0a6f n of the house.o\n+00013f20: 6e74 6f20 6120 706c 616e 6520 7769 7468 nto a plane with\n+00013f30: 2072 6179 7320 7061 7373 696e 6720 7468 rays passing th\n+00013f40: 726f 7567 6820 7468 6520 6f72 6967 696e rough the origin\n+00013f50: 2e20 2042 6567 696e 2062 7920 706f 7369 . Begin by posi\n+00013f60: 7469 6f6e 696e 670a 7468 6520 686f 7573 tioning.the hous\n+00013f70: 6520 7477 656c 7665 2075 6e69 7473 2062 e twelve units b\n+00013f80: 6163 6b20 6f6e 2074 6865 206e 6567 6174 ack on the negat\n+00013f90: 6976 6520 7a2d 6178 6973 2061 6e64 2061 ive z-axis and a\n+00013fa0: 6464 696e 6720 6120 7365 7420 6f66 0a63 dding a set of.c\n+00013fb0: 6f6f 7264 696e 6174 6520 6178 6573 2e20 oordinate axes. \n+00013fc0: 2054 6f20 6d6f 7665 2074 6865 2068 6f75 To move the hou\n+00013fd0: 7365 2077 6520 6e65 6564 206f 6e6c 7920 se we need only \n+00013fe0: 6368 616e 6765 2074 6865 2022 686f 7573 change the \"hous\n+00013ff0: 650a 706f 7369 7469 6f6e 696e 6722 2074 e.positioning\" t\n+00014000: 7261 6e73 666f 726d 2064 6566 696e 6564 ransform defined\n+00014010: 2065 6172 6c69 6572 2e0a 2020 6465 6620 earlier.. def \n+00014020: 6870 2072 6f74 6174 6528 2d34 302c 205b hp rotate(-40, [\n+00014030: 302c 312c 305d 2920 7468 656e 2074 7261 0,1,0]) then tra\n+00014040: 6e73 6c61 7465 285b 302c 302c 2d31 325d nslate([0,0,-12]\n+00014050: 290a 2020 6465 6620 6178 6573 207b 0a20 ). def axes {. \n+00014060: 2020 2064 6566 2073 7a20 310a 2020 2020 def sz 1. \n+00014070: 6c69 6e65 205b 6172 726f 7773 3d3c 2d3e line [arrows=<->\n+00014080: 5d20 2873 7a2c 302c 3029 284f 2928 302c ] (sz,0,0)(O)(0,\n+00014090: 737a 2c30 290a 2020 2020 6c69 6e65 205b sz,0). line [\n+000140a0: 6172 726f 7773 3d2d 3e5d 2020 284f 2928 arrows=->] (O)(\n+000140b0: 302c 302c 737a 290a 2020 2020 6c69 6e65 0,0,sz). line\n+000140c0: 205b 6c69 6e65 7769 6474 683d 2e32 7074 [linewidth=.2pt\n+000140d0: 2c6c 696e 6563 6f6c 6f72 3d62 6c75 652c ,linecolor=blue,\n+000140e0: 6c69 6e65 7374 796c 653d 6461 7368 6564 linestyle=dashed\n+000140f0: 5d20 284f 2928 302c 302c 2d31 3029 0a20 ] (O)(0,0,-10). \n+00014100: 2020 2073 7065 6369 616c 207c 5c75 7075 special |\\upu\n+00014110: 745b 725d 2331 7b24 7824 7d5c 7570 7574 t[r]#1{$x$}\\uput\n+00014120: 5b75 5d23 327b 2479 247d 5c75 7075 745b [u]#2{$y$}\\uput[\n+00014130: 6c5d 2333 7b24 7a24 7d7c 0a20 2020 2020 l]#3{$z$}|. \n+00014140: 2028 737a 2c30 2c30 2928 302c 737a 2c30 (sz,0,0)(0,sz,0\n+00014150: 2928 302c 302c 737a 290a 2020 7d0a 0a20 )(0,0,sz). }.. \n+00014160: 2020 5469 6d65 2066 6f72 2061 6e6f 7468 Time for anoth\n+00014170: 6572 2074 6573 742e 2020 4c65 7427 7320 er test. Let's \n+00014180: 6275 696c 6420 6120 7265 616c 2076 6965 build a real vie\n+00014190: 7720 7472 616e 7366 6f72 6d2c 2063 7265 w transform, cre\n+000141a0: 6174 696e 6720 610a e280 9c76 6972 7475 ating a....virtu\n+000141b0: 616c 2063 616d 6572 61e2 809d 2074 6f20 al camera... to \n+000141c0: 6c6f 6f6b 2061 7420 7468 6520 7363 656e look at the scen\n+000141d0: 6520 7765 2061 7265 2063 6f6e 7374 7275 e we are constru\n+000141e0: 6374 696e 672e 2020 5265 706c 6163 6520 cting. Replace \n+000141f0: 7468 650a e280 9872 6570 6561 74e2 8099 the....repeat...\n+00014200: 2077 6974 680a 2020 6465 6620 6579 6520 with. def eye \n+00014210: 2831 302c 342c 3130 290a 2020 6465 6620 (10,4,10). def \n+00014220: 6c6f 6f6b 5f61 7420 2830 2c30 2c2d 3529 look_at (0,0,-5)\n+00014230: 0a20 2070 7574 207b 2076 6965 7728 2865 . put { view((e\n+00014240: 7965 292c 2028 6c6f 6f6b 5f61 7429 2920 ye), (look_at)) \n+00014250: 7d20 7b20 7b68 6f75 7365 7d7b 6178 6573 } { {house}{axes\n+00014260: 7d20 7d0a 2020 2054 6865 20e2 809c 7669 } }. The ...vi\n+00014270: 6577 2074 7261 6e73 666f 726d e280 9d20 ew transform... \n+00014280: 7265 706f 7369 7469 6f6e 7320 7468 6520 repositions the \n+00014290: 7363 656e 6520 736f 2074 6861 7420 7468 scene so that th\n+000142a0: 6520 706f 696e 7420 e280 9865 7965 e280 e point ...eye..\n+000142b0: 9920 6973 0a61 7420 7468 6520 6f72 6967 . is.at the orig\n+000142c0: 696e 2061 6e64 2074 6865 2064 6972 6563 in and the direc\n+000142d0: 7469 6f6e 2066 726f 6d20 e280 9865 7965 tion from ...eye\n+000142e0: e280 9920 746f 20e2 8098 6c6f 6f6b 5f61 ... to ...look_a\n+000142f0: 74e2 8099 2069 7320 7468 6520 6e65 6761 t... is the nega\n+00014300: 7469 7665 0a7a 2d61 7869 732e 2020 5468 tive.z-axis. Th\n+00014310: 6973 2072 6571 7569 7265 7320 6120 726f is requires a ro\n+00014320: 7461 7469 6f6e 2061 6e64 2061 2074 7261 tation and a tra\n+00014330: 6e73 6c61 7469 6f6e 2074 6861 7420 6172 nslation that ar\n+00014340: 6520 616c 6c20 7061 636b 6564 0a69 6e74 e all packed.int\n+00014350: 6f20 7468 6520 636f 6e73 7472 7563 746f o the constructo\n+00014360: 7220 e280 9876 6965 77e2 8099 2e0a 0a00 r ...view.......\n+00014370: 085b 696d 6167 6520 7372 633d 2265 7831 .[image src=\"ex1\n+00014380: 3530 2e70 6e67 2220 7465 7874 3d22 496d 50.png\" text=\"Im\n+00014390: 6167 6520 6578 3135 3020 6f6d 6974 7465 age ex150 omitte\n+000143a0: 6420 696e 2074 6578 7420 7665 7273 696f d in text versio\n+000143b0: 6e20 6f66 2074 6869 7320 646f 6375 6d65 n of this docume\n+000143c0: 6e74 2e22 0008 5d0a 0a54 6869 7320 6973 nt.\"..]..This is\n+000143d0: 2073 7461 7274 696e 6720 746f 206c 6f6f starting to loo\n+000143e0: 6b20 676f 6f64 2120 2041 6464 2074 6865 k good! Add the\n+000143f0: 2070 726f 6a65 6374 696f 6e20 706c 616e projection plan\n+00014400: 6520 6861 6c66 2077 6179 0a62 6574 7765 e half way.betwe\n+00014410: 656e 2074 6865 206f 7269 6769 6e20 616e en the origin an\n+00014420: 6420 7468 6520 686f 7573 6520 6174 207a d the house at z\n+00014430: 3d2d 352e 2020 5765 276c 6c20 7472 7920 =-5. We'll try \n+00014440: 7468 6520 616e 676c 6520 6172 6775 6d65 the angle argume\n+00014450: 6e74 0a66 6561 7475 7265 206f 6620 e280 nt.feature of ..\n+00014460: 9873 7065 6369 616c e280 9920 746f 2070 .special... to p\n+00014470: 6f73 6974 696f 6e20 6120 6c61 6265 6c2e osition a label.\n+00014480: 0a20 2064 6566 2070 2035 2025 2070 726f . def p 5 % pro\n+00014490: 6a65 6374 696f 6e20 6469 7374 616e 6365 jection distance\n+000144a0: 2028 706c 616e 6520 6174 207a 3d2d 7029 (plane at z=-p)\n+000144b0: 0a20 2064 6566 2070 726f 6a65 6374 696f . def projectio\n+000144c0: 6e5f 706c 616e 6520 7b0a 2020 2020 6465 n_plane {. de\n+000144d0: 6620 737a 2031 2e35 0a20 2020 2070 6f6c f sz 1.5. pol\n+000144e0: 7967 6f6e 2028 2d73 7a2c 2d73 7a2c 2d70 ygon (-sz,-sz,-p\n+000144f0: 2928 737a 2c2d 737a 2c2d 7029 2873 7a2c )(sz,-sz,-p)(sz,\n+00014500: 737a 2c2d 7029 282d 737a 2c73 7a2c 2d70 sz,-p)(-sz,sz,-p\n+00014510: 290a 2020 2020 7370 6563 6961 6c20 7c5c ). special |\\\n+00014520: 7270 7574 5b62 5d23 312d 3223 337b 5c66 rput[b]#1-2#3{\\f\n+00014530: 6f6f 746e 6f74 6573 697a 655c 7366 2070 ootnotesize\\sf p\n+00014540: 726f 6a65 6374 696f 6e20 706c 616e 657d rojection plane}\n+00014550: 7c20 0a20 2020 2020 2028 2d73 7a2c 2d73 | . (-sz,-s\n+00014560: 7a2c 2d70 2928 737a 2c2d 737a 2c2d 7029 z,-p)(sz,-sz,-p)\n+00014570: 2830 2c2d 737a 2b2e 312c 2d70 290a 2020 (0,-sz+.1,-p). \n+00014580: 7d0a 4164 6420 7b70 726f 6a65 6374 696f }.Add {projectio\n+00014590: 6e5f 706c 616e 657d 2074 6f20 7468 6520 n_plane} to the \n+000145a0: 6c69 7374 206f 6620 6f62 6a65 6374 7320 list of objects \n+000145b0: 696e 2074 6865 20e2 8098 7075 74e2 8099 in the ...put...\n+000145c0: 2061 626f 7665 2e0a 0a00 085b 696d 6167 above.....[imag\n+000145d0: 6520 7372 633d 2265 7831 3630 2e70 6e67 e src=\"ex160.png\n+000145e0: 2220 7465 7874 3d22 496d 6167 6520 6578 \" text=\"Image ex\n+000145f0: 3136 3020 6f6d 6974 7465 6420 696e 2074 160 omitted in t\n+00014600: 6578 7420 7665 7273 696f 6e20 6f66 2074 ext version of t\n+00014610: 6869 7320 646f 6375 6d65 6e74 2e22 0008 his document.\"..\n+00014620: 5d0a 0a20 2020 5468 6520 7761 7920 7765 ].. The way we\n+00014630: 2063 6f6e 7374 7275 6374 6564 2074 6865 constructed the\n+00014640: 2070 6f69 6e74 7320 6f66 2074 6865 2068 points of the h\n+00014650: 6f75 7365 206e 6f77 206d 616b 6573 2069 ouse now makes i\n+00014660: 7420 6561 7379 2074 6f0a 6472 6177 2072 t easy to.draw r\n+00014670: 6179 7320 6f66 2070 726f 6a65 6374 696f ays of projectio\n+00014680: 6e2e 2020 5765 276c 6c20 6361 7374 206f n. We'll cast o\n+00014690: 6e65 2072 6179 2066 726f 6d20 6576 6572 ne ray from ever\n+000146a0: 7920 7669 7369 626c 6520 7665 7274 6578 y visible vertex\n+000146b0: 0a6f 6620 7468 6520 686f 7573 6520 616e .of the house an\n+000146c0: 6420 6465 6669 6e65 206f 7074 696f 6e73 d define options\n+000146d0: 2073 6f20 7468 6520 6170 7065 6172 616e so the appearan\n+000146e0: 6365 206f 6620 616c 6c20 7261 7973 2063 ce of all rays c\n+000146f0: 616e 2062 650a 6368 616e 6765 6420 6174 an be.changed at\n+00014700: 2074 6865 2073 616d 6520 7469 6d65 2e0a the same time..\n+00014710: 2020 6465 6620 7072 6f6a 6563 7469 6f6e def projection\n+00014720: 5f72 6179 7320 7b0a 2020 2020 6465 6620 _rays {. def \n+00014730: 7261 796f 7074 205b 6c69 6e65 7769 6474 rayopt [linewidt\n+00014740: 683d 2e33 7074 2c6c 696e 6563 6f6c 6f72 h=.3pt,linecolor\n+00014750: 3d6c 6967 6874 6772 6179 5d0a 2020 2020 =lightgray]. \n+00014760: 6c69 6e65 205b 7261 796f 7074 5d28 4f29 line [rayopt](O)\n+00014770: 2870 5231 2920 6c69 6e65 205b 7261 796f (pR1) line [rayo\n+00014780: 7074 5d28 4f29 2870 5232 2920 6c69 6e65 pt](O)(pR2) line\n+00014790: 5b72 6179 6f70 745d 284f 2928 7052 3329 [rayopt](O)(pR3)\n+000147a0: 0a20 2020 206c 696e 6520 5b72 6179 6f70 . line [rayop\n+000147b0: 745d 284f 2928 7052 3429 206c 696e 6520 t](O)(pR4) line \n+000147c0: 5b72 6179 6f70 745d 284f 2928 7052 3529 [rayopt](O)(pR5)\n+000147d0: 0a20 2020 206c 696e 6520 5b72 6179 6f70 . line [rayop\n+000147e0: 745d 284f 2928 704c 3129 206c 696e 6520 t](O)(pL1) line \n+000147f0: 5b72 6179 6f70 745d 284f 2928 704c 3229 [rayopt](O)(pL2)\n+00014800: 206c 696e 655b 7261 796f 7074 5d28 4f29 line[rayopt](O)\n+00014810: 2870 4c35 290a 2020 2020 6c69 6e65 205b (pL5). line [\n+00014820: 7261 796f 7074 5d28 4f29 2870 4431 2920 rayopt](O)(pD1) \n+00014830: 6c69 6e65 205b 7261 796f 7074 5d28 4f29 line [rayopt](O)\n+00014840: 2870 4432 2920 0a20 2020 206c 696e 6520 (pD2) . line \n+00014850: 5b72 6179 6f70 745d 284f 2928 7044 3329 [rayopt](O)(pD3)\n+00014860: 206c 696e 6520 5b72 6179 6f70 745d 284f line [rayopt](O\n+00014870: 2928 7044 3429 200a 2020 7d0a 5468 6520 )(pD4) . }.The \n+00014880: 7265 7375 6c74 2069 7320 7368 6f77 6e20 result is shown \n+00014890: 6865 7265 2e0a 0a00 085b 696d 6167 6520 here.....[image \n+000148a0: 7372 633d 2265 7831 3730 2e70 6e67 2220 src=\"ex170.png\" \n+000148b0: 7465 7874 3d22 496d 6167 6520 6578 3137 text=\"Image ex17\n+000148c0: 3020 6f6d 6974 7465 6420 696e 2074 6578 0 omitted in tex\n+000148d0: 7420 7665 7273 696f 6e20 6f66 2074 6869 t version of thi\n+000148e0: 7320 646f 6375 6d65 6e74 2e22 0008 5d0a s document.\"..].\n+000148f0: 0a54 6865 2072 6179 7320 7069 6572 6365 .The rays pierce\n+00014900: 2074 6865 2070 726f 6a65 6374 696f 6e20 the projection \n+00014910: 706c 616e 6520 6174 2074 6865 2063 6f72 plane at the cor\n+00014920: 7265 7370 6f6e 6469 6e67 2070 6f69 6e74 responding point\n+00014930: 7320 6f6e 2074 6865 0a70 6572 7370 6563 s on the.perspec\n+00014940: 7469 7665 2069 6d61 6765 2077 6520 6172 tive image we ar\n+00014950: 6520 7472 7969 6e67 2074 6f20 6472 6177 e trying to draw\n+00014960: 2e20 2041 6c62 7265 6368 7420 44c3 bc72 . Albrecht D..r\n+00014970: 6572 2061 6e64 2068 6973 0a52 656e 6169 er and his.Renai\n+00014980: 7373 616e 6365 2063 6f6e 7465 6d70 6f72 ssance contempor\n+00014990: 6172 6965 7320 6861 6420 7468 6520 7361 aries had the sa\n+000149a0: 6d65 2069 6465 6120 696e 2074 6865 2065 me idea in the e\n+000149b0: 6172 6c79 2031 3530 3027 732e 0a0a 0008 arly 1500's.....\n+000149c0: 5b69 6d61 6765 2073 7263 3d22 6475 6572 [image src=\"duer\n+000149d0: 6572 2e70 6e67 2220 7465 7874 3d22 496d er.png\" text=\"Im\n+000149e0: 6167 6520 6f66 2044 7565 7265 7227 7320 age of Duerer's \n+000149f0: 7065 7273 7065 6374 6976 6520 636f 6d70 perspective comp\n+00014a00: 7574 6572 206f 6d69 7474 6564 2069 6e20 uter omitted in \n+00014a10: 7465 7874 2076 6572 7369 6f6e 206f 6620 text version of \n+00014a20: 7468 6973 2064 6f63 756d 656e 742e 2200 this document.\".\n+00014a30: 085d 0a0a 2020 2041 6c6c 2074 6861 7427 .].. All that'\n+00014a40: 7320 6c65 6674 2069 7320 746f 2066 696e s left is to fin\n+00014a50: 6420 6120 7761 7920 746f 2063 6f6e 6e65 d a way to conne\n+00014a60: 6374 2074 6865 2070 6f69 6e74 7320 6f66 ct the points of\n+00014a70: 2074 6865 2068 6f75 7365 0a6f 6e20 7468 the house.on th\n+00014a80: 6520 7072 6f6a 6563 7469 6f6e 2070 6c61 e projection pla\n+00014a90: 6e65 2e20 2057 6520 636f 756c 6420 7075 ne. We could pu\n+00014aa0: 6c6c 206f 7574 2061 2067 6f6f 6420 636f ll out a good co\n+00014ab0: 6d70 7574 6572 2067 7261 7068 6963 730a mputer graphics.\n+00014ac0: 7465 7874 2c20 6669 6e64 2074 6865 206e text, find the n\n+00014ad0: 6563 6573 7361 7279 206d 6174 7269 782c ecessary matrix,\n+00014ae0: 2061 6e64 2065 6e74 6572 2069 7420 6f75 and enter it ou\n+00014af0: 7273 656c 7665 7320 6173 2061 2074 7261 rselves as a tra\n+00014b00: 6e73 666f 726d 0a6c 6974 6572 616c 2e20 nsform.literal. \n+00014b10: 202a 4e6f 7465 2054 7261 6e73 666f 726d *Note Transform\n+00014b20: 206c 6974 6572 616c 733a 3a2e 2020 5468 literals::. Th\n+00014b30: 6174 2077 6f72 6b20 6973 2061 6c72 6561 at work is alrea\n+00014b40: 6479 2064 6f6e 652c 0a68 6f77 6576 6572 dy done,.however\n+00014b50: 2e20 2057 6520 6361 6e20 7573 6520 7468 . We can use th\n+00014b60: 6520 e280 9870 726f 6a65 6374 2870 29e2 e ...project(p).\n+00014b70: 8099 2063 6f6e 7374 7275 6374 6f72 2e0a .. constructor..\n+00014b80: 0a20 2020 5468 6572 6520 6172 6520 7374 . There are st\n+00014b90: 696c 6c20 736f 6d65 2064 6574 6169 6c73 ill some details\n+00014ba0: 2074 6861 7420 7265 7175 6972 6520 6361 that require ca\n+00014bb0: 7265 2e20 2050 726f 6a65 6374 696f 6e20 re. Projection \n+00014bc0: 7769 6c6c 0a66 6c61 7474 656e 2077 6861 will.flatten wha\n+00014bd0: 7465 7665 7220 6973 2074 7261 6e73 666f tever is transfo\n+00014be0: 726d 6564 206f 6e74 6f20 7468 6520 706c rmed onto the pl\n+00014bf0: 616e 6520 7a3d 2d70 2e20 2054 6865 7265 ane z=-p. There\n+00014c00: 666f 7265 2061 6e79 2070 6172 740a 6f66 fore any part.of\n+00014c10: 2074 6865 2068 6f75 7365 2063 6f75 6c64 the house could\n+00014c20: 2064 6973 6170 7065 6172 2062 6568 696e disappear behin\n+00014c30: 6420 7468 6520 7072 6f6a 6563 7469 6f6e d the projection\n+00014c40: 2070 6c61 6e65 2028 7468 6520 6869 6464 plane (the hidd\n+00014c50: 656e 0a73 7572 6661 6365 2061 6c67 6f72 en.surface algor\n+00014c60: 6974 686d 206f 7264 6572 7320 6f62 6a65 ithm orders obje\n+00014c70: 6374 7320 6174 2074 6865 2073 616d 6520 cts at the same \n+00014c80: 6465 7074 6820 6172 6269 7472 6172 696c depth arbitraril\n+00014c90: 7929 2e20 2054 6865 0a64 6f6f 7220 6d61 y). The.door ma\n+00014ca0: 7920 616c 736f 2064 6973 6170 7065 6172 y also disappear\n+00014cb0: 2062 6568 696e 6420 7468 6520 6672 6f6e behind the fron\n+00014cc0: 7420 6f66 2074 6865 2068 6f75 7365 2e20 t of the house. \n+00014cd0: 2054 6f20 6d61 6b65 2073 7572 650a 6576 To make sure.ev\n+00014ce0: 6572 7974 6869 6e67 2072 656d 6169 6e73 erything remains\n+00014cf0: 2076 6973 6962 6c65 2c20 7765 276c 6c20 visible, we'll \n+00014d00: 706c 6163 6520 7468 6520 686f 7573 6520 place the house \n+00014d10: 6120 7469 6e79 2062 6974 2069 6e20 6672 a tiny bit in fr\n+00014d20: 6f6e 7420 6f66 0a74 6865 2070 726f 6a65 ont of.the proje\n+00014d30: 6374 696f 6e20 706c 616e 6520 616e 6420 ction plane and \n+00014d40: 6120 7365 636f 6e64 2063 6f70 7920 6f66 a second copy of\n+00014d50: 2074 6865 2064 6f6f 7220 696e 2066 726f the door in fro\n+00014d60: 6e74 206f 6620 7468 650a 686f 7573 652e nt of the.house.\n+00014d70: 0a20 2064 6566 2070 726f 6a65 6374 696f . def projectio\n+00014d80: 6e20 7b0a 2020 2020 2520 6520 6973 2061 n {. % e is a\n+00014d90: 2073 6d61 6c6c 206e 756d 6265 7220 6465 small number de\n+00014da0: 6669 6e65 6420 6162 6f76 650a 2020 2020 fined above. \n+00014db0: 7075 7420 7b20 7072 6f6a 6563 7428 7029 put { project(p)\n+00014dc0: 2074 6865 6e20 7472 616e 736c 6174 6528 then translate(\n+00014dd0: 5b30 2c30 2c31 2a65 5d29 207d 207b 686f [0,0,1*e]) } {ho\n+00014de0: 7573 657d 0a20 2020 2070 7574 207b 2070 use}. put { p\n+00014df0: 726f 6a65 6374 2870 2920 7468 656e 2074 roject(p) then t\n+00014e00: 7261 6e73 6c61 7465 285b 302c 302c 322a ranslate([0,0,2*\n+00014e10: 655d 2920 7d20 7b64 6f6f 727d 0a20 207d e]) } {door}. }\n+00014e20: 0a0a 0008 5b69 6d61 6765 2073 7263 3d22 ....[image src=\"\n+00014e30: 6578 3138 302e 706e 6722 2074 6578 743d ex180.png\" text=\n+00014e40: 2249 6d61 6765 2065 7831 3830 206f 6d69 \"Image ex180 omi\n+00014e50: 7474 6564 2069 6e20 7465 7874 2076 6572 tted in text ver\n+00014e60: 7369 6f6e 206f 6620 7468 6973 2064 6f63 sion of this doc\n+00014e70: 756d 656e 742e 2200 085d 0a0a 2020 2049 ument.\"..].. I\n+00014e80: 6620 796f 7520 6861 7665 2073 7475 6469 f you have studi\n+00014e90: 6564 2061 6e64 2075 6e64 6572 7374 616e ed and understan\n+00014ea0: 6420 616c 6c20 7468 6973 2c20 796f 7520 d all this, you \n+00014eb0: 6172 6520 7765 6c6c 206f 6e20 7468 6520 are well on the \n+00014ec0: 7761 790a 746f 2073 7563 6365 7373 2077 way.to success w\n+00014ed0: 6974 6820 e280 9873 6b65 7463 68e2 8099 ith ...sketch...\n+00014ee0: 2e20 204e 6f74 2073 686f 776e 2061 7265 . Not shown are\n+00014ef0: 2074 6865 2032 3020 6f72 2073 6f20 6974 the 20 or so it\n+00014f00: 6572 6174 696f 6e73 2074 6861 740a 7765 erations that.we\n+00014f10: 7265 2072 6571 7569 7265 6420 746f 2066 re required to f\n+00014f20: 696e 6420 6120 7265 6173 6f6e 6162 6c65 ind a reasonable\n+00014f30: 2076 6965 7769 6e67 2061 6e67 6c65 2061 viewing angle a\n+00014f40: 6e64 2068 6f75 7365 2070 6f73 6974 696f nd house positio\n+00014f50: 6e2c 0a65 7463 2e20 204e 6f6e 6574 6865 n,.etc. Nonethe\n+00014f60: 6c65 7373 2c20 7468 6973 2064 7261 7769 less, this drawi\n+00014f70: 6e67 2077 6173 2063 6f6d 706c 6574 6564 ng was completed\n+00014f80: 2069 6e20 6162 6f75 7420 616e 2068 6f75 in about an hou\n+00014f90: 722e 2020 5768 696c 6520 610a 4755 4920 r. While a.GUI \n+00014fa0: 746f 6f6c 206d 6179 2068 6176 6520 6265 tool may have be\n+00014fb0: 656e 2061 206c 6974 746c 6520 6661 7374 en a little fast\n+00014fc0: 6572 2c20 6974 2069 7320 756e 6c69 6b65 er, it is unlike\n+00014fd0: 6c79 2074 6861 7420 6120 6e65 770a 6472 ly that a new.dr\n+00014fe0: 6177 696e 672c 2069 7473 656c 6620 6120 awing, itself a \n+00014ff0: 7065 7273 7065 6374 6976 6520 7072 6f6a perspective proj\n+00015000: 6563 7469 6f6e 206f 6620 7468 6520 7363 ection of the sc\n+00015010: 656e 652c 2063 6f75 6c64 2062 650a 6765 ene, could be.ge\n+00015020: 6e65 7261 7465 6420 7769 7468 2074 776f nerated with two\n+00015030: 206d 6f72 6520 6d69 6e75 7465 7327 2077 more minutes' w\n+00015040: 6f72 6b21 2020 4a75 7374 2063 6861 6e67 ork! Just chang\n+00015050: 6520 7468 6520 7669 6577 2074 7261 6e73 e the view trans\n+00015060: 666f 726d 0a74 6f0a 2020 7075 7420 7b20 form.to. put { \n+00015070: 7669 6577 2828 6579 6529 2c20 286c 6f6f view((eye), (loo\n+00015080: 6b5f 6174 2929 2074 6865 6e20 7065 7273 k_at)) then pers\n+00015090: 7065 6374 6976 6528 3929 207d 207b 202e pective(9) } { .\n+000150a0: 2e2e 0a61 6e64 2070 726f 6475 6365 2074 ...and produce t\n+000150b0: 6869 732e 0a0a 0008 5b69 6d61 6765 2073 his.....[image s\n+000150c0: 7263 3d22 6578 3139 302e 706e 6722 2074 rc=\"ex190.png\" t\n+000150d0: 6578 743d 2249 6d61 6765 2065 7831 3930 ext=\"Image ex190\n+000150e0: 206f 6d69 7474 6564 2069 6e20 7465 7874 omitted in text\n+000150f0: 2076 6572 7369 6f6e 206f 6620 7468 6973 version of this\n+00015100: 2064 6f63 756d 656e 742e 2200 085d 0a0a document.\"..]..\n+00015110: 1f0a 4669 6c65 3a20 736b 6574 6368 2e69 ..File: sketch.i\n+00015120: 6e66 6f2c 2020 4e6f 6465 3a20 4120 7465 nfo, Node: A te\n+00015130: 6368 6e69 6361 6c20 6472 6177 696e 672c chnical drawing,\n+00015140: 2020 4e65 7874 3a20 4120 6869 6572 6172 Next: A hierar\n+00015150: 6368 6963 616c 206d 6f64 656c 2c20 2050 chical model, P\n+00015160: 7265 763a 204f 7665 7276 6965 772c 2020 rev: Overview, \n+00015170: 5570 3a20 4275 696c 6469 6e67 2061 2064 Up: Building a d\n+00015180: 7261 7769 6e67 0a0a 342e 3220 4120 7465 rawing..4.2 A te\n+00015190: 6368 6e69 6361 6c20 6472 6177 696e 670a chnical drawing.\n 000151a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================\n-000151b0: 3d3d 3d0a 0a4c 6574 2773 206c 6f6f 6b20 ===..Let's look \n-000151c0: 6174 2061 2064 7261 7769 6e67 2074 6861 at a drawing tha\n-000151d0: 7420 7265 7072 6573 656e 7473 2074 6865 t represents the\n-000151e0: 206b 696e 6420 6f66 2070 726f 626c 656d kind of problem\n-000151f0: 20e2 8098 736b 6574 6368 e280 9920 7761 ...sketch... wa\n-00015200: 730a 6d65 616e 7420 746f 2073 6f6c 7665 s.meant to solve\n-00015210: 2d2d 6120 7061 6972 206f 6620 7465 7874 --a pair of text\n-00015220: 626f 6f6b 2066 6967 7572 6573 2072 6567 book figures reg\n-00015230: 6172 6469 6e67 2061 2070 6f6c 7967 6f6e arding a polygon\n-00015240: 616c 0a61 7070 726f 7869 6d61 7469 6f6e al.approximation\n-00015250: 206f 6620 6120 7472 756e 6361 7465 6420 of a truncated \n-00015260: 636f 6e65 2e20 2048 6572 6520 6172 6520 cone. Here are \n-00015270: 7468 6520 7069 6374 7572 6573 2077 6520 the pictures we \n-00015280: 7769 6c6c 0a70 726f 6475 6365 2e0a 0a00 will.produce....\n-00015290: 085b 696d 6167 6520 7372 633d 2265 7832 .[image src=\"ex2\n-000152a0: 3530 2e70 6e67 2220 7465 7874 3d22 496d 50.png\" text=\"Im\n-000152b0: 6167 6520 6578 3235 3020 6f6d 6974 7465 age ex250 omitte\n-000152c0: 6420 696e 2074 6578 7420 7665 7273 696f d in text versio\n-000152d0: 6e20 6f66 2074 6869 7320 646f 6375 6d65 n of this docume\n-000152e0: 6e74 2e22 0008 5d20 2020 2020 0008 5b69 nt.\"..] ..[i\n-000152f0: 6d61 6765 2073 7263 3d22 6578 3236 302e mage src=\"ex260.\n-00015300: 706e 6722 2074 6578 743d 2249 6d61 6765 png\" text=\"Image\n-00015310: 2065 7832 3630 206f 6d69 7474 6564 2069 ex260 omitted i\n-00015320: 6e20 7465 7874 2076 6572 7369 6f6e 206f n text version o\n-00015330: 6620 7468 6973 2064 6f63 756d 656e 742e f this document.\n-00015340: 2200 085d 0a0a 2020 2054 6865 2063 6f6e \"..].. The con\n-00015350: 6520 7368 6170 6520 6973 206a 7573 7420 e shape is just \n-00015360: 6120 7377 6570 7420 6c69 6e65 2077 6974 a swept line wit\n-00015370: 6820 6e6f 2063 6c6f 7375 7265 2074 6167 h no closure tag\n-00015380: 2061 6e64 2063 756c 6c69 6e67 0a74 7572 and culling.tur\n-00015390: 6e65 6420 6f66 662e 2020 4265 6769 6e20 ned off. Begin \n-000153a0: 6279 2073 6574 7469 6e67 2075 7020 736f by setting up so\n-000153b0: 6d65 2075 7365 6675 6c20 636f 6e73 7461 me useful consta\n-000153c0: 6e74 732e 0a20 2064 6566 204f 2028 302c nts.. def O (0,\n-000153d0: 302c 3029 2064 6566 2049 205b 312c 302c 0,0) def I [1,0,\n-000153e0: 305d 2064 6566 204a 205b 302c 312c 305d 0] def J [0,1,0]\n-000153f0: 2064 6566 204b 205b 302c 302c 315d 0a20 def K [0,0,1]. \n-00015400: 2064 6566 2070 3020 2831 2c32 2920 6465 def p0 (1,2) de\n-00015410: 6620 7031 2028 312e 352c 3029 2064 6566 f p1 (1.5,0) def\n-00015420: 204e 2038 0a20 2064 6566 2073 6567 5f72 N 8. def seg_r\n-00015430: 6f74 2072 6f74 6174 6528 3336 3020 2f20 ot rotate(360 / \n-00015440: 4e2c 205b 4a5d 290a 5468 6520 706f 696e N, [J]).The poin\n-00015450: 7473 20e2 8098 7030 e280 9920 616e 6420 ts ...p0... and \n-00015460: e280 9870 31e2 8099 2061 7265 2074 6865 ...p1... are the\n-00015470: 2065 6e64 2070 6f69 6e74 7320 6f66 2074 end points of t\n-00015480: 6865 206c 696e 6520 746f 2062 6520 7377 he line to be sw\n-00015490: 6570 742e 0a54 6865 2064 6566 696e 6974 ept..The definit\n-000154a0: 696f 6e20 e280 9873 6567 5f72 6f74 e280 ion ...seg_rot..\n-000154b0: 9920 6973 2074 6865 2073 7765 6570 2074 . is the sweep t\n-000154c0: 7261 6e73 666f 726d 6174 696f 6e2e 2020 ransformation. \n-000154d0: 5769 7468 2074 6865 7365 2c20 7468 650a With these, the.\n-000154e0: 636f 6e65 2069 7473 656c 6620 6973 2073 cone itself is s\n-000154f0: 696d 706c 652e 0a20 2073 7765 6570 5b63 imple.. sweep[c\n-00015500: 756c 6c3d 6661 6c73 655d 207b 204e 2c20 ull=false] { N, \n-00015510: 5b5b 7365 675f 726f 745d 5d20 7d20 6c69 [[seg_rot]] } li\n-00015520: 6e65 2870 3029 2870 3129 0a0a 2020 2054 ne(p0)(p1).. T\n-00015530: 6865 2061 7865 7320 6172 6520 6e65 7874 he axes are next\n-00015540: 2061 6e64 2069 6e63 6c75 6465 2061 6e20 and include an \n-00015550: 696e 7465 7265 7369 6e67 2074 7269 636b interesing trick\n-00015560: 2074 6861 7420 7368 6f77 7320 7468 650a that shows the.\n-00015570: 6869 6464 656e 2070 6172 7473 2061 7320 hidden parts as \n-00015580: 646f 7474 6564 206c 696e 6573 2e20 2054 dotted lines. T\n-00015590: 6865 2073 6563 7265 7420 6973 2064 7261 he secret is dra\n-000155a0: 7720 7468 6520 6178 6573 2074 7769 6365 w the axes twice\n-000155b0: 2d2d 736f 6c69 640a 6c69 6e65 7320 7769 --solid.lines wi\n-000155c0: 7468 2074 6865 206e 6f72 6d61 6c20 6869 th the normal hi\n-000155d0: 6464 656e 2073 7572 6661 6365 2061 6c67 dden surface alg\n-000155e0: 6f72 6974 686d 2069 6e20 6566 6665 6374 orithm in effect\n-000155f0: 2c20 616e 6420 7468 656e 0a64 6f74 7465 , and then.dotte\n-00015600: 6420 7769 7468 2074 6865 206f 7074 696f d with the optio\n-00015610: 6e20 e280 986c 6179 3d6f 7665 72e2 8099 n ...lay=over...\n-00015620: 2073 6f20 7468 6174 206e 6f20 706f 6c79 so that no poly\n-00015630: 676f 6e73 2063 616e 2068 6964 6520 7468 gons can hide th\n-00015640: 656d 2e0a 2020 6465 6620 6178 2028 6478 em.. def ax (dx\n-00015650: 2c30 2c30 2920 2520 7469 7073 206f 6620 ,0,0) % tips of \n-00015660: 7468 6520 6178 6573 0a20 2064 6566 2061 the axes. def a\n-00015670: 7920 2830 2c64 792c 3029 0a20 2064 6566 y (0,dy,0). def\n-00015680: 2061 7a20 2830 2c30 2c64 7a29 0a20 206c az (0,0,dz). l\n-00015690: 696e 655b 6172 726f 7773 3d3c 2d3e 2c6c ine[arrows=<->,l\n-000156a0: 696e 6577 6964 7468 3d2e 3470 745d 2861 inewidth=.4pt](a\n-000156b0: 7829 284f 2928 6179 290a 2020 6c69 6e65 x)(O)(ay). line\n-000156c0: 5b61 7272 6f77 733d 2d3e 2c6c 696e 6577 [arrows=->,linew\n-000156d0: 6964 7468 3d2e 3470 745d 284f 2928 617a idth=.4pt](O)(az\n-000156e0: 290a 2020 2520 7265 7065 6174 2064 6f74 ). % repeat dot\n-000156f0: 7465 6420 6173 2061 6e20 6f76 6572 6c61 ted as an overla\n-00015700: 7920 746f 2068 696e 7420 6174 2074 6865 y to hint at the\n-00015710: 2068 6964 6465 6e20 6c69 6e65 730a 2020 hidden lines. \n-00015720: 6c69 6e65 5b6c 6179 3d6f 7665 722c 6c69 line[lay=over,li\n-00015730: 6e65 7374 796c 653d 646f 7474 6564 2c6c nestyle=dotted,l\n-00015740: 696e 6577 6964 7468 3d2e 3470 745d 2861 inewidth=.4pt](a\n-00015750: 7829 284f 2928 6179 290a 2020 6c69 6e65 x)(O)(ay). line\n-00015760: 5b6c 6179 3d6f 7665 722c 6c69 6e65 7374 [lay=over,linest\n-00015770: 796c 653d 646f 7474 6564 2c6c 696e 6577 yle=dotted,linew\n-00015780: 6964 7468 3d2e 3470 745d 284f 2928 617a idth=.4pt](O)(az\n-00015790: 290a 2020 7370 6563 6961 6c7c 5c66 6f6f ). special|\\foo\n-000157a0: 746e 6f74 6573 697a 650a 2020 2020 2020 tnotesize. \n-000157b0: 2020 2020 5c75 7075 745b 645d 2331 7b24 \\uput[d]#1{$\n-000157c0: 7824 7d5c 7570 7574 5b75 5d23 327b 2479 x$}\\uput[u]#2{$y\n-000157d0: 247d 5c75 7075 745b 6c5d 2333 7b24 7a24 $}\\uput[l]#3{$z$\n-000157e0: 7d7c 0a20 2020 2028 6178 2928 6179 2928 }|. (ax)(ay)(\n-000157f0: 617a 290a 5468 6520 6c61 6265 6c73 2061 az).The labels a\n-00015800: 7265 2061 7070 6c69 6564 2077 6974 6820 re applied with \n-00015810: e280 9850 5354 7269 636b 73e2 8099 2073 ...PSTricks... s\n-00015820: 7065 6369 616c 206f 626a 6563 7473 2061 pecial objects a\n-00015830: 7320 7573 7561 6c2e 0a0a 2020 2046 6f72 s usual... For\n-00015840: 2074 6865 2068 6569 6768 7420 6469 6d65 the height dime\n-00015850: 6e73 696f 6e20 6d61 726b 2c20 7468 6520 nsion mark, the \n-00015860: 706f 7765 7220 6f66 2061 6666 696e 6520 power of affine \n-00015870: 6172 6974 6865 7469 6320 6973 2076 6572 arithetic is ver\n-00015880: 790a 6865 6c70 6675 6c2e 0a20 2064 6566 y.helpful.. def\n-00015890: 2068 6469 6d5f 7265 6620 756e 6974 2828 hdim_ref unit((\n-000158a0: 7031 2920 2d20 284f 2929 2074 6865 6e20 p1) - (O)) then \n-000158b0: 5b5b 7365 675f 726f 745d 5d5e 320a 2020 [[seg_rot]]^2. \n-000158c0: 6465 6620 6330 2028 7030 2920 7468 656e def c0 (p0) then\n-000158d0: 2073 6361 6c65 285b 4a5d 290a 2020 6465 scale([J]). de\n-000158e0: 6620 6830 3020 2863 3029 202b 2031 2e31 f h00 (c0) + 1.1\n-000158f0: 202a 205b 6864 696d 5f72 6566 5d0a 2020 * [hdim_ref]. \n-00015900: 6465 6620 6830 3120 2863 3029 202b 2031 def h01 (c0) + 1\n-00015910: 2e39 202a 205b 6864 696d 5f72 6566 5d0a .9 * [hdim_ref].\n-00015920: 2020 6465 6620 6830 3220 2863 3029 202b def h02 (c0) +\n-00015930: 2031 2e38 202a 205b 6864 696d 5f72 6566 1.8 * [hdim_ref\n-00015940: 5d0a 2020 6c69 6e65 2868 3030 2928 6830 ]. line(h00)(h0\n-00015950: 3129 0a20 2064 6566 2068 3130 2028 4f29 1). def h10 (O)\n-00015960: 202b 2031 2e36 202a 205b 6864 696d 5f72 + 1.6 * [hdim_r\n-00015970: 6566 5d0a 2020 6465 6620 6831 3120 284f ef]. def h11 (O\n-00015980: 2920 2b20 312e 3920 2a20 5b68 6469 6d5f ) + 1.9 * [hdim_\n-00015990: 7265 665d 0a20 2064 6566 2068 3132 2028 ref]. def h12 (\n-000159a0: 4f29 202b 2031 2e38 202a 205b 6864 696d O) + 1.8 * [hdim\n-000159b0: 5f72 6566 5d0a 2020 6c69 6e65 2868 3130 _ref]. line(h10\n-000159c0: 2928 6831 3129 0a20 206c 696e 655b 6172 )(h11). line[ar\n-000159d0: 726f 7773 3d3c 2d3e 5d28 6830 3229 2868 rows=<->](h02)(h\n-000159e0: 3132 290a 2020 6465 6620 686d 3220 2828 12). def hm2 ((\n-000159f0: 6830 3229 202d 2028 4f29 202b 2028 6831 h02) - (O) + (h1\n-00015a00: 3229 202d 2028 4f29 2920 2f20 3220 2b20 2) - (O)) / 2 + \n-00015a10: 284f 290a 2020 7370 6563 6961 6c7c 5c66 (O). special|\\f\n-00015a20: 6f6f 746e 6f74 6573 697a 655c 7270 7574 ootnotesize\\rput\n-00015a30: 2a23 317b 2468 247d 7c28 686d 3229 0a20 *#1{$h$}|(hm2). \n-00015a40: 2020 5468 6520 6765 6e65 7261 6c20 6964 The general id\n-00015a50: 6561 2065 6d70 6c6f 7965 6420 6865 7265 ea employed here\n-00015a60: 2069 7320 746f 2063 6f6d 7075 7465 2061 is to compute a\n-00015a70: 2075 6e69 7420 2272 6566 6572 656e 6365 unit \"reference\n-00015a80: 0a76 6563 746f 7222 2070 6172 616c 6c65 .vector\" paralle\n-00015a90: 6c20 746f 2074 6865 2078 7a2d 706c 616e l to the xz-plan\n-00015aa0: 6520 696e 2074 6865 2064 6573 6972 6564 e in the desired\n-00015ab0: 2064 6972 6563 7469 6f6e 206f 6620 7468 direction of th\n-00015ac0: 650a 6469 6d65 6e73 696f 6e20 6672 6f6d e.dimension from\n-00015ad0: 2074 6865 206f 7269 6769 6e2e 2020 5468 the origin. Th\n-00015ae0: 6520 7472 616e 7366 6f72 6d61 7469 6f6e e transformation\n-00015af0: 20e2 8098 5b5b 7365 675f 726f 745d 5d5e ...[[seg_rot]]^\n-00015b00: 32e2 8099 2072 6f74 6174 6573 0a74 776f 2... rotates.two\n-00015b10: 2073 6567 6d65 6e74 7320 6162 6f75 7420 segments about \n-00015b20: 7468 6520 792d 6178 6973 2e20 2057 6865 the y-axis. Whe\n-00015b30: 6e20 6170 706c 6965 6420 746f 20e2 8098 n applied to ...\n-00015b40: 2870 3129 202d 2028 4f29 e280 992c 2074 (p1) - (O)..., t\n-00015b50: 6865 0a72 6573 756c 7469 6e67 2076 6563 he.resulting vec\n-00015b60: 746f 7220 706f 696e 7473 2074 6f20 7468 tor points to th\n-00015b70: 6520 7269 6768 7420 6173 2073 686f 776e e right as shown\n-00015b80: 2e20 2049 6e20 7468 6973 206d 616e 6e65 . In this manne\n-00015b90: 722c 2077 6520 6361 6e0a 7069 636b 2061 r, we can.pick a\n-00015ba0: 6e79 2076 6572 7465 7820 6173 2074 6865 ny vertex as the\n-00015bb0: 206c 6f63 6174 696f 6e20 6f66 2074 6865 location of the\n-00015bc0: 2068 6569 6768 7420 6469 6d65 6e73 696f height dimensio\n-00015bd0: 6e20 6c69 6e65 7320 6279 2076 6172 7969 n lines by varyi\n-00015be0: 6e67 0a74 6865 2065 7870 6f6e 656e 7420 ng.the exponent \n-00015bf0: 6f66 20e2 8098 5b5b 7365 675f 726f 745d of ...[[seg_rot]\n-00015c00: 5de2 8099 2e20 2054 6869 7320 6973 206f ].... This is o\n-00015c10: 6e6c 7920 6f6e 6520 6f66 206d 616e 7920 nly one of many \n-00015c20: 706f 7373 6962 6c65 0a73 7472 6174 6567 possible.strateg\n-00015c30: 6965 732e 0a0a 2020 2054 6865 2063 6f6d ies... The com\n-00015c40: 7075 7461 7469 6f6e 206f 6620 e280 9868 putation of ...h\n-00015c50: 6d32 e280 9920 6973 2061 2075 7365 6675 m2... is a usefu\n-00015c60: 6c20 6964 696f 6d20 666f 7220 6669 6e64 l idiom for find\n-00015c70: 696e 6720 7468 6520 6365 6e74 726f 6964 ing the centroid\n-00015c80: 0a6f 6620 6120 7365 7420 6f66 2070 6f69 .of a set of poi\n-00015c90: 6e74 732e 0a0a 2020 2054 6865 2074 776f nts... The two\n-00015ca0: 2072 6164 6975 7320 6d61 726b 7320 6172 radius marks ar\n-00015cb0: 6520 646f 6e65 2073 696d 696c 6172 6c79 e done similarly\n-00015cc0: 2c20 736f 2077 6520 7072 6573 656e 7420 , so we present \n-00015cd0: 7468 6520 636f 6465 0a77 6974 686f 7574 the code.without\n-00015ce0: 2063 6f6d 6d65 6e74 2e0a 2020 2520 7261 comment.. % ra\n-00015cf0: 6469 7573 206d 6561 7375 7265 6d65 6e74 dius measurement\n-00015d00: 206d 6172 6b73 0a20 2064 6566 2067 6170 marks. def gap\n-00015d10: 205b 302c 2e32 2c30 5d20 2025 2075 7365 [0,.2,0] % use\n-00015d20: 6420 746f 2063 7265 6174 6520 736d 616c d to create smal\n-00015d30: 6c20 7665 7274 6963 616c 2067 6170 730a l vertical gaps.\n-00015d40: 0a20 2025 2066 6972 7374 2072 310a 2020 . % first r1. \n-00015d50: 6465 6620 7570 3120 5b30 2c33 2e31 2c30 def up1 [0,3.1,0\n-00015d60: 5d20 2520 7469 636b 2072 6973 6573 2061 ] % tick rises a\n-00015d70: 626f 7665 2064 696d 656e 7369 6f6e 2061 bove dimension a\n-00015d80: 206c 6974 746c 650a 2020 6465 6620 7231 little. def r1\n-00015d90: 2028 2870 3129 2074 6865 6e20 5b5b 7365 ((p1) then [[se\n-00015da0: 675f 726f 745d 5d5e 2d32 2920 2b20 5b75 g_rot]]^-2) + [u\n-00015db0: 7031 5d0a 2020 6465 6620 7231 6320 2872 p1]. def r1c (r\n-00015dc0: 3129 2074 6865 6e20 7363 616c 6528 5b4a 1) then scale([J\n-00015dd0: 5d29 0a20 2064 6566 2072 3174 2028 7231 ]). def r1t (r1\n-00015de0: 2920 2b20 5b67 6170 5d0a 2020 6465 6620 ) + [gap]. def \n-00015df0: 7231 6220 2828 7231 7429 2074 6865 6e20 r1b ((r1t) then \n-00015e00: 7363 616c 6528 5b31 2c30 2c31 5d29 2920 scale([1,0,1])) \n-00015e10: 2b20 5b67 6170 5d0a 2020 6c69 6e65 5b61 + [gap]. line[a\n-00015e20: 7272 6f77 733d 3c2d 3e5d 2872 3163 2928 rrows=<->](r1c)(\n-00015e30: 7231 2920 2025 2064 696d 656e 7369 6f6e r1) % dimension\n-00015e40: 206c 696e 650a 2020 6c69 6e65 2872 3162 line. line(r1b\n-00015e50: 2928 7231 7429 2020 2020 2020 2020 2020 )(r1t) \n-00015e60: 2020 2025 2074 6963 6b0a 2020 6465 6620 % tick. def \n-00015e70: 7231 6d20 2828 7231 2920 2d20 284f 2920 r1m ((r1) - (O) \n-00015e80: 2b20 2872 3163 2920 2d20 284f 2929 202f + (r1c) - (O)) /\n-00015e90: 2032 202b 2028 4f29 2025 206c 6162 656c 2 + (O) % label\n-00015ea0: 2070 6f73 6974 696f 6e0a 2020 7370 6563 position. spec\n-00015eb0: 6961 6c20 7c5c 666f 6f74 6e6f 7465 7369 ial |\\footnotesi\n-00015ec0: 7a65 5c72 7075 742a 2331 7b24 725f 3124 ze\\rput*#1{$r_1$\n-00015ed0: 7d7c 2872 316d 2920 2025 206c 6162 656c }|(r1m) % label\n-00015ee0: 0a0a 2020 2520 7361 6d65 2064 7269 6c6c .. % same drill\n-00015ef0: 2066 6f72 2072 302c 2062 7574 206d 7573 for r0, but mus\n-00015f00: 7420 7072 6f6a 6563 7420 646f 776e 2066 t project down f\n-00015f10: 6972 7374 0a20 2064 6566 2075 7030 205b irst. def up0 [\n-00015f20: 302c 322e 372c 305d 0a20 2064 6566 2072 0,2.7,0]. def r\n-00015f30: 3020 2828 7030 2920 7468 656e 2073 6361 0 ((p0) then sca\n-00015f40: 6c65 285b 312c 302c 315d 2920 7468 656e le([1,0,1]) then\n-00015f50: 205b 5b73 6567 5f72 6f74 5d5d 5e2d 3229 [[seg_rot]]^-2)\n-00015f60: 202b 205b 7570 305d 0a20 2064 6566 2072 + [up0]. def r\n-00015f70: 3063 2028 7230 2920 7468 656e 2073 6361 0c (r0) then sca\n-00015f80: 6c65 285b 4a5d 290a 2020 6465 6620 7230 le([J]). def r0\n-00015f90: 7420 2872 3029 202b 205b 6761 705d 0a20 t (r0) + [gap]. \n-00015fa0: 2064 6566 2072 3062 2028 2870 3029 2074 def r0b ((p0) t\n-00015fb0: 6865 6e20 5b5b 7365 675f 726f 745d 5d5e hen [[seg_rot]]^\n-00015fc0: 2d32 2920 2b20 5b67 6170 5d0a 2020 6c69 -2) + [gap]. li\n-00015fd0: 6e65 5b61 7272 6f77 733d 3c2d 3e5d 2872 ne[arrows=<->](r\n-00015fe0: 3063 2928 7230 290a 2020 6c69 6e65 2872 0c)(r0). line(r\n-00015ff0: 3062 2928 7230 7429 0a20 2064 6566 2072 0b)(r0t). def r\n-00016000: 306d 2028 2872 3029 202d 2028 4f29 202b 0m ((r0) - (O) +\n-00016010: 2028 7230 6329 202d 2028 4f29 2920 2f20 (r0c) - (O)) / \n-00016020: 3220 2b20 284f 290a 2020 7370 6563 6961 2 + (O). specia\n-00016030: 6c20 7c5c 666f 6f74 6e6f 7465 7369 7a65 l |\\footnotesize\n-00016040: 5c72 7075 742a 2331 7b24 725f 3024 7d7c \\rput*#1{$r_0$}|\n-00016050: 2872 306d 290a 0a20 2020 5468 6520 7365 (r0m).. The se\n-00016060: 636f 6e64 2064 7261 7769 6e67 2075 7365 cond drawing use\n-00016070: 7320 7468 6520 7361 6d65 2074 6563 686e s the same techn\n-00016080: 6971 7565 732e 2020 4f6e 6c79 2074 6865 iques. Only the\n-00016090: 206d 6574 686f 6420 666f 720a 6472 6177 method for.draw\n-000160a0: 696e 6720 7468 6520 656c 6c69 7074 6963 ing the elliptic\n-000160b0: 616c 2061 7263 2069 7320 6e65 772e 2020 al arc is new. \n-000160c0: 4865 7265 2069 7320 7468 6520 636f 6465 Here is the code\n-000160d0: 2e0a 2020 6465 6620 6d69 6420 2828 7030 .. def mid ((p0\n-000160e0: 3029 2d28 4f29 2b28 7031 3029 2d28 4f29 0)-(O)+(p10)-(O)\n-000160f0: 2b28 7031 3129 2d28 4f29 2b28 7030 3129 +(p11)-(O)+(p01)\n-00016100: 2d28 4f29 292f 342b 284f 290a 2020 7370 -(O))/4+(O). sp\n-00016110: 6563 6961 6c7c 5c72 7075 7423 317b 5c70 ecial|\\rput#1{\\p\n-00016120: 7363 7573 746f 6d7b 0a20 2020 205c 7363 scustom{. \\sc\n-00016130: 616c 657b 3120 312e 337d 0a20 2020 205c ale{1 1.3}. \\\n-00016140: 7073 6172 635b 6172 726f 776c 656e 6774 psarc[arrowlengt\n-00016150: 683d 2e35 5d7b 2d3e 7d7b 2e32 357d 7b2d h=.5]{->}{.25}{-\n-00016160: 3630 7d7b 3234 307d 7d7d 7c0a 2020 2020 60}{240}}}|. \n-00016170: 5b6c 6179 3d6f 7665 725d 286d 6964 290a [lay=over](mid).\n-00016180: 5765 2063 6f75 6c64 2068 6176 6520 7377 We could have sw\n-00016190: 6570 7420 6120 706f 696e 7420 746f 206d ept a point to m\n-000161a0: 616b 6520 7468 6520 6172 6320 7769 7468 ake the arc with\n-000161b0: 20e2 8098 736b 6574 6368 e280 992c 2062 ...sketch..., b\n-000161c0: 7574 2075 7369 6e67 2061 0ae2 8098 5053 ut using a....PS\n-000161d0: 5472 6963 6b73 e280 9920 6375 7374 6f6d Tricks... custom\n-000161e0: 2067 7261 7068 6963 2077 6173 2073 696d graphic was sim\n-000161f0: 706c 6572 2e20 2041 6761 696e 2077 6520 pler. Again we \n-00016200: 636f 6d70 7574 6564 2074 6865 2063 656e computed the cen\n-00016210: 7472 6f69 640a 6f66 2074 6865 2071 7561 troid.of the qua\n-00016220: 6472 696c 6174 6572 616c 2062 7920 6176 drilateral by av\n-00016230: 6572 6167 696e 6720 706f 696e 7473 2e20 eraging points. \n-00016240: 204e 6f74 6520 7468 6174 2073 6361 6c69 Note that scali\n-00016250: 6e67 2069 6e0a 506f 7374 7363 7269 7074 ng in.Postscript\n-00016260: 2064 6973 746f 7274 7320 7468 6520 6172 distorts the ar\n-00016270: 726f 7768 6561 642c 2062 7574 2069 6e20 rowhead, but in \n-00016280: 7468 6973 2063 6173 6520 7468 6520 6469 this case the di\n-00016290: 7374 6f72 7469 6f6e 0a61 6374 7561 6c6c stortion.actuall\n-000162a0: 7920 6c6f 6f6b 7320 6265 7474 6572 2069 y looks better i\n-000162b0: 6e20 7468 6520 7072 6f6a 6563 7469 6f6e n the projection\n-000162c0: 206f 6620 7468 6520 736c 616e 7465 6420 of the slanted \n-000162d0: 6661 6365 2e20 2041 20e2 8098 736b 6574 face. A ...sket\n-000162e0: 6368 e280 990a 6172 726f 7768 6561 6420 ch....arrowhead \n-000162f0: 776f 756c 6420 6e6f 7420 6861 7665 2062 would not have b\n-00016300: 6565 6e20 6469 7374 6f72 7465 642e 0a0a een distorted...\n-00016310: 2020 2054 6865 2063 6f6d 706c 6574 6520 The complete \n-00016320: 636f 6465 2066 6f72 2074 6869 7320 6578 code for this ex\n-00016330: 616d 706c 652c 2077 6869 6368 2064 7261 ample, which dra\n-00016340: 7773 2065 6974 6865 7220 6669 6775 7265 ws either figure\n-00016350: 0a64 6570 656e 6469 6e67 206f 6e20 7468 .depending on th\n-00016360: 6520 6465 6669 6e69 7469 6f6e 206f 6620 e definition of \n-00016370: 7468 6520 7461 6720 e280 983c 6c61 6265 the tag ......, is incl\n-00016390: 7564 6564 2069 6e20 7468 650a e280 9873 uded in the....s\n-000163a0: 6b65 7463 68e2 8099 2064 6973 7472 6962 ketch... distrib\n-000163b0: 7574 696f 6e20 696e 2074 6865 2066 696c ution in the fil\n-000163c0: 6520 e280 9874 7275 6e63 6174 6564 636f e ...truncatedco\n-000163d0: 6e65 2e73 6be2 8099 2e0a 0a1f 0a46 696c ne.sk........Fil\n-000163e0: 653a 2073 6b65 7463 682e 696e 666f 2c20 e: sketch.info, \n-000163f0: 204e 6f64 653a 2041 2068 6965 7261 7263 Node: A hierarc\n-00016400: 6869 6361 6c20 6d6f 6465 6c2c 2020 4e65 hical model, Ne\n-00016410: 7874 3a20 4361 7665 6174 732c 2020 5072 xt: Caveats, Pr\n-00016420: 6576 3a20 4120 7465 6368 6e69 6361 6c20 ev: A technical \n-00016430: 6472 6177 696e 672c 2020 5570 3a20 4275 drawing, Up: Bu\n-00016440: 696c 6469 6e67 2061 2064 7261 7769 6e67 ilding a drawing\n-00016450: 0a0a 342e 3320 4120 6869 6572 6172 6368 ..4.3 A hierarch\n-00016460: 6963 616c 206d 6f64 656c 0a3d 3d3d 3d3d ical model.=====\n+000151b0: 3d3d 3d3d 3d3d 3d0a 0a4c 6574 2773 206c =======..Let's l\n+000151c0: 6f6f 6b20 6174 2061 2064 7261 7769 6e67 ook at a drawing\n+000151d0: 2074 6861 7420 7265 7072 6573 656e 7473 that represents\n+000151e0: 2074 6865 206b 696e 6420 6f66 2070 726f the kind of pro\n+000151f0: 626c 656d 20e2 8098 736b 6574 6368 e280 blem ...sketch..\n+00015200: 9920 7761 730a 6d65 616e 7420 746f 2073 . was.meant to s\n+00015210: 6f6c 7665 2d2d 6120 7061 6972 206f 6620 olve--a pair of \n+00015220: 7465 7874 626f 6f6b 2066 6967 7572 6573 textbook figures\n+00015230: 2072 6567 6172 6469 6e67 2061 2070 6f6c regarding a pol\n+00015240: 7967 6f6e 616c 0a61 7070 726f 7869 6d61 ygonal.approxima\n+00015250: 7469 6f6e 206f 6620 6120 7472 756e 6361 tion of a trunca\n+00015260: 7465 6420 636f 6e65 2e20 2048 6572 6520 ted cone. Here \n+00015270: 6172 6520 7468 6520 7069 6374 7572 6573 are the pictures\n+00015280: 2077 6520 7769 6c6c 0a70 726f 6475 6365 we will.produce\n+00015290: 2e0a 0a00 085b 696d 6167 6520 7372 633d .....[image src=\n+000152a0: 2265 7832 3530 2e70 6e67 2220 7465 7874 \"ex250.png\" text\n+000152b0: 3d22 496d 6167 6520 6578 3235 3020 6f6d =\"Image ex250 om\n+000152c0: 6974 7465 6420 696e 2074 6578 7420 7665 itted in text ve\n+000152d0: 7273 696f 6e20 6f66 2074 6869 7320 646f rsion of this do\n+000152e0: 6375 6d65 6e74 2e22 0008 5d20 2020 2020 cument.\"..] \n+000152f0: 0008 5b69 6d61 6765 2073 7263 3d22 6578 ..[image src=\"ex\n+00015300: 3236 302e 706e 6722 2074 6578 743d 2249 260.png\" text=\"I\n+00015310: 6d61 6765 2065 7832 3630 206f 6d69 7474 mage ex260 omitt\n+00015320: 6564 2069 6e20 7465 7874 2076 6572 7369 ed in text versi\n+00015330: 6f6e 206f 6620 7468 6973 2064 6f63 756d on of this docum\n+00015340: 656e 742e 2200 085d 0a0a 2020 2054 6865 ent.\"..].. The\n+00015350: 2063 6f6e 6520 7368 6170 6520 6973 206a cone shape is j\n+00015360: 7573 7420 6120 7377 6570 7420 6c69 6e65 ust a swept line\n+00015370: 2077 6974 6820 6e6f 2063 6c6f 7375 7265 with no closure\n+00015380: 2074 6167 2061 6e64 2063 756c 6c69 6e67 tag and culling\n+00015390: 0a74 7572 6e65 6420 6f66 662e 2020 4265 .turned off. Be\n+000153a0: 6769 6e20 6279 2073 6574 7469 6e67 2075 gin by setting u\n+000153b0: 7020 736f 6d65 2075 7365 6675 6c20 636f p some useful co\n+000153c0: 6e73 7461 6e74 732e 0a20 2064 6566 204f nstants.. def O\n+000153d0: 2028 302c 302c 3029 2064 6566 2049 205b (0,0,0) def I [\n+000153e0: 312c 302c 305d 2064 6566 204a 205b 302c 1,0,0] def J [0,\n+000153f0: 312c 305d 2064 6566 204b 205b 302c 302c 1,0] def K [0,0,\n+00015400: 315d 0a20 2064 6566 2070 3020 2831 2c32 1]. def p0 (1,2\n+00015410: 2920 6465 6620 7031 2028 312e 352c 3029 ) def p1 (1.5,0)\n+00015420: 2064 6566 204e 2038 0a20 2064 6566 2073 def N 8. def s\n+00015430: 6567 5f72 6f74 2072 6f74 6174 6528 3336 eg_rot rotate(36\n+00015440: 3020 2f20 4e2c 205b 4a5d 290a 5468 6520 0 / N, [J]).The \n+00015450: 706f 696e 7473 20e2 8098 7030 e280 9920 points ...p0... \n+00015460: 616e 6420 e280 9870 31e2 8099 2061 7265 and ...p1... are\n+00015470: 2074 6865 2065 6e64 2070 6f69 6e74 7320 the end points \n+00015480: 6f66 2074 6865 206c 696e 6520 746f 2062 of the line to b\n+00015490: 6520 7377 6570 742e 0a54 6865 2064 6566 e swept..The def\n+000154a0: 696e 6974 696f 6e20 e280 9873 6567 5f72 inition ...seg_r\n+000154b0: 6f74 e280 9920 6973 2074 6865 2073 7765 ot... is the swe\n+000154c0: 6570 2074 7261 6e73 666f 726d 6174 696f ep transformatio\n+000154d0: 6e2e 2020 5769 7468 2074 6865 7365 2c20 n. With these, \n+000154e0: 7468 650a 636f 6e65 2069 7473 656c 6620 the.cone itself \n+000154f0: 6973 2073 696d 706c 652e 0a20 2073 7765 is simple.. swe\n+00015500: 6570 5b63 756c 6c3d 6661 6c73 655d 207b ep[cull=false] {\n+00015510: 204e 2c20 5b5b 7365 675f 726f 745d 5d20 N, [[seg_rot]] \n+00015520: 7d20 6c69 6e65 2870 3029 2870 3129 0a0a } line(p0)(p1)..\n+00015530: 2020 2054 6865 2061 7865 7320 6172 6520 The axes are \n+00015540: 6e65 7874 2061 6e64 2069 6e63 6c75 6465 next and include\n+00015550: 2061 6e20 696e 7465 7265 7369 6e67 2074 an interesing t\n+00015560: 7269 636b 2074 6861 7420 7368 6f77 7320 rick that shows \n+00015570: 7468 650a 6869 6464 656e 2070 6172 7473 the.hidden parts\n+00015580: 2061 7320 646f 7474 6564 206c 696e 6573 as dotted lines\n+00015590: 2e20 2054 6865 2073 6563 7265 7420 6973 . The secret is\n+000155a0: 2064 7261 7720 7468 6520 6178 6573 2074 draw the axes t\n+000155b0: 7769 6365 2d2d 736f 6c69 640a 6c69 6e65 wice--solid.line\n+000155c0: 7320 7769 7468 2074 6865 206e 6f72 6d61 s with the norma\n+000155d0: 6c20 6869 6464 656e 2073 7572 6661 6365 l hidden surface\n+000155e0: 2061 6c67 6f72 6974 686d 2069 6e20 6566 algorithm in ef\n+000155f0: 6665 6374 2c20 616e 6420 7468 656e 0a64 fect, and then.d\n+00015600: 6f74 7465 6420 7769 7468 2074 6865 206f otted with the o\n+00015610: 7074 696f 6e20 e280 986c 6179 3d6f 7665 ption ...lay=ove\n+00015620: 72e2 8099 2073 6f20 7468 6174 206e 6f20 r... so that no \n+00015630: 706f 6c79 676f 6e73 2063 616e 2068 6964 polygons can hid\n+00015640: 6520 7468 656d 2e0a 2020 6465 6620 6178 e them.. def ax\n+00015650: 2028 6478 2c30 2c30 2920 2520 7469 7073 (dx,0,0) % tips\n+00015660: 206f 6620 7468 6520 6178 6573 0a20 2064 of the axes. d\n+00015670: 6566 2061 7920 2830 2c64 792c 3029 0a20 ef ay (0,dy,0). \n+00015680: 2064 6566 2061 7a20 2830 2c30 2c64 7a29 def az (0,0,dz)\n+00015690: 0a20 206c 696e 655b 6172 726f 7773 3d3c . line[arrows=<\n+000156a0: 2d3e 2c6c 696e 6577 6964 7468 3d2e 3470 ->,linewidth=.4p\n+000156b0: 745d 2861 7829 284f 2928 6179 290a 2020 t](ax)(O)(ay). \n+000156c0: 6c69 6e65 5b61 7272 6f77 733d 2d3e 2c6c line[arrows=->,l\n+000156d0: 696e 6577 6964 7468 3d2e 3470 745d 284f inewidth=.4pt](O\n+000156e0: 2928 617a 290a 2020 2520 7265 7065 6174 )(az). % repeat\n+000156f0: 2064 6f74 7465 6420 6173 2061 6e20 6f76 dotted as an ov\n+00015700: 6572 6c61 7920 746f 2068 696e 7420 6174 erlay to hint at\n+00015710: 2074 6865 2068 6964 6465 6e20 6c69 6e65 the hidden line\n+00015720: 730a 2020 6c69 6e65 5b6c 6179 3d6f 7665 s. line[lay=ove\n+00015730: 722c 6c69 6e65 7374 796c 653d 646f 7474 r,linestyle=dott\n+00015740: 6564 2c6c 696e 6577 6964 7468 3d2e 3470 ed,linewidth=.4p\n+00015750: 745d 2861 7829 284f 2928 6179 290a 2020 t](ax)(O)(ay). \n+00015760: 6c69 6e65 5b6c 6179 3d6f 7665 722c 6c69 line[lay=over,li\n+00015770: 6e65 7374 796c 653d 646f 7474 6564 2c6c nestyle=dotted,l\n+00015780: 696e 6577 6964 7468 3d2e 3470 745d 284f inewidth=.4pt](O\n+00015790: 2928 617a 290a 2020 7370 6563 6961 6c7c )(az). special|\n+000157a0: 5c66 6f6f 746e 6f74 6573 697a 650a 2020 \\footnotesize. \n+000157b0: 2020 2020 2020 2020 5c75 7075 745b 645d \\uput[d]\n+000157c0: 2331 7b24 7824 7d5c 7570 7574 5b75 5d23 #1{$x$}\\uput[u]#\n+000157d0: 327b 2479 247d 5c75 7075 745b 6c5d 2333 2{$y$}\\uput[l]#3\n+000157e0: 7b24 7a24 7d7c 0a20 2020 2028 6178 2928 {$z$}|. (ax)(\n+000157f0: 6179 2928 617a 290a 5468 6520 6c61 6265 ay)(az).The labe\n+00015800: 6c73 2061 7265 2061 7070 6c69 6564 2077 ls are applied w\n+00015810: 6974 6820 e280 9850 5354 7269 636b 73e2 ith ...PSTricks.\n+00015820: 8099 2073 7065 6369 616c 206f 626a 6563 .. special objec\n+00015830: 7473 2061 7320 7573 7561 6c2e 0a0a 2020 ts as usual... \n+00015840: 2046 6f72 2074 6865 2068 6569 6768 7420 For the height \n+00015850: 6469 6d65 6e73 696f 6e20 6d61 726b 2c20 dimension mark, \n+00015860: 7468 6520 706f 7765 7220 6f66 2061 6666 the power of aff\n+00015870: 696e 6520 6172 6974 6865 7469 6320 6973 ine arithetic is\n+00015880: 2076 6572 790a 6865 6c70 6675 6c2e 0a20 very.helpful.. \n+00015890: 2064 6566 2068 6469 6d5f 7265 6620 756e def hdim_ref un\n+000158a0: 6974 2828 7031 2920 2d20 284f 2929 2074 it((p1) - (O)) t\n+000158b0: 6865 6e20 5b5b 7365 675f 726f 745d 5d5e hen [[seg_rot]]^\n+000158c0: 320a 2020 6465 6620 6330 2028 7030 2920 2. def c0 (p0) \n+000158d0: 7468 656e 2073 6361 6c65 285b 4a5d 290a then scale([J]).\n+000158e0: 2020 6465 6620 6830 3020 2863 3029 202b def h00 (c0) +\n+000158f0: 2031 2e31 202a 205b 6864 696d 5f72 6566 1.1 * [hdim_ref\n+00015900: 5d0a 2020 6465 6620 6830 3120 2863 3029 ]. def h01 (c0)\n+00015910: 202b 2031 2e39 202a 205b 6864 696d 5f72 + 1.9 * [hdim_r\n+00015920: 6566 5d0a 2020 6465 6620 6830 3220 2863 ef]. def h02 (c\n+00015930: 3029 202b 2031 2e38 202a 205b 6864 696d 0) + 1.8 * [hdim\n+00015940: 5f72 6566 5d0a 2020 6c69 6e65 2868 3030 _ref]. line(h00\n+00015950: 2928 6830 3129 0a20 2064 6566 2068 3130 )(h01). def h10\n+00015960: 2028 4f29 202b 2031 2e36 202a 205b 6864 (O) + 1.6 * [hd\n+00015970: 696d 5f72 6566 5d0a 2020 6465 6620 6831 im_ref]. def h1\n+00015980: 3120 284f 2920 2b20 312e 3920 2a20 5b68 1 (O) + 1.9 * [h\n+00015990: 6469 6d5f 7265 665d 0a20 2064 6566 2068 dim_ref]. def h\n+000159a0: 3132 2028 4f29 202b 2031 2e38 202a 205b 12 (O) + 1.8 * [\n+000159b0: 6864 696d 5f72 6566 5d0a 2020 6c69 6e65 hdim_ref]. line\n+000159c0: 2868 3130 2928 6831 3129 0a20 206c 696e (h10)(h11). lin\n+000159d0: 655b 6172 726f 7773 3d3c 2d3e 5d28 6830 e[arrows=<->](h0\n+000159e0: 3229 2868 3132 290a 2020 6465 6620 686d 2)(h12). def hm\n+000159f0: 3220 2828 6830 3229 202d 2028 4f29 202b 2 ((h02) - (O) +\n+00015a00: 2028 6831 3229 202d 2028 4f29 2920 2f20 (h12) - (O)) / \n+00015a10: 3220 2b20 284f 290a 2020 7370 6563 6961 2 + (O). specia\n+00015a20: 6c7c 5c66 6f6f 746e 6f74 6573 697a 655c l|\\footnotesize\\\n+00015a30: 7270 7574 2a23 317b 2468 247d 7c28 686d rput*#1{$h$}|(hm\n+00015a40: 3229 0a20 2020 5468 6520 6765 6e65 7261 2). The genera\n+00015a50: 6c20 6964 6561 2065 6d70 6c6f 7965 6420 l idea employed \n+00015a60: 6865 7265 2069 7320 746f 2063 6f6d 7075 here is to compu\n+00015a70: 7465 2061 2075 6e69 7420 2272 6566 6572 te a unit \"refer\n+00015a80: 656e 6365 0a76 6563 746f 7222 2070 6172 ence.vector\" par\n+00015a90: 616c 6c65 6c20 746f 2074 6865 2078 7a2d allel to the xz-\n+00015aa0: 706c 616e 6520 696e 2074 6865 2064 6573 plane in the des\n+00015ab0: 6972 6564 2064 6972 6563 7469 6f6e 206f ired direction o\n+00015ac0: 6620 7468 650a 6469 6d65 6e73 696f 6e20 f the.dimension \n+00015ad0: 6672 6f6d 2074 6865 206f 7269 6769 6e2e from the origin.\n+00015ae0: 2020 5468 6520 7472 616e 7366 6f72 6d61 The transforma\n+00015af0: 7469 6f6e 20e2 8098 5b5b 7365 675f 726f tion ...[[seg_ro\n+00015b00: 745d 5d5e 32e2 8099 2072 6f74 6174 6573 t]]^2... rotates\n+00015b10: 0a74 776f 2073 6567 6d65 6e74 7320 6162 .two segments ab\n+00015b20: 6f75 7420 7468 6520 792d 6178 6973 2e20 out the y-axis. \n+00015b30: 2057 6865 6e20 6170 706c 6965 6420 746f When applied to\n+00015b40: 20e2 8098 2870 3129 202d 2028 4f29 e280 ...(p1) - (O)..\n+00015b50: 992c 2074 6865 0a72 6573 756c 7469 6e67 ., the.resulting\n+00015b60: 2076 6563 746f 7220 706f 696e 7473 2074 vector points t\n+00015b70: 6f20 7468 6520 7269 6768 7420 6173 2073 o the right as s\n+00015b80: 686f 776e 2e20 2049 6e20 7468 6973 206d hown. In this m\n+00015b90: 616e 6e65 722c 2077 6520 6361 6e0a 7069 anner, we can.pi\n+00015ba0: 636b 2061 6e79 2076 6572 7465 7820 6173 ck any vertex as\n+00015bb0: 2074 6865 206c 6f63 6174 696f 6e20 6f66 the location of\n+00015bc0: 2074 6865 2068 6569 6768 7420 6469 6d65 the height dime\n+00015bd0: 6e73 696f 6e20 6c69 6e65 7320 6279 2076 nsion lines by v\n+00015be0: 6172 7969 6e67 0a74 6865 2065 7870 6f6e arying.the expon\n+00015bf0: 656e 7420 6f66 20e2 8098 5b5b 7365 675f ent of ...[[seg_\n+00015c00: 726f 745d 5de2 8099 2e20 2054 6869 7320 rot]].... This \n+00015c10: 6973 206f 6e6c 7920 6f6e 6520 6f66 206d is only one of m\n+00015c20: 616e 7920 706f 7373 6962 6c65 0a73 7472 any possible.str\n+00015c30: 6174 6567 6965 732e 0a0a 2020 2054 6865 ategies... The\n+00015c40: 2063 6f6d 7075 7461 7469 6f6e 206f 6620 computation of \n+00015c50: e280 9868 6d32 e280 9920 6973 2061 2075 ...hm2... is a u\n+00015c60: 7365 6675 6c20 6964 696f 6d20 666f 7220 seful idiom for \n+00015c70: 6669 6e64 696e 6720 7468 6520 6365 6e74 finding the cent\n+00015c80: 726f 6964 0a6f 6620 6120 7365 7420 6f66 roid.of a set of\n+00015c90: 2070 6f69 6e74 732e 0a0a 2020 2054 6865 points... The\n+00015ca0: 2074 776f 2072 6164 6975 7320 6d61 726b two radius mark\n+00015cb0: 7320 6172 6520 646f 6e65 2073 696d 696c s are done simil\n+00015cc0: 6172 6c79 2c20 736f 2077 6520 7072 6573 arly, so we pres\n+00015cd0: 656e 7420 7468 6520 636f 6465 0a77 6974 ent the code.wit\n+00015ce0: 686f 7574 2063 6f6d 6d65 6e74 2e0a 2020 hout comment.. \n+00015cf0: 2520 7261 6469 7573 206d 6561 7375 7265 % radius measure\n+00015d00: 6d65 6e74 206d 6172 6b73 0a20 2064 6566 ment marks. def\n+00015d10: 2067 6170 205b 302c 2e32 2c30 5d20 2025 gap [0,.2,0] %\n+00015d20: 2075 7365 6420 746f 2063 7265 6174 6520 used to create \n+00015d30: 736d 616c 6c20 7665 7274 6963 616c 2067 small vertical g\n+00015d40: 6170 730a 0a20 2025 2066 6972 7374 2072 aps.. % first r\n+00015d50: 310a 2020 6465 6620 7570 3120 5b30 2c33 1. def up1 [0,3\n+00015d60: 2e31 2c30 5d20 2520 7469 636b 2072 6973 .1,0] % tick ris\n+00015d70: 6573 2061 626f 7665 2064 696d 656e 7369 es above dimensi\n+00015d80: 6f6e 2061 206c 6974 746c 650a 2020 6465 on a little. de\n+00015d90: 6620 7231 2028 2870 3129 2074 6865 6e20 f r1 ((p1) then \n+00015da0: 5b5b 7365 675f 726f 745d 5d5e 2d32 2920 [[seg_rot]]^-2) \n+00015db0: 2b20 5b75 7031 5d0a 2020 6465 6620 7231 + [up1]. def r1\n+00015dc0: 6320 2872 3129 2074 6865 6e20 7363 616c c (r1) then scal\n+00015dd0: 6528 5b4a 5d29 0a20 2064 6566 2072 3174 e([J]). def r1t\n+00015de0: 2028 7231 2920 2b20 5b67 6170 5d0a 2020 (r1) + [gap]. \n+00015df0: 6465 6620 7231 6220 2828 7231 7429 2074 def r1b ((r1t) t\n+00015e00: 6865 6e20 7363 616c 6528 5b31 2c30 2c31 hen scale([1,0,1\n+00015e10: 5d29 2920 2b20 5b67 6170 5d0a 2020 6c69 ])) + [gap]. li\n+00015e20: 6e65 5b61 7272 6f77 733d 3c2d 3e5d 2872 ne[arrows=<->](r\n+00015e30: 3163 2928 7231 2920 2025 2064 696d 656e 1c)(r1) % dimen\n+00015e40: 7369 6f6e 206c 696e 650a 2020 6c69 6e65 sion line. line\n+00015e50: 2872 3162 2928 7231 7429 2020 2020 2020 (r1b)(r1t) \n+00015e60: 2020 2020 2020 2025 2074 6963 6b0a 2020 % tick. \n+00015e70: 6465 6620 7231 6d20 2828 7231 2920 2d20 def r1m ((r1) - \n+00015e80: 284f 2920 2b20 2872 3163 2920 2d20 284f (O) + (r1c) - (O\n+00015e90: 2929 202f 2032 202b 2028 4f29 2025 206c )) / 2 + (O) % l\n+00015ea0: 6162 656c 2070 6f73 6974 696f 6e0a 2020 abel position. \n+00015eb0: 7370 6563 6961 6c20 7c5c 666f 6f74 6e6f special |\\footno\n+00015ec0: 7465 7369 7a65 5c72 7075 742a 2331 7b24 tesize\\rput*#1{$\n+00015ed0: 725f 3124 7d7c 2872 316d 2920 2025 206c r_1$}|(r1m) % l\n+00015ee0: 6162 656c 0a0a 2020 2520 7361 6d65 2064 abel.. % same d\n+00015ef0: 7269 6c6c 2066 6f72 2072 302c 2062 7574 rill for r0, but\n+00015f00: 206d 7573 7420 7072 6f6a 6563 7420 646f must project do\n+00015f10: 776e 2066 6972 7374 0a20 2064 6566 2075 wn first. def u\n+00015f20: 7030 205b 302c 322e 372c 305d 0a20 2064 p0 [0,2.7,0]. d\n+00015f30: 6566 2072 3020 2828 7030 2920 7468 656e ef r0 ((p0) then\n+00015f40: 2073 6361 6c65 285b 312c 302c 315d 2920 scale([1,0,1]) \n+00015f50: 7468 656e 205b 5b73 6567 5f72 6f74 5d5d then [[seg_rot]]\n+00015f60: 5e2d 3229 202b 205b 7570 305d 0a20 2064 ^-2) + [up0]. d\n+00015f70: 6566 2072 3063 2028 7230 2920 7468 656e ef r0c (r0) then\n+00015f80: 2073 6361 6c65 285b 4a5d 290a 2020 6465 scale([J]). de\n+00015f90: 6620 7230 7420 2872 3029 202b 205b 6761 f r0t (r0) + [ga\n+00015fa0: 705d 0a20 2064 6566 2072 3062 2028 2870 p]. def r0b ((p\n+00015fb0: 3029 2074 6865 6e20 5b5b 7365 675f 726f 0) then [[seg_ro\n+00015fc0: 745d 5d5e 2d32 2920 2b20 5b67 6170 5d0a t]]^-2) + [gap].\n+00015fd0: 2020 6c69 6e65 5b61 7272 6f77 733d 3c2d line[arrows=<-\n+00015fe0: 3e5d 2872 3063 2928 7230 290a 2020 6c69 >](r0c)(r0). li\n+00015ff0: 6e65 2872 3062 2928 7230 7429 0a20 2064 ne(r0b)(r0t). d\n+00016000: 6566 2072 306d 2028 2872 3029 202d 2028 ef r0m ((r0) - (\n+00016010: 4f29 202b 2028 7230 6329 202d 2028 4f29 O) + (r0c) - (O)\n+00016020: 2920 2f20 3220 2b20 284f 290a 2020 7370 ) / 2 + (O). sp\n+00016030: 6563 6961 6c20 7c5c 666f 6f74 6e6f 7465 ecial |\\footnote\n+00016040: 7369 7a65 5c72 7075 742a 2331 7b24 725f size\\rput*#1{$r_\n+00016050: 3024 7d7c 2872 306d 290a 0a20 2020 5468 0$}|(r0m).. Th\n+00016060: 6520 7365 636f 6e64 2064 7261 7769 6e67 e second drawing\n+00016070: 2075 7365 7320 7468 6520 7361 6d65 2074 uses the same t\n+00016080: 6563 686e 6971 7565 732e 2020 4f6e 6c79 echniques. Only\n+00016090: 2074 6865 206d 6574 686f 6420 666f 720a the method for.\n+000160a0: 6472 6177 696e 6720 7468 6520 656c 6c69 drawing the elli\n+000160b0: 7074 6963 616c 2061 7263 2069 7320 6e65 ptical arc is ne\n+000160c0: 772e 2020 4865 7265 2069 7320 7468 6520 w. Here is the \n+000160d0: 636f 6465 2e0a 2020 6465 6620 6d69 6420 code.. def mid \n+000160e0: 2828 7030 3029 2d28 4f29 2b28 7031 3029 ((p00)-(O)+(p10)\n+000160f0: 2d28 4f29 2b28 7031 3129 2d28 4f29 2b28 -(O)+(p11)-(O)+(\n+00016100: 7030 3129 2d28 4f29 292f 342b 284f 290a p01)-(O))/4+(O).\n+00016110: 2020 7370 6563 6961 6c7c 5c72 7075 7423 special|\\rput#\n+00016120: 317b 5c70 7363 7573 746f 6d7b 0a20 2020 1{\\pscustom{. \n+00016130: 205c 7363 616c 657b 3120 312e 337d 0a20 \\scale{1 1.3}. \n+00016140: 2020 205c 7073 6172 635b 6172 726f 776c \\psarc[arrowl\n+00016150: 656e 6774 683d 2e35 5d7b 2d3e 7d7b 2e32 ength=.5]{->}{.2\n+00016160: 357d 7b2d 3630 7d7b 3234 307d 7d7d 7c0a 5}{-60}{240}}}|.\n+00016170: 2020 2020 5b6c 6179 3d6f 7665 725d 286d [lay=over](m\n+00016180: 6964 290a 5765 2063 6f75 6c64 2068 6176 id).We could hav\n+00016190: 6520 7377 6570 7420 6120 706f 696e 7420 e swept a point \n+000161a0: 746f 206d 616b 6520 7468 6520 6172 6320 to make the arc \n+000161b0: 7769 7468 20e2 8098 736b 6574 6368 e280 with ...sketch..\n+000161c0: 992c 2062 7574 2075 7369 6e67 2061 0ae2 ., but using a..\n+000161d0: 8098 5053 5472 6963 6b73 e280 9920 6375 ..PSTricks... cu\n+000161e0: 7374 6f6d 2067 7261 7068 6963 2077 6173 stom graphic was\n+000161f0: 2073 696d 706c 6572 2e20 2041 6761 696e simpler. Again\n+00016200: 2077 6520 636f 6d70 7574 6564 2074 6865 we computed the\n+00016210: 2063 656e 7472 6f69 640a 6f66 2074 6865 centroid.of the\n+00016220: 2071 7561 6472 696c 6174 6572 616c 2062 quadrilateral b\n+00016230: 7920 6176 6572 6167 696e 6720 706f 696e y averaging poin\n+00016240: 7473 2e20 204e 6f74 6520 7468 6174 2073 ts. Note that s\n+00016250: 6361 6c69 6e67 2069 6e0a 506f 7374 7363 caling in.Postsc\n+00016260: 7269 7074 2064 6973 746f 7274 7320 7468 ript distorts th\n+00016270: 6520 6172 726f 7768 6561 642c 2062 7574 e arrowhead, but\n+00016280: 2069 6e20 7468 6973 2063 6173 6520 7468 in this case th\n+00016290: 6520 6469 7374 6f72 7469 6f6e 0a61 6374 e distortion.act\n+000162a0: 7561 6c6c 7920 6c6f 6f6b 7320 6265 7474 ually looks bett\n+000162b0: 6572 2069 6e20 7468 6520 7072 6f6a 6563 er in the projec\n+000162c0: 7469 6f6e 206f 6620 7468 6520 736c 616e tion of the slan\n+000162d0: 7465 6420 6661 6365 2e20 2041 20e2 8098 ted face. A ...\n+000162e0: 736b 6574 6368 e280 990a 6172 726f 7768 sketch....arrowh\n+000162f0: 6561 6420 776f 756c 6420 6e6f 7420 6861 ead would not ha\n+00016300: 7665 2062 6565 6e20 6469 7374 6f72 7465 ve been distorte\n+00016310: 642e 0a0a 2020 2054 6865 2063 6f6d 706c d... The compl\n+00016320: 6574 6520 636f 6465 2066 6f72 2074 6869 ete code for thi\n+00016330: 7320 6578 616d 706c 652c 2077 6869 6368 s example, which\n+00016340: 2064 7261 7773 2065 6974 6865 7220 6669 draws either fi\n+00016350: 6775 7265 0a64 6570 656e 6469 6e67 206f gure.depending o\n+00016360: 6e20 7468 6520 6465 6669 6e69 7469 6f6e n the definition\n+00016370: 206f 6620 7468 6520 7461 6720 e280 983c of the tag ...<\n+00016380: 6c61 6265 6c65 643e e280 992c 2069 7320 labeled>..., is \n+00016390: 696e 636c 7564 6564 2069 6e20 7468 650a included in the.\n+000163a0: e280 9873 6b65 7463 68e2 8099 2064 6973 ...sketch... dis\n+000163b0: 7472 6962 7574 696f 6e20 696e 2074 6865 tribution in the\n+000163c0: 2066 696c 6520 e280 9874 7275 6e63 6174 file ...truncat\n+000163d0: 6564 636f 6e65 2e73 6be2 8099 2e0a 0a1f edcone.sk.......\n+000163e0: 0a46 696c 653a 2073 6b65 7463 682e 696e .File: sketch.in\n+000163f0: 666f 2c20 204e 6f64 653a 2041 2068 6965 fo, Node: A hie\n+00016400: 7261 7263 6869 6361 6c20 6d6f 6465 6c2c rarchical model,\n+00016410: 2020 4e65 7874 3a20 4361 7665 6174 732c Next: Caveats,\n+00016420: 2020 5072 6576 3a20 4120 7465 6368 6e69 Prev: A techni\n+00016430: 6361 6c20 6472 6177 696e 672c 2020 5570 cal drawing, Up\n+00016440: 3a20 4275 696c 6469 6e67 2061 2064 7261 : Building a dra\n+00016450: 7769 6e67 0a0a 342e 3320 4120 6869 6572 wing..4.3 A hier\n+00016460: 6172 6368 6963 616c 206d 6f64 656c 0a3d archical model.=\n 00016470: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================\n-00016480: 3d3d 3d0a 0a57 6869 6c65 20e2 8098 736b ===..While ...sk\n-00016490: 6574 6368 e280 9920 7761 7320 6e65 7665 etch... was neve\n-000164a0: 7220 6d65 616e 7420 746f 2062 6520 6120 r meant to be a \n-000164b0: 6765 6f6d 6574 7269 6320 6d6f 6465 6c69 geometric modeli\n-000164c0: 6e67 206c 616e 6775 6167 652c 2069 740a ng language, it.\n-000164d0: 636f 6d65 7320 6661 6972 6c79 2063 6c6f comes fairly clo\n-000164e0: 7365 2e20 2054 6865 2066 6f6c 6c6f 7769 se. The followi\n-000164f0: 6e67 2065 7861 6d70 6c65 2070 7574 7320 ng example puts \n-00016500: 616c 6c20 7765 2068 6176 6520 7365 656e all we have seen\n-00016510: 2074 6f20 776f 726b 0a69 6e20 6120 7665 to work.in a ve\n-00016520: 7279 2073 696d 706c 6520 6d6f 6465 6c20 ry simple model \n-00016530: 6f66 2074 6865 2068 756d 616e 2068 616e of the human han\n-00016540: 642e 2020 5374 6172 7420 6279 2073 7765 d. Start by swe\n-00016550: 6570 696e 6720 6120 6c69 6e65 2074 6f0a eping a line to.\n-00016560: 6d61 6b65 2061 2074 7275 6e63 6174 6564 make a truncated\n-00016570: 2063 6f6e 652c 2077 6869 6368 2077 696c cone, which wil\n-00016580: 6c20 6265 2063 6f70 6965 6420 6f76 6572 l be copied over\n-00016590: 2061 6e64 206f 7665 7220 6167 6169 6e20 and over again \n-000165a0: 746f 206d 616b 650a 7468 6520 7365 676d to make.the segm\n-000165b0: 656e 7473 206f 6620 6669 6e67 6572 732e ents of fingers.\n-000165c0: 0a20 2064 6566 204f 2028 302c 302c 3029 . def O (0,0,0)\n-000165d0: 2025 206f 7269 6769 6e0a 2020 6465 6620 % origin. def \n-000165e0: 4920 5b31 2c30 2c30 5d20 6465 6620 4a20 I [1,0,0] def J \n-000165f0: 5b30 2c31 2c30 5d20 6465 6620 4b20 5b30 [0,1,0] def K [0\n-00016600: 2c30 2c31 5d20 2520 6361 6e6f 6e69 6361 ,0,1] % canonica\n-00016610: 6c20 756e 6974 2076 6563 746f 7273 0a20 l unit vectors. \n-00016620: 2064 6566 2073 6567 6d65 6e74 207b 0a20 def segment {. \n-00016630: 2020 2064 6566 206e 5f66 6163 6573 2038 def n_faces 8\n-00016640: 0a20 2020 2073 7765 6570 207b 206e 5f66 . sweep { n_f\n-00016650: 6163 6573 3c3e 2c20 726f 7461 7465 2833 aces<>, rotate(3\n-00016660: 3630 202f 206e 5f66 6163 6573 2c20 5b4a 60 / n_faces, [J\n-00016670: 5d29 207d 200a 2020 2020 2020 6c69 6e65 ]) } . line\n-00016680: 2870 726f 7869 6d61 6c5f 7261 642c 2030 (proximal_rad, 0\n-00016690: 2928 6469 7374 616c 5f72 6164 2c20 6469 )(distal_rad, di\n-000166a0: 7374 616c 5f6c 656e 290a 2020 7d0a 496e stal_len). }.In\n-000166b0: 2068 616e 6420 616e 6174 6f6d 792c 205f hand anatomy, _\n-000166c0: 6469 7374 616c 5f20 6973 2022 6174 2074 distal_ is \"at t\n-000166d0: 6865 2074 6970 2220 616e 6420 5f70 726f he tip\" and _pro\n-000166e0: 7869 6d61 6c5f 2069 7320 2269 6e20 7468 ximal_ is \"in th\n-000166f0: 6520 6172 6561 0a6f 6620 7468 6520 7061 e area.of the pa\n-00016700: 6c6d 2e22 2020 5765 2068 6176 6520 6f6d lm.\" We have om\n-00016710: 6974 7465 6420 616c 6c20 7468 6520 7363 itted all the sc\n-00016720: 616c 6172 2063 6f6e 7374 616e 7473 2e20 alar constants. \n-00016730: 2059 6f75 2063 616e 2066 696e 640a 7468 You can find.th\n-00016740: 656d 2069 6e20 e280 9868 616e 642e 736b em in ...hand.sk\n-00016750: e280 992c 2077 6869 6368 2069 7320 7072 ..., which is pr\n-00016760: 6f76 6964 6564 2069 6e20 7468 6520 e280 ovided in the ..\n-00016770: 9873 6b65 7463 68e2 8099 2064 6973 7472 .sketch... distr\n-00016780: 6962 7574 696f 6e2e 0a0a 2020 2057 6520 ibution... We \n-00016790: 616c 736f 206e 6565 6420 6120 7072 6f74 also need a prot\n-000167a0: 6f74 7970 6963 616c 2073 7068 6572 6520 otypical sphere \n-000167b0: 746f 2075 7365 2066 6f72 2074 6865 206a to use for the j\n-000167c0: 6f69 6e74 7320 7468 656d 7365 6c76 6573 oints themselves\n-000167d0: 2e0a 2020 6465 6620 6a6f 696e 745f 7370 .. def joint_sp\n-000167e0: 6865 7265 207b 0a20 2020 2064 6566 206e here {. def n\n-000167f0: 5f6a 6f69 6e74 5f66 6163 6573 2038 0a20 _joint_faces 8. \n-00016800: 2020 2073 7765 6570 205b 6669 6c6c 636f sweep [fillco\n-00016810: 6c6f 723d 7265 645d 207b 206e 5f6a 6f69 lor=red] { n_joi\n-00016820: 6e74 5f66 6163 6573 2c20 726f 7461 7465 nt_faces, rotate\n-00016830: 2833 3630 202f 206e 5f6a 6f69 6e74 5f66 (360 / n_joint_f\n-00016840: 6163 6573 2c20 5b4a 5d29 207d 0a20 2020 aces, [J]) }. \n-00016850: 2020 2073 7765 6570 207b 206e 5f6a 6f69 sweep { n_joi\n-00016860: 6e74 5f66 6163 6573 2c20 726f 7461 7465 nt_faces, rotate\n-00016870: 2831 3830 202f 206e 5f6a 6f69 6e74 5f66 (180 / n_joint_f\n-00016880: 6163 6573 2920 7d20 0a20 2020 2020 2020 aces) } . \n-00016890: 2028 302c 202d 6a6f 696e 745f 7261 6429 (0, -joint_rad)\n-000168a0: 0a20 207d 0a0a 2020 2057 6527 6c6c 206e . }.. We'll n\n-000168b0: 6f77 2064 6573 6967 6e20 7468 6520 696e ow design the in\n-000168c0: 6465 7820 6669 6e67 6572 2028 6e75 6d62 dex finger (numb\n-000168d0: 6572 2031 2069 6e20 6f75 7220 6e6f 7461 er 1 in our nota\n-000168e0: 7469 6f6e 616c 0a63 6f6e 7665 6e74 696f tional.conventio\n-000168f0: 6e3b 2066 696e 6765 7220 3020 6973 2074 n; finger 0 is t\n-00016900: 6865 2074 6875 6d62 292e 2020 5468 6520 he thumb). The \n-00016910: 6469 7374 616c 2072 6f74 6174 696f 6e20 distal rotation \n-00016920: 666f 7220 7468 6520 6669 6e67 6572 0a61 for the finger.a\n-00016930: 7070 6c69 6573 206f 6e6c 7920 746f 2074 pplies only to t\n-00016940: 6865 2074 6970 2c20 736f 2077 6520 6465 he tip, so we de\n-00016950: 6669 6e65 2074 6865 2066 6f6c 6c6f 7769 fine the followi\n-00016960: 6e67 2e0a 2020 6465 6620 6469 7374 616c ng.. def distal\n-00016970: 5f31 207b 0a20 2020 2070 7574 207b 2074 _1 {. put { t\n-00016980: 7261 6e73 6c61 7465 286a 6f69 6e74 5f67 ranslate(joint_g\n-00016990: 6170 202a 206a 6f69 6e74 5f72 6164 202a ap * joint_rad *\n-000169a0: 205b 4a5d 2920 0a20 2020 2020 2020 2020 [J]) . \n-000169b0: 2074 6865 6e20 726f 7461 7465 2864 6973 then rotate(dis\n-000169c0: 7461 6c5f 315f 726f 742c 205b 495d 2920 tal_1_rot, [I]) \n-000169d0: 0a20 2020 2020 2020 2020 2074 6865 6e20 . then \n-000169e0: 7472 616e 736c 6174 6528 2864 6973 7461 translate((dista\n-000169f0: 6c5f 6c65 6e20 2b20 6a6f 696e 745f 6761 l_len + joint_ga\n-00016a00: 7020 2a20 6a6f 696e 745f 7261 6429 202a p * joint_rad) *\n-00016a10: 205b 4a5d 2920 7d0a 2020 2020 2020 7b73 [J]) }. {s\n-00016a20: 6567 6d65 6e74 7d0a 2020 2020 7075 7420 egment}. put \n-00016a30: 7b20 726f 7461 7465 2864 6973 7461 6c5f { rotate(distal_\n-00016a40: 315f 726f 7420 2f20 322c 205b 495d 290a 1_rot / 2, [I]).\n-00016a50: 2020 2020 2020 2020 2020 7468 656e 2074 then t\n-00016a60: 7261 6e73 6c61 7465 2828 6469 7374 616c ranslate((distal\n-00016a70: 5f6c 656e 202b 206a 6f69 6e74 5f67 6170 _len + joint_gap\n-00016a80: 202a 206a 6f69 6e74 5f72 6164 2920 2a20 * joint_rad) * \n-00016a90: 5b4a 5d29 207d 200a 2020 2020 2020 7b6a [J]) } . {j\n-00016aa0: 6f69 6e74 5f73 7068 6572 657d 0a20 2020 oint_sphere}. \n-00016ab0: 2070 7574 207b 2073 6361 6c65 2820 5b4a put { scale( [J\n-00016ac0: 5d20 2b20 7072 6f78 696d 616c 5f64 6973 ] + proximal_dis\n-00016ad0: 7461 6c5f 7261 7469 6f20 2a20 285b 495d tal_ratio * ([I]\n-00016ae0: 2b5b 4b5d 2920 2920 7d0a 2020 2020 2020 +[K]) ) }. \n-00016af0: 7b73 6567 6d65 6e74 7d0a 2020 7d20 200a {segment}. } .\n-00016b00: 5468 6520 6964 656e 7469 6669 6572 7320 The identifiers \n-00016b10: 6865 7265 2061 7265 2066 6f72 2073 697a here are for siz\n-00016b20: 6520 616e 6420 6c6f 6361 7469 6f6e 2063 e and location c\n-00016b30: 6f6e 7374 616e 7473 2e20 2054 6865 2065 onstants. The e\n-00016b40: 7863 6570 7469 6f6e 0a69 7320 e280 9864 xception.is ...d\n-00016b50: 6973 7461 6c5f 726f 745f 31e2 8099 2e20 istal_rot_1.... \n-00016b60: 2054 6869 7320 726f 7461 7469 6f6e 2070 This rotation p\n-00016b70: 6172 616d 6574 6572 206d 6f64 656c 7320 arameter models \n-00016b80: 7468 6520 666c 6578 696e 6720 6f66 2074 the flexing of t\n-00016b90: 6865 0a66 696e 6765 7220 7469 702e 2020 he.finger tip. \n-00016ba0: 5468 6520 6669 7273 7420 e280 9870 7574 The first ...put\n-00016bb0: e280 9920 6d61 6b65 7320 6120 636f 7079 ... makes a copy\n-00016bc0: 206f 6620 7468 6520 6669 6e67 6572 2073 of the finger s\n-00016bd0: 6567 6d65 6e74 2074 6861 7420 6973 0a74 egment that is.t\n-00016be0: 7261 6e73 6c61 7465 6420 7570 7761 7264 ranslated upward\n-00016bf0: 206a 7573 7420 6661 7220 656e 6f75 6768 just far enough\n-00016c00: 2074 6f20 6d61 6b65 2072 6f6f 6d20 666f to make room fo\n-00016c10: 7220 7468 6520 7370 6865 7269 6361 6c20 r the spherical \n-00016c20: 6a6f 696e 742e 0a54 6865 6e20 6974 2061 joint..Then it a\n-00016c30: 7070 6c69 6573 2074 6865 2064 6973 7461 pplies the dista\n-00016c40: 6c20 726f 7461 7469 6f6e 2e20 2046 696e l rotation. Fin\n-00016c50: 616c 6c79 2069 7420 7472 616e 736c 6174 ally it translat\n-00016c60: 6573 2074 6865 2077 686f 6c65 0a61 7373 es the whole.ass\n-00016c70: 656d 626c 7920 7570 7761 7264 2061 6761 embly upward aga\n-00016c80: 696e 2074 6f20 6d61 6b65 2072 6f6f 6d20 in to make room \n-00016c90: 666f 7220 7468 6520 6d69 6464 6c65 2070 for the middle p\n-00016ca0: 686c 616e 6765 7320 2874 6865 206e 6578 hlanges (the nex\n-00016cb0: 740a 626f 6e65 2074 6f77 6172 6420 7468 t.bone toward th\n-00016cc0: 6520 7061 6c6d 292e 2020 5468 6520 7365 e palm). The se\n-00016cd0: 636f 6e64 20e2 8098 7075 74e2 8099 2070 cond ...put... p\n-00016ce0: 6f73 6974 696f 6e73 2074 6865 2073 7068 ositions the sph\n-00016cf0: 6572 652e 2020 5468 6572 6520 6973 0a61 ere. There is.a\n-00016d00: 2072 6f74 6174 696f 6e20 746f 2070 6c61 rotation to pla\n-00016d10: 6365 2074 6865 2067 7269 6420 6f6e 2074 ce the grid on t\n-00016d20: 6865 2073 7068 6572 6520 7375 7266 6163 he sphere surfac\n-00016d30: 6520 6174 2061 6e20 6e69 6365 2061 6e67 e at an nice ang\n-00016d40: 6c65 2c0a 7468 656e 2061 2074 7261 6e73 le,.then a trans\n-00016d50: 6c61 7469 6f6e 2074 6f20 7468 6520 6261 lation to the ba\n-00016d60: 7365 206f 6620 7468 6520 6469 7374 616c se of the distal\n-00016d70: 2070 686c 616e 6765 732c 2077 6869 6368 phlanges, which\n-00016d80: 2069 7320 616c 736f 0a63 656e 7465 7220 is also.center \n-00016d90: 6f66 2069 7473 2072 6f74 6174 696f 6e2e of its rotation.\n-00016da0: 2020 4669 6e61 6c6c 792c 2074 6865 206c Finally, the l\n-00016db0: 6173 7420 e280 9870 7574 e280 9920 706f ast ...put... po\n-00016dc0: 7369 7469 6f6e 7320 7468 6520 6d69 6464 sitions the midd\n-00016dd0: 6c65 0a73 6567 6d65 6e74 2069 7473 656c le.segment itsel\n-00016de0: 662e 0a0a 2020 2054 6865 206d 6964 646c f... The middl\n-00016df0: 6520 6a6f 696e 7420 6973 2074 6865 206e e joint is the n\n-00016e00: 6578 7420 6f6e 6520 646f 776e 2c20 7769 ext one down, wi\n-00016e10: 7468 2072 6f74 6174 696f 6e20 616e 676c th rotation angl\n-00016e20: 650a e280 986d 6964 646c 655f 726f 745f e....middle_rot_\n-00016e30: 31e2 8099 2e20 2057 6865 6e20 7468 6973 1.... When this\n-00016e40: 2061 6e67 6c65 2063 6861 6e67 6573 2c20 angle changes, \n-00016e50: 7765 206e 6565 6420 616c 6c20 7468 6520 we need all the \n-00016e60: 6f62 6a65 6374 7320 696e 0ae2 8098 6469 objects in....di\n-00016e70: 7374 616c 5f31 e280 9920 746f 2072 6f74 stal_1... to rot\n-00016e80: 6174 6520 6173 2061 2075 6e69 742e 2020 ate as a unit. \n-00016e90: 5468 6973 2069 7320 7468 6520 7265 6173 This is the reas\n-00016ea0: 6f6e 696e 6720 6265 6869 6e64 2074 6865 oning behind the\n-00016eb0: 206e 6578 740a 6465 6669 6e69 7469 6f6e next.definition\n-00016ec0: 2e0a 2020 6465 6620 6669 6e67 6572 5f31 .. def finger_1\n-00016ed0: 207b 0a20 2020 2070 7574 207b 2074 7261 {. put { tra\n-00016ee0: 6e73 6c61 7465 286a 6f69 6e74 5f67 6170 nslate(joint_gap\n-00016ef0: 202a 206a 6f69 6e74 5f72 6164 202a 205b * joint_rad * [\n-00016f00: 4a5d 290a 2020 2020 2020 2020 2020 7468 J]). th\n-00016f10: 656e 2072 6f74 6174 6528 6d69 6464 6c65 en rotate(middle\n-00016f20: 5f31 5f72 6f74 2c20 5b49 5d29 0a20 2020 _1_rot, [I]). \n-00016f30: 2020 2020 2020 2074 6865 6e20 7472 616e then tran\n-00016f40: 736c 6174 6528 286d 6964 646c 655f 7261 slate((middle_ra\n-00016f50: 7469 6f20 2a20 6469 7374 616c 5f6c 656e tio * distal_len\n-00016f60: 202b 200a 2020 2020 2020 2020 2020 2020 + . \n-00016f70: 2020 2020 2020 2020 2020 2020 2020 6a6f jo\n-00016f80: 696e 745f 6761 7020 2a20 6a6f 696e 745f int_gap * joint_\n-00016f90: 7261 6429 202a 205b 4a5d 2920 7d0a 2020 rad) * [J]) }. \n-00016fa0: 2020 2020 7b64 6973 7461 6c5f 317d 0a20 {distal_1}. \n-00016fb0: 2020 2070 7574 207b 2073 6361 6c65 2870 put { scale(p\n-00016fc0: 726f 7869 6d61 6c5f 6469 7374 616c 5f72 roximal_distal_r\n-00016fd0: 6174 696f 290a 2020 2020 2020 2020 2020 atio). \n-00016fe0: 7468 656e 2072 6f74 6174 6528 6d69 6464 then rotate(midd\n-00016ff0: 6c65 5f31 5f72 6f74 202f 2032 2c20 5b49 le_1_rot / 2, [I\n-00017000: 5d29 0a20 2020 2020 2020 2020 2074 6865 ]). the\n-00017010: 6e20 7472 616e 736c 6174 6528 286d 6964 n translate((mid\n-00017020: 646c 655f 7261 7469 6f20 2a20 6469 7374 dle_ratio * dist\n-00017030: 616c 5f6c 656e 202b 200a 2020 2020 2020 al_len + . \n+00016480: 3d3d 3d3d 3d3d 3d0a 0a57 6869 6c65 20e2 =======..While .\n+00016490: 8098 736b 6574 6368 e280 9920 7761 7320 ..sketch... was \n+000164a0: 6e65 7665 7220 6d65 616e 7420 746f 2062 never meant to b\n+000164b0: 6520 6120 6765 6f6d 6574 7269 6320 6d6f e a geometric mo\n+000164c0: 6465 6c69 6e67 206c 616e 6775 6167 652c deling language,\n+000164d0: 2069 740a 636f 6d65 7320 6661 6972 6c79 it.comes fairly\n+000164e0: 2063 6c6f 7365 2e20 2054 6865 2066 6f6c close. The fol\n+000164f0: 6c6f 7769 6e67 2065 7861 6d70 6c65 2070 lowing example p\n+00016500: 7574 7320 616c 6c20 7765 2068 6176 6520 uts all we have \n+00016510: 7365 656e 2074 6f20 776f 726b 0a69 6e20 seen to work.in \n+00016520: 6120 7665 7279 2073 696d 706c 6520 6d6f a very simple mo\n+00016530: 6465 6c20 6f66 2074 6865 2068 756d 616e del of the human\n+00016540: 2068 616e 642e 2020 5374 6172 7420 6279 hand. Start by\n+00016550: 2073 7765 6570 696e 6720 6120 6c69 6e65 sweeping a line\n+00016560: 2074 6f0a 6d61 6b65 2061 2074 7275 6e63 to.make a trunc\n+00016570: 6174 6564 2063 6f6e 652c 2077 6869 6368 ated cone, which\n+00016580: 2077 696c 6c20 6265 2063 6f70 6965 6420 will be copied \n+00016590: 6f76 6572 2061 6e64 206f 7665 7220 6167 over and over ag\n+000165a0: 6169 6e20 746f 206d 616b 650a 7468 6520 ain to make.the \n+000165b0: 7365 676d 656e 7473 206f 6620 6669 6e67 segments of fing\n+000165c0: 6572 732e 0a20 2064 6566 204f 2028 302c ers.. def O (0,\n+000165d0: 302c 3029 2025 206f 7269 6769 6e0a 2020 0,0) % origin. \n+000165e0: 6465 6620 4920 5b31 2c30 2c30 5d20 6465 def I [1,0,0] de\n+000165f0: 6620 4a20 5b30 2c31 2c30 5d20 6465 6620 f J [0,1,0] def \n+00016600: 4b20 5b30 2c30 2c31 5d20 2520 6361 6e6f K [0,0,1] % cano\n+00016610: 6e69 6361 6c20 756e 6974 2076 6563 746f nical unit vecto\n+00016620: 7273 0a20 2064 6566 2073 6567 6d65 6e74 rs. def segment\n+00016630: 207b 0a20 2020 2064 6566 206e 5f66 6163 {. def n_fac\n+00016640: 6573 2038 0a20 2020 2073 7765 6570 207b es 8. sweep {\n+00016650: 206e 5f66 6163 6573 3c3e 2c20 726f 7461 n_faces<>, rota\n+00016660: 7465 2833 3630 202f 206e 5f66 6163 6573 te(360 / n_faces\n+00016670: 2c20 5b4a 5d29 207d 200a 2020 2020 2020 , [J]) } . \n+00016680: 6c69 6e65 2870 726f 7869 6d61 6c5f 7261 line(proximal_ra\n+00016690: 642c 2030 2928 6469 7374 616c 5f72 6164 d, 0)(distal_rad\n+000166a0: 2c20 6469 7374 616c 5f6c 656e 290a 2020 , distal_len). \n+000166b0: 7d0a 496e 2068 616e 6420 616e 6174 6f6d }.In hand anatom\n+000166c0: 792c 205f 6469 7374 616c 5f20 6973 2022 y, _distal_ is \"\n+000166d0: 6174 2074 6865 2074 6970 2220 616e 6420 at the tip\" and \n+000166e0: 5f70 726f 7869 6d61 6c5f 2069 7320 2269 _proximal_ is \"i\n+000166f0: 6e20 7468 6520 6172 6561 0a6f 6620 7468 n the area.of th\n+00016700: 6520 7061 6c6d 2e22 2020 5765 2068 6176 e palm.\" We hav\n+00016710: 6520 6f6d 6974 7465 6420 616c 6c20 7468 e omitted all th\n+00016720: 6520 7363 616c 6172 2063 6f6e 7374 616e e scalar constan\n+00016730: 7473 2e20 2059 6f75 2063 616e 2066 696e ts. You can fin\n+00016740: 640a 7468 656d 2069 6e20 e280 9868 616e d.them in ...han\n+00016750: 642e 736b e280 992c 2077 6869 6368 2069 d.sk..., which i\n+00016760: 7320 7072 6f76 6964 6564 2069 6e20 7468 s provided in th\n+00016770: 6520 e280 9873 6b65 7463 68e2 8099 2064 e ...sketch... d\n+00016780: 6973 7472 6962 7574 696f 6e2e 0a0a 2020 istribution... \n+00016790: 2057 6520 616c 736f 206e 6565 6420 6120 We also need a \n+000167a0: 7072 6f74 6f74 7970 6963 616c 2073 7068 prototypical sph\n+000167b0: 6572 6520 746f 2075 7365 2066 6f72 2074 ere to use for t\n+000167c0: 6865 206a 6f69 6e74 7320 7468 656d 7365 he joints themse\n+000167d0: 6c76 6573 2e0a 2020 6465 6620 6a6f 696e lves.. def join\n+000167e0: 745f 7370 6865 7265 207b 0a20 2020 2064 t_sphere {. d\n+000167f0: 6566 206e 5f6a 6f69 6e74 5f66 6163 6573 ef n_joint_faces\n+00016800: 2038 0a20 2020 2073 7765 6570 205b 6669 8. sweep [fi\n+00016810: 6c6c 636f 6c6f 723d 7265 645d 207b 206e llcolor=red] { n\n+00016820: 5f6a 6f69 6e74 5f66 6163 6573 2c20 726f _joint_faces, ro\n+00016830: 7461 7465 2833 3630 202f 206e 5f6a 6f69 tate(360 / n_joi\n+00016840: 6e74 5f66 6163 6573 2c20 5b4a 5d29 207d nt_faces, [J]) }\n+00016850: 0a20 2020 2020 2073 7765 6570 207b 206e . sweep { n\n+00016860: 5f6a 6f69 6e74 5f66 6163 6573 2c20 726f _joint_faces, ro\n+00016870: 7461 7465 2831 3830 202f 206e 5f6a 6f69 tate(180 / n_joi\n+00016880: 6e74 5f66 6163 6573 2920 7d20 0a20 2020 nt_faces) } . \n+00016890: 2020 2020 2028 302c 202d 6a6f 696e 745f (0, -joint_\n+000168a0: 7261 6429 0a20 207d 0a0a 2020 2057 6527 rad). }.. We'\n+000168b0: 6c6c 206e 6f77 2064 6573 6967 6e20 7468 ll now design th\n+000168c0: 6520 696e 6465 7820 6669 6e67 6572 2028 e index finger (\n+000168d0: 6e75 6d62 6572 2031 2069 6e20 6f75 7220 number 1 in our \n+000168e0: 6e6f 7461 7469 6f6e 616c 0a63 6f6e 7665 notational.conve\n+000168f0: 6e74 696f 6e3b 2066 696e 6765 7220 3020 ntion; finger 0 \n+00016900: 6973 2074 6865 2074 6875 6d62 292e 2020 is the thumb). \n+00016910: 5468 6520 6469 7374 616c 2072 6f74 6174 The distal rotat\n+00016920: 696f 6e20 666f 7220 7468 6520 6669 6e67 ion for the fing\n+00016930: 6572 0a61 7070 6c69 6573 206f 6e6c 7920 er.applies only \n+00016940: 746f 2074 6865 2074 6970 2c20 736f 2077 to the tip, so w\n+00016950: 6520 6465 6669 6e65 2074 6865 2066 6f6c e define the fol\n+00016960: 6c6f 7769 6e67 2e0a 2020 6465 6620 6469 lowing.. def di\n+00016970: 7374 616c 5f31 207b 0a20 2020 2070 7574 stal_1 {. put\n+00016980: 207b 2074 7261 6e73 6c61 7465 286a 6f69 { translate(joi\n+00016990: 6e74 5f67 6170 202a 206a 6f69 6e74 5f72 nt_gap * joint_r\n+000169a0: 6164 202a 205b 4a5d 2920 0a20 2020 2020 ad * [J]) . \n+000169b0: 2020 2020 2074 6865 6e20 726f 7461 7465 then rotate\n+000169c0: 2864 6973 7461 6c5f 315f 726f 742c 205b (distal_1_rot, [\n+000169d0: 495d 2920 0a20 2020 2020 2020 2020 2074 I]) . t\n+000169e0: 6865 6e20 7472 616e 736c 6174 6528 2864 hen translate((d\n+000169f0: 6973 7461 6c5f 6c65 6e20 2b20 6a6f 696e istal_len + join\n+00016a00: 745f 6761 7020 2a20 6a6f 696e 745f 7261 t_gap * joint_ra\n+00016a10: 6429 202a 205b 4a5d 2920 7d0a 2020 2020 d) * [J]) }. \n+00016a20: 2020 7b73 6567 6d65 6e74 7d0a 2020 2020 {segment}. \n+00016a30: 7075 7420 7b20 726f 7461 7465 2864 6973 put { rotate(dis\n+00016a40: 7461 6c5f 315f 726f 7420 2f20 322c 205b tal_1_rot / 2, [\n+00016a50: 495d 290a 2020 2020 2020 2020 2020 7468 I]). th\n+00016a60: 656e 2074 7261 6e73 6c61 7465 2828 6469 en translate((di\n+00016a70: 7374 616c 5f6c 656e 202b 206a 6f69 6e74 stal_len + joint\n+00016a80: 5f67 6170 202a 206a 6f69 6e74 5f72 6164 _gap * joint_rad\n+00016a90: 2920 2a20 5b4a 5d29 207d 200a 2020 2020 ) * [J]) } . \n+00016aa0: 2020 7b6a 6f69 6e74 5f73 7068 6572 657d {joint_sphere}\n+00016ab0: 0a20 2020 2070 7574 207b 2073 6361 6c65 . put { scale\n+00016ac0: 2820 5b4a 5d20 2b20 7072 6f78 696d 616c ( [J] + proximal\n+00016ad0: 5f64 6973 7461 6c5f 7261 7469 6f20 2a20 _distal_ratio * \n+00016ae0: 285b 495d 2b5b 4b5d 2920 2920 7d0a 2020 ([I]+[K]) ) }. \n+00016af0: 2020 2020 7b73 6567 6d65 6e74 7d0a 2020 {segment}. \n+00016b00: 7d20 200a 5468 6520 6964 656e 7469 6669 } .The identifi\n+00016b10: 6572 7320 6865 7265 2061 7265 2066 6f72 ers here are for\n+00016b20: 2073 697a 6520 616e 6420 6c6f 6361 7469 size and locati\n+00016b30: 6f6e 2063 6f6e 7374 616e 7473 2e20 2054 on constants. T\n+00016b40: 6865 2065 7863 6570 7469 6f6e 0a69 7320 he exception.is \n+00016b50: e280 9864 6973 7461 6c5f 726f 745f 31e2 ...distal_rot_1.\n+00016b60: 8099 2e20 2054 6869 7320 726f 7461 7469 ... This rotati\n+00016b70: 6f6e 2070 6172 616d 6574 6572 206d 6f64 on parameter mod\n+00016b80: 656c 7320 7468 6520 666c 6578 696e 6720 els the flexing \n+00016b90: 6f66 2074 6865 0a66 696e 6765 7220 7469 of the.finger ti\n+00016ba0: 702e 2020 5468 6520 6669 7273 7420 e280 p. The first ..\n+00016bb0: 9870 7574 e280 9920 6d61 6b65 7320 6120 .put... makes a \n+00016bc0: 636f 7079 206f 6620 7468 6520 6669 6e67 copy of the fing\n+00016bd0: 6572 2073 6567 6d65 6e74 2074 6861 7420 er segment that \n+00016be0: 6973 0a74 7261 6e73 6c61 7465 6420 7570 is.translated up\n+00016bf0: 7761 7264 206a 7573 7420 6661 7220 656e ward just far en\n+00016c00: 6f75 6768 2074 6f20 6d61 6b65 2072 6f6f ough to make roo\n+00016c10: 6d20 666f 7220 7468 6520 7370 6865 7269 m for the spheri\n+00016c20: 6361 6c20 6a6f 696e 742e 0a54 6865 6e20 cal joint..Then \n+00016c30: 6974 2061 7070 6c69 6573 2074 6865 2064 it applies the d\n+00016c40: 6973 7461 6c20 726f 7461 7469 6f6e 2e20 istal rotation. \n+00016c50: 2046 696e 616c 6c79 2069 7420 7472 616e Finally it tran\n+00016c60: 736c 6174 6573 2074 6865 2077 686f 6c65 slates the whole\n+00016c70: 0a61 7373 656d 626c 7920 7570 7761 7264 .assembly upward\n+00016c80: 2061 6761 696e 2074 6f20 6d61 6b65 2072 again to make r\n+00016c90: 6f6f 6d20 666f 7220 7468 6520 6d69 6464 oom for the midd\n+00016ca0: 6c65 2070 686c 616e 6765 7320 2874 6865 le phlanges (the\n+00016cb0: 206e 6578 740a 626f 6e65 2074 6f77 6172 next.bone towar\n+00016cc0: 6420 7468 6520 7061 6c6d 292e 2020 5468 d the palm). Th\n+00016cd0: 6520 7365 636f 6e64 20e2 8098 7075 74e2 e second ...put.\n+00016ce0: 8099 2070 6f73 6974 696f 6e73 2074 6865 .. positions the\n+00016cf0: 2073 7068 6572 652e 2020 5468 6572 6520 sphere. There \n+00016d00: 6973 0a61 2072 6f74 6174 696f 6e20 746f is.a rotation to\n+00016d10: 2070 6c61 6365 2074 6865 2067 7269 6420 place the grid \n+00016d20: 6f6e 2074 6865 2073 7068 6572 6520 7375 on the sphere su\n+00016d30: 7266 6163 6520 6174 2061 6e20 6e69 6365 rface at an nice\n+00016d40: 2061 6e67 6c65 2c0a 7468 656e 2061 2074 angle,.then a t\n+00016d50: 7261 6e73 6c61 7469 6f6e 2074 6f20 7468 ranslation to th\n+00016d60: 6520 6261 7365 206f 6620 7468 6520 6469 e base of the di\n+00016d70: 7374 616c 2070 686c 616e 6765 732c 2077 stal phlanges, w\n+00016d80: 6869 6368 2069 7320 616c 736f 0a63 656e hich is also.cen\n+00016d90: 7465 7220 6f66 2069 7473 2072 6f74 6174 ter of its rotat\n+00016da0: 696f 6e2e 2020 4669 6e61 6c6c 792c 2074 ion. Finally, t\n+00016db0: 6865 206c 6173 7420 e280 9870 7574 e280 he last ...put..\n+00016dc0: 9920 706f 7369 7469 6f6e 7320 7468 6520 . positions the \n+00016dd0: 6d69 6464 6c65 0a73 6567 6d65 6e74 2069 middle.segment i\n+00016de0: 7473 656c 662e 0a0a 2020 2054 6865 206d tself... The m\n+00016df0: 6964 646c 6520 6a6f 696e 7420 6973 2074 iddle joint is t\n+00016e00: 6865 206e 6578 7420 6f6e 6520 646f 776e he next one down\n+00016e10: 2c20 7769 7468 2072 6f74 6174 696f 6e20 , with rotation \n+00016e20: 616e 676c 650a e280 986d 6964 646c 655f angle....middle_\n+00016e30: 726f 745f 31e2 8099 2e20 2057 6865 6e20 rot_1.... When \n+00016e40: 7468 6973 2061 6e67 6c65 2063 6861 6e67 this angle chang\n+00016e50: 6573 2c20 7765 206e 6565 6420 616c 6c20 es, we need all \n+00016e60: 7468 6520 6f62 6a65 6374 7320 696e 0ae2 the objects in..\n+00016e70: 8098 6469 7374 616c 5f31 e280 9920 746f ..distal_1... to\n+00016e80: 2072 6f74 6174 6520 6173 2061 2075 6e69 rotate as a uni\n+00016e90: 742e 2020 5468 6973 2069 7320 7468 6520 t. This is the \n+00016ea0: 7265 6173 6f6e 696e 6720 6265 6869 6e64 reasoning behind\n+00016eb0: 2074 6865 206e 6578 740a 6465 6669 6e69 the next.defini\n+00016ec0: 7469 6f6e 2e0a 2020 6465 6620 6669 6e67 tion.. def fing\n+00016ed0: 6572 5f31 207b 0a20 2020 2070 7574 207b er_1 {. put {\n+00016ee0: 2074 7261 6e73 6c61 7465 286a 6f69 6e74 translate(joint\n+00016ef0: 5f67 6170 202a 206a 6f69 6e74 5f72 6164 _gap * joint_rad\n+00016f00: 202a 205b 4a5d 290a 2020 2020 2020 2020 * [J]). \n+00016f10: 2020 7468 656e 2072 6f74 6174 6528 6d69 then rotate(mi\n+00016f20: 6464 6c65 5f31 5f72 6f74 2c20 5b49 5d29 ddle_1_rot, [I])\n+00016f30: 0a20 2020 2020 2020 2020 2074 6865 6e20 . then \n+00016f40: 7472 616e 736c 6174 6528 286d 6964 646c translate((middl\n+00016f50: 655f 7261 7469 6f20 2a20 6469 7374 616c e_ratio * distal\n+00016f60: 5f6c 656e 202b 200a 2020 2020 2020 2020 _len + . \n+00016f70: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00016f80: 2020 6a6f 696e 745f 6761 7020 2a20 6a6f joint_gap * jo\n+00016f90: 696e 745f 7261 6429 202a 205b 4a5d 2920 int_rad) * [J]) \n+00016fa0: 7d0a 2020 2020 2020 7b64 6973 7461 6c5f }. {distal_\n+00016fb0: 317d 0a20 2020 2070 7574 207b 2073 6361 1}. put { sca\n+00016fc0: 6c65 2870 726f 7869 6d61 6c5f 6469 7374 le(proximal_dist\n+00016fd0: 616c 5f72 6174 696f 290a 2020 2020 2020 al_ratio). \n+00016fe0: 2020 2020 7468 656e 2072 6f74 6174 6528 then rotate(\n+00016ff0: 6d69 6464 6c65 5f31 5f72 6f74 202f 2032 middle_1_rot / 2\n+00017000: 2c20 5b49 5d29 0a20 2020 2020 2020 2020 , [I]). \n+00017010: 2074 6865 6e20 7472 616e 736c 6174 6528 then translate(\n+00017020: 286d 6964 646c 655f 7261 7469 6f20 2a20 (middle_ratio * \n+00017030: 6469 7374 616c 5f6c 656e 202b 200a 2020 distal_len + . \n 00017040: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00017050: 2020 2020 6a6f 696e 745f 6761 7020 2a20 joint_gap * \n-00017060: 6a6f 696e 745f 7261 6429 202a 205b 4a5d joint_rad) * [J]\n-00017070: 2920 7d20 0a20 2020 2020 207b 6a6f 696e ) } . {join\n-00017080: 745f 7370 6865 7265 7d0a 2020 2020 7075 t_sphere}. pu\n-00017090: 7420 7b20 7363 616c 6528 206d 6964 646c t { scale( middl\n-000170a0: 655f 7261 7469 6f20 2a20 5b4a 5d20 2b20 e_ratio * [J] + \n-000170b0: 0a20 2020 2020 2020 2020 2020 2020 2020 . \n-000170c0: 2020 7072 6f78 696d 616c 5f64 6973 7461 proximal_dista\n-000170d0: 6c5f 7261 7469 6f5e 3220 2a20 285b 495d l_ratio^2 * ([I]\n-000170e0: 2b5b 4b5d 2920 2920 7d0a 2020 2020 2020 +[K]) ) }. \n-000170f0: 7b73 6567 6d65 6e74 7d0a 2020 7d0a 5468 {segment}. }.Th\n-00017100: 6973 206c 6f6f 6b73 2076 6572 7920 7369 is looks very si\n-00017110: 6d69 6c61 7220 746f 2074 6865 2070 7265 milar to the pre\n-00017120: 7669 6f75 7320 6465 6669 6e69 7469 6f6e vious definition\n-00017130: 2c20 616e 6420 6974 2069 732e 2020 5468 , and it is. Th\n-00017140: 650a 696d 706f 7274 616e 7420 6469 6666 e.important diff\n-00017150: 6572 656e 6365 2069 7320 7468 6174 2072 erence is that r\n-00017160: 6174 6865 7220 7468 616e 2070 6f73 6974 ather than posit\n-00017170: 696f 6e69 6e67 2061 6e64 2072 6f74 6174 ioning and rotat\n-00017180: 696e 6720 610a 7369 6e67 6c65 2073 6567 ing a.single seg\n-00017190: 6d65 6e74 2c20 7765 2070 6f73 6974 696f ment, we positio\n-000171a0: 6e20 616e 6420 726f 7461 7465 2074 6865 n and rotate the\n-000171b0: 2065 6e74 6972 6520 2261 7373 656d 626c entire \"assembl\n-000171c0: 7922 2064 6566 696e 6564 2061 730a e280 y\" defined as...\n-000171d0: 9864 6973 7461 6c5f 31e2 8099 2e20 2054 .distal_1.... T\n-000171e0: 6865 2072 6573 7420 6973 206a 7573 7420 he rest is just \n-000171f0: 6172 6974 686d 6574 6963 2074 6f20 636f arithmetic to co\n-00017200: 6d70 7574 6520 7369 7a65 7320 616e 6420 mpute sizes and \n-00017210: 706f 7369 7469 6f6e 730a 7468 6174 206c positions.that l\n-00017220: 6f6f 6b20 6e69 6365 2e20 2054 6865 206c ook nice. The l\n-00017230: 6173 7420 e280 9870 7574 e280 9920 706c ast ...put... pl\n-00017240: 6163 6573 2061 6e20 6170 7072 6f70 7269 aces an appropri\n-00017250: 6174 656c 7920 7368 6170 6564 2073 6567 ately shaped seg\n-00017260: 6d65 6e74 0a74 6861 7420 6973 2074 6865 ment.that is the\n-00017270: 205f 7072 6f78 696d 616c 2070 686c 616e _proximal phlan\n-00017280: 6765 735f 2c20 7468 6520 626f 6e65 2074 ges_, the bone t\n-00017290: 6861 7420 6a6f 696e 7320 7468 6520 7061 hat joins the pa\n-000172a0: 6c6d 206f 6620 7468 650a 6861 6e64 2e20 lm of the.hand. \n-000172b0: 2054 6869 7320 636f 6d70 6c65 7465 7320 This completes \n-000172c0: 7468 6520 6669 6e67 6572 2069 7473 656c the finger itsel\n-000172d0: 662e 0a0a 2020 2041 6c6c 2074 6865 206f f... All the o\n-000172e0: 7468 6572 2066 696e 6765 7273 2061 7265 ther fingers are\n-000172f0: 2064 6573 6372 6962 6564 2069 6465 6e74 described ident\n-00017300: 6963 616c 6c79 2074 6f20 7468 6973 206f ically to this o\n-00017310: 6e65 2e20 2057 650a 6163 636f 756e 7420 ne. We.account \n-00017320: 666f 7220 7468 6520 6661 6374 2074 6861 for the fact tha\n-00017330: 7420 7265 616c 2066 696e 6765 7273 2061 t real fingers a\n-00017340: 7265 2064 6966 6665 7265 6e74 2073 697a re different siz\n-00017350: 6573 2069 6e20 7468 6520 6e65 7874 0a73 es in the next.s\n-00017360: 7465 702c 2077 6869 6368 2069 7320 746f tep, which is to\n-00017370: 2062 7569 6c64 2074 6865 2065 6e74 6972 build the entir\n-00017380: 6520 6861 6e64 2e0a 0a20 2020 5468 6520 e hand... The \n-00017390: e280 9868 616e 64e2 8099 2064 6566 696e ...hand... defin\n-000173a0: 6974 696f 6e20 7468 6174 2066 6f6c 6c6f ition that follo\n-000173b0: 7773 2069 6e63 6c75 6465 7320 6120 7365 ws includes a se\n-000173c0: 6374 696f 6e20 666f 7220 6561 6368 0a66 ction for each.f\n-000173d0: 696e 6765 722e 2020 5765 276c 6c20 636f inger. We'll co\n-000173e0: 6e74 696e 7565 2077 6974 6820 6669 6e67 ntinue with fing\n-000173f0: 6572 2031 2061 6e64 206f 6d69 7420 616c er 1 and omit al\n-00017400: 6c20 7468 6520 6f74 6865 7273 2e20 2028 l the others. (\n-00017410: 4f66 206e 6f74 650a 6973 2074 6861 7420 Of note.is that \n-00017420: 7468 6520 7468 756d 6220 6e65 6564 7320 the thumb needs \n-00017430: 736c 6967 6874 6c79 2073 7065 6369 616c slightly special\n-00017440: 2074 7265 6174 6d65 6e74 2d2d 616e 2065 treatment--an e\n-00017450: 7874 7261 2072 6f74 6174 696f 6e20 746f xtra rotation to\n-00017460: 0a61 6363 6f75 6e74 2066 6f72 2069 7473 .account for its\n-00017470: 206f 7070 6f73 696e 6720 616e 676c 652e opposing angle.\n-00017480: 2020 5468 6973 2069 7320 636c 6561 7220 This is clear \n-00017490: 696e 2074 6865 2066 756c 6c20 736f 7572 in the full sour\n-000174a0: 6365 2063 6f64 652e 290a 4e6f 7420 7375 ce code.).Not su\n-000174b0: 7270 7269 7369 6e67 6c79 2c20 7468 6520 rprisingly, the \n-000174c0: e280 9868 616e 64e2 8099 2064 6566 696e ...hand... defin\n-000174d0: 6974 696f 6e20 6c6f 6f6b 7320 7665 7279 ition looks very\n-000174e0: 206d 7563 6820 6c69 6b65 2074 6865 0a70 much like the.p\n-000174f0: 7265 7669 6f75 7320 7477 6f2e 2020 4974 revious two. It\n-00017500: 2073 686f 756c 6420 6265 206e 6f20 7375 should be no su\n-00017510: 7270 7269 7365 2074 6861 7420 7768 656e rprise that when\n-00017520: 2074 6865 2072 6f74 6174 696f 6e20 7061 the rotation pa\n-00017530: 7261 6d65 7465 720a e280 986d 6574 615f rameter....meta_\n-00017540: 315f 726f 74e2 8099 2063 6861 6e67 6573 1_rot... changes\n-00017550: 2c20 7468 6520 656e 7469 7265 2066 696e , the entire fin\n-00017560: 6765 7220 726f 7461 7465 7321 2020 5468 ger rotates! Th\n-00017570: 6572 6520 6973 2061 6e20 6164 6469 7469 ere is an additi\n-00017580: 6f6e 616c 0a72 6f74 6174 696f 6e20 7468 onal.rotation th\n-00017590: 6174 2061 6c6c 6f77 7320 7468 6520 6669 at allows the fi\n-000175a0: 6e67 6572 7320 746f 2073 7072 6561 6420 ngers to spread \n-000175b0: 6c61 7465 7261 6c6c 792e 2020 5765 2073 laterally. We s\n-000175c0: 6179 2074 6865 7365 0a6a 6f69 6e74 7320 ay these.joints \n-000175d0: 6f66 2074 6865 2070 726f 7869 6d61 6c20 of the proximal \n-000175e0: 7068 6c61 6e67 6573 2068 6176 6520 7477 phlanges have tw\n-000175f0: 6f20 5f64 6567 7265 6573 206f 6620 6672 o _degrees of fr\n-00017600: 6565 646f 6d5f 2e20 2054 6865 0a6a 6f69 eedom_. The.joi\n-00017610: 6e74 7320 6869 6768 6572 206f 6e20 7468 nts higher on th\n-00017620: 6520 6669 6e67 6572 2068 6176 6520 6f6e e finger have on\n-00017630: 6c79 206f 6e65 2e20 2046 696e 616c 6c79 ly one. Finally\n-00017640: 2c20 6561 6368 2066 696e 6765 7220 6973 , each finger is\n-00017650: 0a73 6361 6c65 6420 6279 2061 2066 6163 .scaled by a fac\n-00017660: 746f 7220 746f 206c 656e 6420 6974 2070 tor to lend it p\n-00017670: 726f 706f 7274 696f 6e2e 0a20 2064 6566 roportion.. def\n-00017680: 2068 616e 6420 7b0a 2020 2020 2520 6669 hand {. % fi\n-00017690: 6e67 6572 2031 205b 616c 6c20 6f74 6865 nger 1 [all othe\n-000176a0: 7220 6669 6e67 6572 7320 6f6d 6974 7465 r fingers omitte\n-000176b0: 645d 0a20 2020 2064 6566 2073 6361 6c65 d]. def scale\n-000176c0: 5f31 202e 3835 0a20 2020 2070 7574 207b _1 .85. put {\n-000176d0: 2073 6361 6c65 2873 6361 6c65 5f31 2920 scale(scale_1) \n-000176e0: 0a20 2020 2020 2020 2020 2074 6865 6e20 . then \n-000176f0: 7472 616e 736c 6174 6528 286a 6f69 6e74 translate((joint\n-00017700: 5f67 6170 202a 206a 6f69 6e74 5f72 6164 _gap * joint_rad\n-00017710: 2920 2a20 5b4a 5d29 0a09 2020 7468 656e ) * [J]).. then\n-00017720: 2072 6f74 6174 6528 6d65 7461 5f31 5f72 rotate(meta_1_r\n-00017730: 6f74 2c20 5b49 5d29 0a20 2020 2020 2020 ot, [I]). \n-00017740: 2020 2074 6865 6e20 726f 7461 7465 282d then rotate(-\n-00017750: 7370 7265 6164 5f72 6f74 2c20 5b4b 5d29 spread_rot, [K])\n-00017760: 0a20 2020 2020 2020 2020 2074 6865 6e20 . then \n-00017770: 7472 616e 736c 6174 6528 2870 726f 7869 translate((proxi\n-00017780: 6d61 6c5f 315f 6c6f 6329 202d 2028 4f29 mal_1_loc) - (O)\n-00017790: 2920 7d20 0a20 2020 2020 207b 6669 6e67 ) } . {fing\n-000177a0: 6572 5f31 7d0a 2020 2020 7075 7420 7b20 er_1}. put { \n-000177b0: 7363 616c 6528 7363 616c 655f 3120 2a20 scale(scale_1 * \n-000177c0: 7072 6f78 696d 616c 5f64 6973 7461 6c5f proximal_distal_\n-000177d0: 7261 7469 6f5e 3229 0a20 2020 2020 2020 ratio^2). \n-000177e0: 2020 2074 6865 6e20 726f 7461 7465 286d then rotate(m\n-000177f0: 6574 615f 315f 726f 7420 2f20 322c 205b eta_1_rot / 2, [\n-00017800: 495d 290a 2020 2020 2020 2020 2020 7468 I]). th\n-00017810: 656e 2072 6f74 6174 6528 2d73 7072 6561 en rotate(-sprea\n-00017820: 645f 726f 742c 205b 4b5d 290a 2020 2020 d_rot, [K]). \n-00017830: 2020 2020 2020 7468 656e 2074 7261 6e73 then trans\n-00017840: 6c61 7465 2828 7072 6f78 696d 616c 5f31 late((proximal_1\n-00017850: 5f6c 6f63 2920 2d20 284f 2929 207d 200a _loc) - (O)) } .\n-00017860: 2020 2020 2020 7b6a 6f69 6e74 5f73 7068 {joint_sph\n-00017870: 6572 657d 0a0a 2020 2020 2520 7061 6c6d ere}.. % palm\n-00017880: 0a20 2020 2073 7765 6570 207b 2031 2c20 . sweep { 1, \n-00017890: 726f 7461 7465 2836 2c20 2830 2c31 352c rotate(6, (0,15,\n-000178a0: 3029 2c20 5b49 5d29 207d 0a20 2020 2020 0), [I]) }. \n-000178b0: 2070 7574 207b 2072 6f74 6174 6528 2d33 put { rotate(-3\n-000178c0: 2c20 2830 2c31 352c 3029 2c20 5b49 5d29 , (0,15,0), [I])\n-000178d0: 207d 207b 0a20 2020 2020 2020 2070 6f6c } {. pol\n-000178e0: 7967 6f6e 2870 726f 7869 6d61 6c5f 315f ygon(proximal_1_\n-000178f0: 6c6f 6329 2870 726f 7869 6d61 6c5f 325f loc)(proximal_2_\n-00017900: 6c6f 6329 0a20 2020 2020 2020 2020 2020 loc). \n-00017910: 2020 2020 2870 726f 7869 6d61 6c5f 335f (proximal_3_\n-00017920: 6c6f 6329 2870 726f 7869 6d61 6c5f 345f loc)(proximal_4_\n-00017930: 6c6f 6329 0a20 2020 2020 2020 2020 2020 loc). \n-00017940: 2020 2020 2868 3529 2868 3629 2868 3661 (h5)(h6)(h6a\n-00017950: 2928 6839 2928 6831 3029 0a20 2020 2020 )(h9)(h10). \n-00017960: 2020 2070 6f6c 7967 6f6e 2868 3661 2928 polygon(h6a)(\n-00017970: 6837 2928 6838 2928 6839 290a 2020 207d h7)(h8)(h9). }\n-00017980: 2020 7d0a 5468 6520 6c61 7374 2073 6563 }.The last sec\n-00017990: 7469 6f6e 206f 6620 7468 6520 6465 6669 tion of the defi\n-000179a0: 6e69 7469 6f6e 2063 7265 6174 6573 2074 nition creates t\n-000179b0: 6865 2070 6f6c 7974 6f70 6520 666f 7220 he polytope for \n-000179c0: 7468 6520 7061 6c6d 206f 660a 7468 6520 the palm of.the \n-000179d0: 6861 6e64 2062 7920 e280 9873 7765 6570 hand by ...sweep\n-000179e0: e280 9969 6e67 2061 2031 302d 7369 6465 ...ing a 10-side\n-000179f0: 6420 706f 6c79 676f 6e20 7468 726f 7567 d polygon throug\n-00017a00: 6820 6120 7665 7279 2073 686f 7274 2061 h a very short a\n-00017a10: 7263 0a28 3920 6465 6772 6565 7329 2e20 rc.(9 degrees). \n-00017a20: 2054 6869 7320 7072 6f76 6964 6573 2061 This provides a\n-00017a30: 2077 6564 6765 2d73 6861 7065 6420 7072 wedge-shaped pr\n-00017a40: 6f66 696c 6520 7768 656e 2076 6965 7765 ofile when viewe\n-00017a50: 6420 6672 6f6d 2074 6865 0a73 6964 652e d from the.side.\n-00017a60: 2020 5468 6520 7468 6963 6b20 656e 6420 The thick end \n-00017a70: 6f66 2074 6865 2077 6564 6765 2069 7320 of the wedge is \n-00017a80: 7468 6520 7772 6973 742e 2020 4265 6361 the wrist. Beca\n-00017a90: 7573 6520 7468 6520 706f 6c79 676f 6e20 use the polygon \n-00017aa0: 6973 0a63 6f6e 6361 7665 2c20 6974 2069 is.concave, it i\n-00017ab0: 7320 7370 6c69 7420 696e 746f 2069 6e74 s split into int\n-00017ac0: 6f20 7477 6f20 636f 6e76 6578 2073 6861 o two convex sha\n-00017ad0: 7065 7320 7769 7468 206e 696e 6520 616e pes with nine an\n-00017ae0: 6420 666f 7572 0a76 6572 7469 6365 732e d four.vertices.\n-00017af0: 0a0a 2020 2057 6520 6361 6e20 6e6f 7720 .. We can now \n-00017b00: 6861 7665 2066 756e 2070 6f73 6974 696f have fun positio\n-00017b10: 6e69 6e67 2074 6865 2068 616e 6420 6279 ning the hand by\n-00017b20: 2061 646a 7573 7469 6e67 2074 6865 2076 adjusting the v\n-00017b30: 6172 696f 7573 0a72 6f74 6174 696f 6e20 arious.rotation \n-00017b40: 616e 676c 6573 2e20 2054 6865 2063 6f6d angles. The com\n-00017b50: 706c 6574 6520 736f 7572 6365 2069 6e63 plete source inc\n-00017b60: 6c75 6465 7320 6465 6669 6e69 7469 6f6e ludes definition\n-00017b70: 7320 7769 7468 0a61 6c74 6572 6e61 7469 s with.alternati\n-00017b80: 7665 7320 7468 6174 2069 6e63 6c75 6465 ves that include\n-00017b90: 2074 6865 2066 6f6c 6c6f 7769 6e67 2076 the following v\n-00017ba0: 6965 7773 2061 6e64 206d 6f72 652e 0a0a iews and more...\n-00017bb0: 0008 5b69 6d61 6765 2073 7263 3d22 6578 ..[image src=\"ex\n-00017bc0: 3231 302e 706e 6722 2074 6578 743d 2249 210.png\" text=\"I\n-00017bd0: 6d61 6765 2065 7832 3130 206f 6d69 7474 mage ex210 omitt\n-00017be0: 6564 2069 6e20 7465 7874 2076 6572 7369 ed in text versi\n-00017bf0: 6f6e 206f 6620 7468 6973 2064 6f63 756d on of this docum\n-00017c00: 656e 742e 2200 085d 0008 5b69 6d61 6765 ent.\"..]..[image\n-00017c10: 2073 7263 3d22 6578 3232 302e 706e 6722 src=\"ex220.png\"\n-00017c20: 2074 6578 743d 2249 6d61 6765 2065 7832 text=\"Image ex2\n-00017c30: 3230 206f 6d69 7474 6564 2069 6e20 7465 20 omitted in te\n-00017c40: 7874 2076 6572 7369 6f6e 206f 6620 7468 xt version of th\n-00017c50: 6973 2064 6f63 756d 656e 742e 2200 085d is document.\"..]\n-00017c60: 0008 5b69 6d61 6765 2073 7263 3d22 6578 ..[image src=\"ex\n-00017c70: 3233 302e 706e 6722 2074 6578 743d 2249 230.png\" text=\"I\n-00017c80: 6d61 6765 2065 7832 3330 206f 6d69 7474 mage ex230 omitt\n-00017c90: 6564 2069 6e20 7465 7874 2076 6572 7369 ed in text versi\n-00017ca0: 6f6e 206f 6620 7468 6973 2064 6f63 756d on of this docum\n-00017cb0: 656e 742e 2200 085d 0008 5b69 6d61 6765 ent.\"..]..[image\n-00017cc0: 2073 7263 3d22 6578 3234 302e 706e 6722 src=\"ex240.png\"\n-00017cd0: 2074 6578 743d 2249 6d61 6765 2065 7832 text=\"Image ex2\n-00017ce0: 3430 206f 6d69 7474 6564 2069 6e20 7465 40 omitted in te\n-00017cf0: 7874 2076 6572 7369 6f6e 206f 6620 7468 xt version of th\n-00017d00: 6973 2064 6f63 756d 656e 742e 2200 085d is document.\"..]\n-00017d10: 0a0a 1f0a 4669 6c65 3a20 736b 6574 6368 ....File: sketch\n-00017d20: 2e69 6e66 6f2c 2020 4e6f 6465 3a20 4361 .info, Node: Ca\n-00017d30: 7665 6174 732c 2020 5072 6576 3a20 4120 veats, Prev: A \n-00017d40: 6869 6572 6172 6368 6963 616c 206d 6f64 hierarchical mod\n-00017d50: 656c 2c20 2055 703a 2042 7569 6c64 696e el, Up: Buildin\n-00017d60: 6720 6120 6472 6177 696e 670a 0a34 2e34 g a drawing..4.4\n-00017d70: 2043 6176 6561 7473 0a3d 3d3d 3d3d 3d3d Caveats.=======\n-00017d80: 3d3d 3d3d 0a0a e280 9853 6b65 7463 68e2 ====.....Sketch.\n-00017d90: 8099 2069 7320 6120 6661 6972 6c79 2070 .. is a fairly p\n-00017da0: 6f77 6572 6675 6c20 746f 6f6c 2066 6f72 owerful tool for\n-00017db0: 2064 7261 7769 6e67 2c20 6275 742c 206a drawing, but, j\n-00017dc0: 7573 7420 6173 2077 6974 6820 5465 582c ust as with TeX,\n-00017dd0: 0a74 6865 2070 6f77 6572 2074 6f20 6372 .the power to cr\n-00017de0: 6561 7465 2062 6561 7574 6966 756c 2072 eate beautiful r\n-00017df0: 6573 756c 7473 2063 6f6d 6573 2061 6c6f esults comes alo\n-00017e00: 6e67 2077 6974 6820 7468 6520 706f 7765 ng with the powe\n-00017e10: 7220 746f 206d 616b 650a 6d69 7374 616b r to make.mistak\n-00017e20: 6573 2e20 2054 6865 2066 6f6c 6c6f 7769 es. The followi\n-00017e30: 6e67 2061 7265 2073 6f6d 6520 706f 696e ng are some poin\n-00017e40: 7473 2077 6865 7265 2063 6172 6520 6973 ts where care is\n-00017e50: 206e 6563 6573 7361 7279 2061 6e64 0a77 necessary and.w\n-00017e60: 6865 7265 2074 6865 2063 7572 7265 6e74 here the current\n-00017e70: 2076 6572 7369 6f6e 206f 6620 e280 9873 version of ...s\n-00017e80: 6b65 7463 68e2 8099 2069 7320 6c69 6d69 ketch... is limi\n-00017e90: 7465 6420 6f72 2068 6173 206b 6e6f 776e ted or has known\n-00017ea0: 2062 7567 732e 0a0a 2a20 4d65 6e75 3a0a bugs...* Menu:.\n-00017eb0: 0a2a 204c 696d 6974 7320 6f6e 2065 7272 .* Limits on err\n-00017ec0: 6f72 2064 6574 6563 7469 6f6e 3a3a 2020 or detection:: \n-00017ed0: 2057 6861 7420 736b 6574 6368 2064 6f65 What sketch doe\n-00017ee0: 736e 2774 2064 6f2e 0a2a 2043 6c69 7070 sn't do..* Clipp\n-00017ef0: 696e 673a 3a20 2020 2020 2020 2020 2020 ing:: \n-00017f00: 2020 2020 2020 2020 204e 6f20 636c 6970 No clip\n-00017f10: 7069 6e67 2061 7420 7072 6573 656e 742e ping at present.\n-00017f20: 0a2a 2048 6964 6465 6e20 7375 7266 6163 .* Hidden surfac\n-00017f30: 6520 7265 6d6f 7661 6c3a 3a20 2020 2020 e removal:: \n-00017f40: 2049 6d70 6572 6665 6374 696f 6e73 2074 Imperfections t\n-00017f50: 6f20 6669 782e 0a0a 1f0a 4669 6c65 3a20 o fix.....File: \n-00017f60: 736b 6574 6368 2e69 6e66 6f2c 2020 4e6f sketch.info, No\n-00017f70: 6465 3a20 4c69 6d69 7473 206f 6e20 6572 de: Limits on er\n-00017f80: 726f 7220 6465 7465 6374 696f 6e2c 2020 ror detection, \n-00017f90: 4e65 7874 3a20 436c 6970 7069 6e67 2c20 Next: Clipping, \n-00017fa0: 2050 7265 763a 2043 6176 6561 7473 2c20 Prev: Caveats, \n-00017fb0: 2055 703a 2043 6176 6561 7473 0a0a 342e Up: Caveats..4.\n-00017fc0: 342e 3120 4c69 6d69 7473 206f 6e20 e280 4.1 Limits on ..\n-00017fd0: 9873 6b65 7463 68e2 8099 2065 7272 6f72 .sketch... error\n-00017fe0: 2064 6574 6563 7469 6f6e 0a2d 2d2d 2d2d detection.-----\n+00017050: 2020 2020 2020 2020 6a6f 696e 745f 6761 joint_ga\n+00017060: 7020 2a20 6a6f 696e 745f 7261 6429 202a p * joint_rad) *\n+00017070: 205b 4a5d 2920 7d20 0a20 2020 2020 207b [J]) } . {\n+00017080: 6a6f 696e 745f 7370 6865 7265 7d0a 2020 joint_sphere}. \n+00017090: 2020 7075 7420 7b20 7363 616c 6528 206d put { scale( m\n+000170a0: 6964 646c 655f 7261 7469 6f20 2a20 5b4a iddle_ratio * [J\n+000170b0: 5d20 2b20 0a20 2020 2020 2020 2020 2020 ] + . \n+000170c0: 2020 2020 2020 7072 6f78 696d 616c 5f64 proximal_d\n+000170d0: 6973 7461 6c5f 7261 7469 6f5e 3220 2a20 istal_ratio^2 * \n+000170e0: 285b 495d 2b5b 4b5d 2920 2920 7d0a 2020 ([I]+[K]) ) }. \n+000170f0: 2020 2020 7b73 6567 6d65 6e74 7d0a 2020 {segment}. \n+00017100: 7d0a 5468 6973 206c 6f6f 6b73 2076 6572 }.This looks ver\n+00017110: 7920 7369 6d69 6c61 7220 746f 2074 6865 y similar to the\n+00017120: 2070 7265 7669 6f75 7320 6465 6669 6e69 previous defini\n+00017130: 7469 6f6e 2c20 616e 6420 6974 2069 732e tion, and it is.\n+00017140: 2020 5468 650a 696d 706f 7274 616e 7420 The.important \n+00017150: 6469 6666 6572 656e 6365 2069 7320 7468 difference is th\n+00017160: 6174 2072 6174 6865 7220 7468 616e 2070 at rather than p\n+00017170: 6f73 6974 696f 6e69 6e67 2061 6e64 2072 ositioning and r\n+00017180: 6f74 6174 696e 6720 610a 7369 6e67 6c65 otating a.single\n+00017190: 2073 6567 6d65 6e74 2c20 7765 2070 6f73 segment, we pos\n+000171a0: 6974 696f 6e20 616e 6420 726f 7461 7465 ition and rotate\n+000171b0: 2074 6865 2065 6e74 6972 6520 2261 7373 the entire \"ass\n+000171c0: 656d 626c 7922 2064 6566 696e 6564 2061 embly\" defined a\n+000171d0: 730a e280 9864 6973 7461 6c5f 31e2 8099 s....distal_1...\n+000171e0: 2e20 2054 6865 2072 6573 7420 6973 206a . The rest is j\n+000171f0: 7573 7420 6172 6974 686d 6574 6963 2074 ust arithmetic t\n+00017200: 6f20 636f 6d70 7574 6520 7369 7a65 7320 o compute sizes \n+00017210: 616e 6420 706f 7369 7469 6f6e 730a 7468 and positions.th\n+00017220: 6174 206c 6f6f 6b20 6e69 6365 2e20 2054 at look nice. T\n+00017230: 6865 206c 6173 7420 e280 9870 7574 e280 he last ...put..\n+00017240: 9920 706c 6163 6573 2061 6e20 6170 7072 . places an appr\n+00017250: 6f70 7269 6174 656c 7920 7368 6170 6564 opriately shaped\n+00017260: 2073 6567 6d65 6e74 0a74 6861 7420 6973 segment.that is\n+00017270: 2074 6865 205f 7072 6f78 696d 616c 2070 the _proximal p\n+00017280: 686c 616e 6765 735f 2c20 7468 6520 626f hlanges_, the bo\n+00017290: 6e65 2074 6861 7420 6a6f 696e 7320 7468 ne that joins th\n+000172a0: 6520 7061 6c6d 206f 6620 7468 650a 6861 e palm of the.ha\n+000172b0: 6e64 2e20 2054 6869 7320 636f 6d70 6c65 nd. This comple\n+000172c0: 7465 7320 7468 6520 6669 6e67 6572 2069 tes the finger i\n+000172d0: 7473 656c 662e 0a0a 2020 2041 6c6c 2074 tself... All t\n+000172e0: 6865 206f 7468 6572 2066 696e 6765 7273 he other fingers\n+000172f0: 2061 7265 2064 6573 6372 6962 6564 2069 are described i\n+00017300: 6465 6e74 6963 616c 6c79 2074 6f20 7468 dentically to th\n+00017310: 6973 206f 6e65 2e20 2057 650a 6163 636f is one. We.acco\n+00017320: 756e 7420 666f 7220 7468 6520 6661 6374 unt for the fact\n+00017330: 2074 6861 7420 7265 616c 2066 696e 6765 that real finge\n+00017340: 7273 2061 7265 2064 6966 6665 7265 6e74 rs are different\n+00017350: 2073 697a 6573 2069 6e20 7468 6520 6e65 sizes in the ne\n+00017360: 7874 0a73 7465 702c 2077 6869 6368 2069 xt.step, which i\n+00017370: 7320 746f 2062 7569 6c64 2074 6865 2065 s to build the e\n+00017380: 6e74 6972 6520 6861 6e64 2e0a 0a20 2020 ntire hand... \n+00017390: 5468 6520 e280 9868 616e 64e2 8099 2064 The ...hand... d\n+000173a0: 6566 696e 6974 696f 6e20 7468 6174 2066 efinition that f\n+000173b0: 6f6c 6c6f 7773 2069 6e63 6c75 6465 7320 ollows includes \n+000173c0: 6120 7365 6374 696f 6e20 666f 7220 6561 a section for ea\n+000173d0: 6368 0a66 696e 6765 722e 2020 5765 276c ch.finger. We'l\n+000173e0: 6c20 636f 6e74 696e 7565 2077 6974 6820 l continue with \n+000173f0: 6669 6e67 6572 2031 2061 6e64 206f 6d69 finger 1 and omi\n+00017400: 7420 616c 6c20 7468 6520 6f74 6865 7273 t all the others\n+00017410: 2e20 2028 4f66 206e 6f74 650a 6973 2074 . (Of note.is t\n+00017420: 6861 7420 7468 6520 7468 756d 6220 6e65 hat the thumb ne\n+00017430: 6564 7320 736c 6967 6874 6c79 2073 7065 eds slightly spe\n+00017440: 6369 616c 2074 7265 6174 6d65 6e74 2d2d cial treatment--\n+00017450: 616e 2065 7874 7261 2072 6f74 6174 696f an extra rotatio\n+00017460: 6e20 746f 0a61 6363 6f75 6e74 2066 6f72 n to.account for\n+00017470: 2069 7473 206f 7070 6f73 696e 6720 616e its opposing an\n+00017480: 676c 652e 2020 5468 6973 2069 7320 636c gle. This is cl\n+00017490: 6561 7220 696e 2074 6865 2066 756c 6c20 ear in the full \n+000174a0: 736f 7572 6365 2063 6f64 652e 290a 4e6f source code.).No\n+000174b0: 7420 7375 7270 7269 7369 6e67 6c79 2c20 t surprisingly, \n+000174c0: 7468 6520 e280 9868 616e 64e2 8099 2064 the ...hand... d\n+000174d0: 6566 696e 6974 696f 6e20 6c6f 6f6b 7320 efinition looks \n+000174e0: 7665 7279 206d 7563 6820 6c69 6b65 2074 very much like t\n+000174f0: 6865 0a70 7265 7669 6f75 7320 7477 6f2e he.previous two.\n+00017500: 2020 4974 2073 686f 756c 6420 6265 206e It should be n\n+00017510: 6f20 7375 7270 7269 7365 2074 6861 7420 o surprise that \n+00017520: 7768 656e 2074 6865 2072 6f74 6174 696f when the rotatio\n+00017530: 6e20 7061 7261 6d65 7465 720a e280 986d n parameter....m\n+00017540: 6574 615f 315f 726f 74e2 8099 2063 6861 eta_1_rot... cha\n+00017550: 6e67 6573 2c20 7468 6520 656e 7469 7265 nges, the entire\n+00017560: 2066 696e 6765 7220 726f 7461 7465 7321 finger rotates!\n+00017570: 2020 5468 6572 6520 6973 2061 6e20 6164 There is an ad\n+00017580: 6469 7469 6f6e 616c 0a72 6f74 6174 696f ditional.rotatio\n+00017590: 6e20 7468 6174 2061 6c6c 6f77 7320 7468 n that allows th\n+000175a0: 6520 6669 6e67 6572 7320 746f 2073 7072 e fingers to spr\n+000175b0: 6561 6420 6c61 7465 7261 6c6c 792e 2020 ead laterally. \n+000175c0: 5765 2073 6179 2074 6865 7365 0a6a 6f69 We say these.joi\n+000175d0: 6e74 7320 6f66 2074 6865 2070 726f 7869 nts of the proxi\n+000175e0: 6d61 6c20 7068 6c61 6e67 6573 2068 6176 mal phlanges hav\n+000175f0: 6520 7477 6f20 5f64 6567 7265 6573 206f e two _degrees o\n+00017600: 6620 6672 6565 646f 6d5f 2e20 2054 6865 f freedom_. The\n+00017610: 0a6a 6f69 6e74 7320 6869 6768 6572 206f .joints higher o\n+00017620: 6e20 7468 6520 6669 6e67 6572 2068 6176 n the finger hav\n+00017630: 6520 6f6e 6c79 206f 6e65 2e20 2046 696e e only one. Fin\n+00017640: 616c 6c79 2c20 6561 6368 2066 696e 6765 ally, each finge\n+00017650: 7220 6973 0a73 6361 6c65 6420 6279 2061 r is.scaled by a\n+00017660: 2066 6163 746f 7220 746f 206c 656e 6420 factor to lend \n+00017670: 6974 2070 726f 706f 7274 696f 6e2e 0a20 it proportion.. \n+00017680: 2064 6566 2068 616e 6420 7b0a 2020 2020 def hand {. \n+00017690: 2520 6669 6e67 6572 2031 205b 616c 6c20 % finger 1 [all \n+000176a0: 6f74 6865 7220 6669 6e67 6572 7320 6f6d other fingers om\n+000176b0: 6974 7465 645d 0a20 2020 2064 6566 2073 itted]. def s\n+000176c0: 6361 6c65 5f31 202e 3835 0a20 2020 2070 cale_1 .85. p\n+000176d0: 7574 207b 2073 6361 6c65 2873 6361 6c65 ut { scale(scale\n+000176e0: 5f31 2920 0a20 2020 2020 2020 2020 2074 _1) . t\n+000176f0: 6865 6e20 7472 616e 736c 6174 6528 286a hen translate((j\n+00017700: 6f69 6e74 5f67 6170 202a 206a 6f69 6e74 oint_gap * joint\n+00017710: 5f72 6164 2920 2a20 5b4a 5d29 0a09 2020 _rad) * [J]).. \n+00017720: 7468 656e 2072 6f74 6174 6528 6d65 7461 then rotate(meta\n+00017730: 5f31 5f72 6f74 2c20 5b49 5d29 0a20 2020 _1_rot, [I]). \n+00017740: 2020 2020 2020 2074 6865 6e20 726f 7461 then rota\n+00017750: 7465 282d 7370 7265 6164 5f72 6f74 2c20 te(-spread_rot, \n+00017760: 5b4b 5d29 0a20 2020 2020 2020 2020 2074 [K]). t\n+00017770: 6865 6e20 7472 616e 736c 6174 6528 2870 hen translate((p\n+00017780: 726f 7869 6d61 6c5f 315f 6c6f 6329 202d roximal_1_loc) -\n+00017790: 2028 4f29 2920 7d20 0a20 2020 2020 207b (O)) } . {\n+000177a0: 6669 6e67 6572 5f31 7d0a 2020 2020 7075 finger_1}. pu\n+000177b0: 7420 7b20 7363 616c 6528 7363 616c 655f t { scale(scale_\n+000177c0: 3120 2a20 7072 6f78 696d 616c 5f64 6973 1 * proximal_dis\n+000177d0: 7461 6c5f 7261 7469 6f5e 3229 0a20 2020 tal_ratio^2). \n+000177e0: 2020 2020 2020 2074 6865 6e20 726f 7461 then rota\n+000177f0: 7465 286d 6574 615f 315f 726f 7420 2f20 te(meta_1_rot / \n+00017800: 322c 205b 495d 290a 2020 2020 2020 2020 2, [I]). \n+00017810: 2020 7468 656e 2072 6f74 6174 6528 2d73 then rotate(-s\n+00017820: 7072 6561 645f 726f 742c 205b 4b5d 290a pread_rot, [K]).\n+00017830: 2020 2020 2020 2020 2020 7468 656e 2074 then t\n+00017840: 7261 6e73 6c61 7465 2828 7072 6f78 696d ranslate((proxim\n+00017850: 616c 5f31 5f6c 6f63 2920 2d20 284f 2929 al_1_loc) - (O))\n+00017860: 207d 200a 2020 2020 2020 7b6a 6f69 6e74 } . {joint\n+00017870: 5f73 7068 6572 657d 0a0a 2020 2020 2520 _sphere}.. % \n+00017880: 7061 6c6d 0a20 2020 2073 7765 6570 207b palm. sweep {\n+00017890: 2031 2c20 726f 7461 7465 2836 2c20 2830 1, rotate(6, (0\n+000178a0: 2c31 352c 3029 2c20 5b49 5d29 207d 0a20 ,15,0), [I]) }. \n+000178b0: 2020 2020 2070 7574 207b 2072 6f74 6174 put { rotat\n+000178c0: 6528 2d33 2c20 2830 2c31 352c 3029 2c20 e(-3, (0,15,0), \n+000178d0: 5b49 5d29 207d 207b 0a20 2020 2020 2020 [I]) } {. \n+000178e0: 2070 6f6c 7967 6f6e 2870 726f 7869 6d61 polygon(proxima\n+000178f0: 6c5f 315f 6c6f 6329 2870 726f 7869 6d61 l_1_loc)(proxima\n+00017900: 6c5f 325f 6c6f 6329 0a20 2020 2020 2020 l_2_loc). \n+00017910: 2020 2020 2020 2020 2870 726f 7869 6d61 (proxima\n+00017920: 6c5f 335f 6c6f 6329 2870 726f 7869 6d61 l_3_loc)(proxima\n+00017930: 6c5f 345f 6c6f 6329 0a20 2020 2020 2020 l_4_loc). \n+00017940: 2020 2020 2020 2020 2868 3529 2868 3629 (h5)(h6)\n+00017950: 2868 3661 2928 6839 2928 6831 3029 0a20 (h6a)(h9)(h10). \n+00017960: 2020 2020 2020 2070 6f6c 7967 6f6e 2868 polygon(h\n+00017970: 3661 2928 6837 2928 6838 2928 6839 290a 6a)(h7)(h8)(h9).\n+00017980: 2020 207d 2020 7d0a 5468 6520 6c61 7374 } }.The last\n+00017990: 2073 6563 7469 6f6e 206f 6620 7468 6520 section of the \n+000179a0: 6465 6669 6e69 7469 6f6e 2063 7265 6174 definition creat\n+000179b0: 6573 2074 6865 2070 6f6c 7974 6f70 6520 es the polytope \n+000179c0: 666f 7220 7468 6520 7061 6c6d 206f 660a for the palm of.\n+000179d0: 7468 6520 6861 6e64 2062 7920 e280 9873 the hand by ...s\n+000179e0: 7765 6570 e280 9969 6e67 2061 2031 302d weep...ing a 10-\n+000179f0: 7369 6465 6420 706f 6c79 676f 6e20 7468 sided polygon th\n+00017a00: 726f 7567 6820 6120 7665 7279 2073 686f rough a very sho\n+00017a10: 7274 2061 7263 0a28 3920 6465 6772 6565 rt arc.(9 degree\n+00017a20: 7329 2e20 2054 6869 7320 7072 6f76 6964 s). This provid\n+00017a30: 6573 2061 2077 6564 6765 2d73 6861 7065 es a wedge-shape\n+00017a40: 6420 7072 6f66 696c 6520 7768 656e 2076 d profile when v\n+00017a50: 6965 7765 6420 6672 6f6d 2074 6865 0a73 iewed from the.s\n+00017a60: 6964 652e 2020 5468 6520 7468 6963 6b20 ide. The thick \n+00017a70: 656e 6420 6f66 2074 6865 2077 6564 6765 end of the wedge\n+00017a80: 2069 7320 7468 6520 7772 6973 742e 2020 is the wrist. \n+00017a90: 4265 6361 7573 6520 7468 6520 706f 6c79 Because the poly\n+00017aa0: 676f 6e20 6973 0a63 6f6e 6361 7665 2c20 gon is.concave, \n+00017ab0: 6974 2069 7320 7370 6c69 7420 696e 746f it is split into\n+00017ac0: 2069 6e74 6f20 7477 6f20 636f 6e76 6578 into two convex\n+00017ad0: 2073 6861 7065 7320 7769 7468 206e 696e shapes with nin\n+00017ae0: 6520 616e 6420 666f 7572 0a76 6572 7469 e and four.verti\n+00017af0: 6365 732e 0a0a 2020 2057 6520 6361 6e20 ces... We can \n+00017b00: 6e6f 7720 6861 7665 2066 756e 2070 6f73 now have fun pos\n+00017b10: 6974 696f 6e69 6e67 2074 6865 2068 616e itioning the han\n+00017b20: 6420 6279 2061 646a 7573 7469 6e67 2074 d by adjusting t\n+00017b30: 6865 2076 6172 696f 7573 0a72 6f74 6174 he various.rotat\n+00017b40: 696f 6e20 616e 676c 6573 2e20 2054 6865 ion angles. The\n+00017b50: 2063 6f6d 706c 6574 6520 736f 7572 6365 complete source\n+00017b60: 2069 6e63 6c75 6465 7320 6465 6669 6e69 includes defini\n+00017b70: 7469 6f6e 7320 7769 7468 0a61 6c74 6572 tions with.alter\n+00017b80: 6e61 7469 7665 7320 7468 6174 2069 6e63 natives that inc\n+00017b90: 6c75 6465 2074 6865 2066 6f6c 6c6f 7769 lude the followi\n+00017ba0: 6e67 2076 6965 7773 2061 6e64 206d 6f72 ng views and mor\n+00017bb0: 652e 0a0a 0008 5b69 6d61 6765 2073 7263 e.....[image src\n+00017bc0: 3d22 6578 3231 302e 706e 6722 2074 6578 =\"ex210.png\" tex\n+00017bd0: 743d 2249 6d61 6765 2065 7832 3130 206f t=\"Image ex210 o\n+00017be0: 6d69 7474 6564 2069 6e20 7465 7874 2076 mitted in text v\n+00017bf0: 6572 7369 6f6e 206f 6620 7468 6973 2064 ersion of this d\n+00017c00: 6f63 756d 656e 742e 2200 085d 0008 5b69 ocument.\"..]..[i\n+00017c10: 6d61 6765 2073 7263 3d22 6578 3232 302e mage src=\"ex220.\n+00017c20: 706e 6722 2074 6578 743d 2249 6d61 6765 png\" text=\"Image\n+00017c30: 2065 7832 3230 206f 6d69 7474 6564 2069 ex220 omitted i\n+00017c40: 6e20 7465 7874 2076 6572 7369 6f6e 206f n text version o\n+00017c50: 6620 7468 6973 2064 6f63 756d 656e 742e f this document.\n+00017c60: 2200 085d 0008 5b69 6d61 6765 2073 7263 \"..]..[image src\n+00017c70: 3d22 6578 3233 302e 706e 6722 2074 6578 =\"ex230.png\" tex\n+00017c80: 743d 2249 6d61 6765 2065 7832 3330 206f t=\"Image ex230 o\n+00017c90: 6d69 7474 6564 2069 6e20 7465 7874 2076 mitted in text v\n+00017ca0: 6572 7369 6f6e 206f 6620 7468 6973 2064 ersion of this d\n+00017cb0: 6f63 756d 656e 742e 2200 085d 0008 5b69 ocument.\"..]..[i\n+00017cc0: 6d61 6765 2073 7263 3d22 6578 3234 302e mage src=\"ex240.\n+00017cd0: 706e 6722 2074 6578 743d 2249 6d61 6765 png\" text=\"Image\n+00017ce0: 2065 7832 3430 206f 6d69 7474 6564 2069 ex240 omitted i\n+00017cf0: 6e20 7465 7874 2076 6572 7369 6f6e 206f n text version o\n+00017d00: 6620 7468 6973 2064 6f63 756d 656e 742e f this document.\n+00017d10: 2200 085d 0a0a 1f0a 4669 6c65 3a20 736b \"..]....File: sk\n+00017d20: 6574 6368 2e69 6e66 6f2c 2020 4e6f 6465 etch.info, Node\n+00017d30: 3a20 4361 7665 6174 732c 2020 5072 6576 : Caveats, Prev\n+00017d40: 3a20 4120 6869 6572 6172 6368 6963 616c : A hierarchical\n+00017d50: 206d 6f64 656c 2c20 2055 703a 2042 7569 model, Up: Bui\n+00017d60: 6c64 696e 6720 6120 6472 6177 696e 670a lding a drawing.\n+00017d70: 0a34 2e34 2043 6176 6561 7473 0a3d 3d3d .4.4 Caveats.===\n+00017d80: 3d3d 3d3d 3d3d 3d3d 0a0a e280 9853 6b65 ========.....Ske\n+00017d90: 7463 68e2 8099 2069 7320 6120 6661 6972 tch... is a fair\n+00017da0: 6c79 2070 6f77 6572 6675 6c20 746f 6f6c ly powerful tool\n+00017db0: 2066 6f72 2064 7261 7769 6e67 2c20 6275 for drawing, bu\n+00017dc0: 742c 206a 7573 7420 6173 2077 6974 6820 t, just as with \n+00017dd0: 5465 582c 0a74 6865 2070 6f77 6572 2074 TeX,.the power t\n+00017de0: 6f20 6372 6561 7465 2062 6561 7574 6966 o create beautif\n+00017df0: 756c 2072 6573 756c 7473 2063 6f6d 6573 ul results comes\n+00017e00: 2061 6c6f 6e67 2077 6974 6820 7468 6520 along with the \n+00017e10: 706f 7765 7220 746f 206d 616b 650a 6d69 power to make.mi\n+00017e20: 7374 616b 6573 2e20 2054 6865 2066 6f6c stakes. The fol\n+00017e30: 6c6f 7769 6e67 2061 7265 2073 6f6d 6520 lowing are some \n+00017e40: 706f 696e 7473 2077 6865 7265 2063 6172 points where car\n+00017e50: 6520 6973 206e 6563 6573 7361 7279 2061 e is necessary a\n+00017e60: 6e64 0a77 6865 7265 2074 6865 2063 7572 nd.where the cur\n+00017e70: 7265 6e74 2076 6572 7369 6f6e 206f 6620 rent version of \n+00017e80: e280 9873 6b65 7463 68e2 8099 2069 7320 ...sketch... is \n+00017e90: 6c69 6d69 7465 6420 6f72 2068 6173 206b limited or has k\n+00017ea0: 6e6f 776e 2062 7567 732e 0a0a 2a20 4d65 nown bugs...* Me\n+00017eb0: 6e75 3a0a 0a2a 204c 696d 6974 7320 6f6e nu:..* Limits on\n+00017ec0: 2065 7272 6f72 2064 6574 6563 7469 6f6e error detection\n+00017ed0: 3a3a 2020 2057 6861 7420 736b 6574 6368 :: What sketch\n+00017ee0: 2064 6f65 736e 2774 2064 6f2e 0a2a 2043 doesn't do..* C\n+00017ef0: 6c69 7070 696e 673a 3a20 2020 2020 2020 lipping:: \n+00017f00: 2020 2020 2020 2020 2020 2020 204e 6f20 No \n+00017f10: 636c 6970 7069 6e67 2061 7420 7072 6573 clipping at pres\n+00017f20: 656e 742e 0a2a 2048 6964 6465 6e20 7375 ent..* Hidden su\n+00017f30: 7266 6163 6520 7265 6d6f 7661 6c3a 3a20 rface removal:: \n+00017f40: 2020 2020 2049 6d70 6572 6665 6374 696f Imperfectio\n+00017f50: 6e73 2074 6f20 6669 782e 0a0a 1f0a 4669 ns to fix.....Fi\n+00017f60: 6c65 3a20 736b 6574 6368 2e69 6e66 6f2c le: sketch.info,\n+00017f70: 2020 4e6f 6465 3a20 4c69 6d69 7473 206f Node: Limits o\n+00017f80: 6e20 6572 726f 7220 6465 7465 6374 696f n error detectio\n+00017f90: 6e2c 2020 4e65 7874 3a20 436c 6970 7069 n, Next: Clippi\n+00017fa0: 6e67 2c20 2050 7265 763a 2043 6176 6561 ng, Prev: Cavea\n+00017fb0: 7473 2c20 2055 703a 2043 6176 6561 7473 ts, Up: Caveats\n+00017fc0: 0a0a 342e 342e 3120 4c69 6d69 7473 206f ..4.4.1 Limits o\n+00017fd0: 6e20 e280 9873 6b65 7463 68e2 8099 2065 n ...sketch... e\n+00017fe0: 7272 6f72 2064 6574 6563 7469 6f6e 0a2d rror detection.-\n 00017ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ----------------\n 00018000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ----------------\n-00018010: 2d2d 2d0a 0ae2 8098 536b 6574 6368 e280 ---.....Sketch..\n-00018020: 9920 6361 7463 6865 7320 6d61 6e79 206b . catches many k\n-00018030: 696e 6473 206f 6620 6572 726f 7273 2c20 inds of errors, \n-00018040: 6275 7420 6e6f 7420 616c 6c2e 2020 466f but not all. Fo\n-00018050: 7220 6578 616d 706c 652c 0a6f 7074 696f r example,.optio\n-00018060: 6e73 2074 6861 7420 736b 6574 6368 2064 ns that sketch d\n-00018070: 6f65 7320 6e6f 7420 7265 636f 676e 697a oes not recogniz\n-00018080: 652c 2065 7665 6e20 696e 636f 7272 6563 e, even incorrec\n-00018090: 7420 6f6e 6573 2c20 6172 6520 7175 6965 t ones, are quie\n-000180a0: 746c 790a 636f 7069 6564 2074 6f20 e280 tly.copied to ..\n-000180b0: 9850 5354 7269 636b 73e2 8099 2063 6f6d .PSTricks... com\n-000180c0: 6d61 6e64 7320 696e 2074 6865 206f 7574 mands in the out\n-000180d0: 7075 742e 2020 4974 2069 7320 616c 736f put. It is also\n-000180e0: 2075 6e66 6f72 7475 6e61 7465 6c79 0a65 unfortunately.e\n-000180f0: 6173 7920 746f 2070 726f 6475 6365 20e2 asy to produce .\n-00018100: 8098 736b 6574 6368 e280 9920 696e 7075 ..sketch... inpu\n-00018110: 7473 2074 6861 7420 6c65 6164 2074 6f20 ts that lead to \n-00018120: 6e6f 2070 6963 7475 7265 2061 7420 616c no picture at al\n-00018130: 6c20 2869 6d70 726f 7065 720a 7665 7274 l (improper.vert\n-00018140: 6578 206f 7264 6572 696e 6720 6361 7573 ex ordering caus\n-00018150: 6573 2065 7665 7279 7468 696e 6720 746f es everything to\n-00018160: 2062 6520 6375 6c6c 6564 292c 2074 6f20 be culled), to \n-00018170: 7069 6374 7572 6573 2074 6861 7420 6172 pictures that ar\n-00018180: 650a 746f 6f20 6269 6720 6f72 2074 6f6f e.too big or too\n-00018190: 2073 6d61 6c6c 2066 6f72 20e2 8098 5053 small for ...PS\n-000181a0: 5472 6963 6b73 e280 9920 746f 2064 7261 Tricks... to dra\n-000181b0: 7720 2864 7565 2074 6f20 6c69 6d69 7473 w (due to limits\n-000181c0: 206f 6620 5465 5820 6d61 7468 292c 0a61 of TeX math),.a\n-000181d0: 6e64 2070 6963 7475 7265 7320 7468 6174 nd pictures that\n-000181e0: 206c 6f6f 6b20 6e6f 7468 696e 6720 6c69 look nothing li\n-000181f0: 6b65 2077 6861 7420 7761 7320 696e 7465 ke what was inte\n-00018200: 6e64 6564 2e20 2041 2070 6963 7475 7265 nded. A picture\n-00018210: 2077 6974 680a 6f6e 6520 6f66 2074 6865 with.one of the\n-00018220: 7365 2070 726f 626c 656d 7320 6361 6e20 se problems can \n-00018230: 6265 2064 6966 6669 6375 6c74 2074 6f20 be difficult to \n-00018240: 2264 6562 7567 2e22 2020 5765 206f 6666 \"debug.\" We off\n-00018250: 6572 2074 6865 0a66 6f6c 6c6f 7769 6e67 er the.following\n-00018260: 2073 7567 6765 7374 696f 6e73 2e0a 2020 suggestions.. \n-00018270: 20e2 80a2 2046 6f6c 6c6f 7720 7468 6520 ... Follow the \n-00018280: 7375 6767 6573 7465 6420 696e 6372 656d suggested increm\n-00018290: 656e 7461 6c20 6465 7665 6c6f 706d 656e ental developmen\n-000182a0: 7420 6d65 7468 6f64 2064 6573 6372 6962 t method describ\n-000182b0: 6564 2069 6e0a 2020 2020 202a 6e6f 7465 ed in. *note\n-000182c0: 204f 7665 7276 6965 773a 3a2e 0a20 2020 Overview::.. \n-000182d0: e280 a220 416c 7761 7973 206d 6169 6e74 ... Always maint\n-000182e0: 6169 6e20 6f6e 6520 6f72 2074 776f 2062 ain one or two b\n-000182f0: 6163 6b2d 7665 7273 696f 6e73 206f 6620 ack-versions of \n-00018300: 6120 6472 6177 696e 6720 736f 2074 6861 a drawing so tha\n-00018310: 7420 6974 2069 730a 2020 2020 2065 6173 t it is. eas\n-00018320: 7920 746f 2066 616c 6c20 6261 636b 2074 y to fall back t\n-00018330: 6f20 6120 6b6e 6f77 6e2d 676f 6f64 2069 o a known-good i\n-00018340: 6e63 7265 6d65 6e74 2e0a 2020 20e2 80a2 ncrement.. ...\n-00018350: 2057 6865 6e20 7573 696e 6720 e280 9870 When using ...p\n-00018360: 6572 7370 6563 7469 7665 e280 992c 2065 erspective..., e\n-00018370: 6e73 7572 6520 616c 6c20 6669 6e61 6c6c nsure all finall\n-00018380: 7920 7472 616e 7366 6f72 6d65 6420 6f62 y transformed ob\n-00018390: 6a65 6374 730a 2020 2020 2073 6174 6973 jects. satis\n-000183a0: 6679 207a 3c30 2061 6e64 2c20 696e 2066 fy z<0 and, in f\n-000183b0: 6163 742c 2064 6f20 6e6f 7420 636f 6d65 act, do not come\n-000183c0: 2076 6572 7920 636c 6f73 6520 746f 2074 very close to t\n-000183d0: 6865 206f 7269 6769 6e20 6174 0a20 2020 he origin at. \n-000183e0: 2020 616c 6c2e 0a20 2020 e280 a220 5465 all.. ... Te\n-000183f0: 6d70 6f72 6172 696c 7920 7573 6520 e280 mporarily use ..\n-00018400: 9863 756c 6c3d 6661 6c73 65e2 8099 2074 .cull=false... t\n-00018410: 6f20 7365 6520 7768 6572 6520 7665 7274 o see where vert\n-00018420: 6578 206f 7264 6572 696e 6720 7072 6f62 ex ordering prob\n-00018430: 6c65 6d73 0a20 2020 2020 6c69 652e 0a20 lems. lie.. \n-00018440: 2020 e280 a220 5573 6520 7465 6d70 6f72 ... Use tempor\n-00018450: 6172 7920 6368 616e 6765 7320 6f66 2063 ary changes of c\n-00018460: 6f6c 6f72 206f 6620 6f6e 6520 6f72 206d olor of one or m\n-00018470: 6f72 6520 6f62 6a65 6374 7320 746f 2065 ore objects to e\n-00018480: 6e73 7572 650a 2020 2020 2074 6861 7420 nsure. that \n-00018490: 796f 7572 2075 6e64 6572 7374 616e 6469 your understandi\n-000184a0: 6e67 206f 6620 7468 6520 7363 656e 6520 ng of the scene \n-000184b0: 6765 6f6d 6574 7279 2069 7320 636f 7272 geometry is corr\n-000184c0: 6563 742e 0a20 2020 e280 a220 4966 20e2 ect.. ... If .\n-000184d0: 8098 5053 5472 6963 6b73 e280 9920 636f ..PSTricks... co\n-000184e0: 6d70 6c61 696e 7320 6162 6f75 7420 736f mplains about so\n-000184f0: 6d65 7468 696e 672c 2069 6e73 7065 6374 mething, inspect\n-00018500: 2074 6865 206f 7574 7075 740a 2020 2020 the output. \n-00018510: 2064 6972 6563 746c 7920 666f 7220 636c directly for cl\n-00018520: 7565 732e 0a0a 1f0a 4669 6c65 3a20 736b ues.....File: sk\n-00018530: 6574 6368 2e69 6e66 6f2c 2020 4e6f 6465 etch.info, Node\n-00018540: 3a20 436c 6970 7069 6e67 2c20 204e 6578 : Clipping, Nex\n-00018550: 743a 2048 6964 6465 6e20 7375 7266 6163 t: Hidden surfac\n-00018560: 6520 7265 6d6f 7661 6c2c 2020 5072 6576 e removal, Prev\n-00018570: 3a20 4c69 6d69 7473 206f 6e20 6572 726f : Limits on erro\n-00018580: 7220 6465 7465 6374 696f 6e2c 2020 5570 r detection, Up\n-00018590: 3a20 4361 7665 6174 730a 0a34 2e34 2e32 : Caveats..4.4.2\n-000185a0: 2043 6c69 7070 696e 670a 2d2d 2d2d 2d2d Clipping.------\n-000185b0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 6375 --------..The cu\n-000185c0: 7272 656e 7420 7665 7273 696f 6e20 6f66 rrent version of\n-000185d0: 20e2 8098 736b 6574 6368 e280 9920 6861 ...sketch... ha\n-000185e0: 7320 6e6f 2063 6c69 7070 696e 6720 6f70 s no clipping op\n-000185f0: 6572 6174 696f 6e73 2e20 2054 6865 2065 erations. The e\n-00018600: 6e74 6972 650a 7363 656e 6520 6973 2061 ntire.scene is a\n-00018610: 6c77 6179 7320 6472 6177 6e2e 2020 5468 lways drawn. Th\n-00018620: 6973 206d 6561 6e73 2074 6861 7420 7768 is means that wh\n-00018630: 656e 2061 2070 6572 7370 6563 7469 7665 en a perspective\n-00018640: 2074 7261 6e73 666f 726d 2069 730a 656d transform is.em\n-00018650: 706c 6f79 6564 2c20 6974 2069 7320 7468 ployed, it is th\n-00018660: 6520 7573 6572 2773 2072 6573 706f 6e73 e user's respons\n-00018670: 6962 696c 6974 7920 746f 206d 616b 6520 ibility to make \n-00018680: 7375 7265 2074 6865 2065 6e74 6972 6520 sure the entire \n-00018690: 7363 656e 650a 7265 6d61 696e 7320 696e scene.remains in\n-000186a0: 2066 726f 6e74 206f 6620 7468 6520 7669 front of the vi\n-000186b0: 6577 6572 2c20 7468 6520 7265 6769 6f6e ewer, the region\n-000186c0: 207a 3c30 2e0a 0a1f 0a46 696c 653a 2073 z<0.....File: s\n-000186d0: 6b65 7463 682e 696e 666f 2c20 204e 6f64 ketch.info, Nod\n-000186e0: 653a 2048 6964 6465 6e20 7375 7266 6163 e: Hidden surfac\n-000186f0: 6520 7265 6d6f 7661 6c2c 2020 5072 6576 e removal, Prev\n-00018700: 3a20 436c 6970 7069 6e67 2c20 2055 703a : Clipping, Up:\n-00018710: 2043 6176 6561 7473 0a0a 342e 342e 3320 Caveats..4.4.3 \n-00018720: 4869 6464 656e 2073 7572 6661 6365 2072 Hidden surface r\n-00018730: 656d 6f76 616c 2061 6e64 2070 6f6c 7967 emoval and polyg\n-00018740: 6f6e 2073 706c 6974 7469 6e67 0a2d 2d2d on splitting.---\n-00018750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ----------------\n+00018010: 2d2d 2d2d 2d2d 2d0a 0ae2 8098 536b 6574 -------.....Sket\n+00018020: 6368 e280 9920 6361 7463 6865 7320 6d61 ch... catches ma\n+00018030: 6e79 206b 696e 6473 206f 6620 6572 726f ny kinds of erro\n+00018040: 7273 2c20 6275 7420 6e6f 7420 616c 6c2e rs, but not all.\n+00018050: 2020 466f 7220 6578 616d 706c 652c 0a6f For example,.o\n+00018060: 7074 696f 6e73 2074 6861 7420 736b 6574 ptions that sket\n+00018070: 6368 2064 6f65 7320 6e6f 7420 7265 636f ch does not reco\n+00018080: 676e 697a 652c 2065 7665 6e20 696e 636f gnize, even inco\n+00018090: 7272 6563 7420 6f6e 6573 2c20 6172 6520 rrect ones, are \n+000180a0: 7175 6965 746c 790a 636f 7069 6564 2074 quietly.copied t\n+000180b0: 6f20 e280 9850 5354 7269 636b 73e2 8099 o ...PSTricks...\n+000180c0: 2063 6f6d 6d61 6e64 7320 696e 2074 6865 commands in the\n+000180d0: 206f 7574 7075 742e 2020 4974 2069 7320 output. It is \n+000180e0: 616c 736f 2075 6e66 6f72 7475 6e61 7465 also unfortunate\n+000180f0: 6c79 0a65 6173 7920 746f 2070 726f 6475 ly.easy to produ\n+00018100: 6365 20e2 8098 736b 6574 6368 e280 9920 ce ...sketch... \n+00018110: 696e 7075 7473 2074 6861 7420 6c65 6164 inputs that lead\n+00018120: 2074 6f20 6e6f 2070 6963 7475 7265 2061 to no picture a\n+00018130: 7420 616c 6c20 2869 6d70 726f 7065 720a t all (improper.\n+00018140: 7665 7274 6578 206f 7264 6572 696e 6720 vertex ordering \n+00018150: 6361 7573 6573 2065 7665 7279 7468 696e causes everythin\n+00018160: 6720 746f 2062 6520 6375 6c6c 6564 292c g to be culled),\n+00018170: 2074 6f20 7069 6374 7572 6573 2074 6861 to pictures tha\n+00018180: 7420 6172 650a 746f 6f20 6269 6720 6f72 t are.too big or\n+00018190: 2074 6f6f 2073 6d61 6c6c 2066 6f72 20e2 too small for .\n+000181a0: 8098 5053 5472 6963 6b73 e280 9920 746f ..PSTricks... to\n+000181b0: 2064 7261 7720 2864 7565 2074 6f20 6c69 draw (due to li\n+000181c0: 6d69 7473 206f 6620 5465 5820 6d61 7468 mits of TeX math\n+000181d0: 292c 0a61 6e64 2070 6963 7475 7265 7320 ),.and pictures \n+000181e0: 7468 6174 206c 6f6f 6b20 6e6f 7468 696e that look nothin\n+000181f0: 6720 6c69 6b65 2077 6861 7420 7761 7320 g like what was \n+00018200: 696e 7465 6e64 6564 2e20 2041 2070 6963 intended. A pic\n+00018210: 7475 7265 2077 6974 680a 6f6e 6520 6f66 ture with.one of\n+00018220: 2074 6865 7365 2070 726f 626c 656d 7320 these problems \n+00018230: 6361 6e20 6265 2064 6966 6669 6375 6c74 can be difficult\n+00018240: 2074 6f20 2264 6562 7567 2e22 2020 5765 to \"debug.\" We\n+00018250: 206f 6666 6572 2074 6865 0a66 6f6c 6c6f offer the.follo\n+00018260: 7769 6e67 2073 7567 6765 7374 696f 6e73 wing suggestions\n+00018270: 2e0a 2020 20e2 80a2 2046 6f6c 6c6f 7720 .. ... Follow \n+00018280: 7468 6520 7375 6767 6573 7465 6420 696e the suggested in\n+00018290: 6372 656d 656e 7461 6c20 6465 7665 6c6f cremental develo\n+000182a0: 706d 656e 7420 6d65 7468 6f64 2064 6573 pment method des\n+000182b0: 6372 6962 6564 2069 6e0a 2020 2020 202a cribed in. *\n+000182c0: 6e6f 7465 204f 7665 7276 6965 773a 3a2e note Overview::.\n+000182d0: 0a20 2020 e280 a220 416c 7761 7973 206d . ... Always m\n+000182e0: 6169 6e74 6169 6e20 6f6e 6520 6f72 2074 aintain one or t\n+000182f0: 776f 2062 6163 6b2d 7665 7273 696f 6e73 wo back-versions\n+00018300: 206f 6620 6120 6472 6177 696e 6720 736f of a drawing so\n+00018310: 2074 6861 7420 6974 2069 730a 2020 2020 that it is. \n+00018320: 2065 6173 7920 746f 2066 616c 6c20 6261 easy to fall ba\n+00018330: 636b 2074 6f20 6120 6b6e 6f77 6e2d 676f ck to a known-go\n+00018340: 6f64 2069 6e63 7265 6d65 6e74 2e0a 2020 od increment.. \n+00018350: 20e2 80a2 2057 6865 6e20 7573 696e 6720 ... When using \n+00018360: e280 9870 6572 7370 6563 7469 7665 e280 ...perspective..\n+00018370: 992c 2065 6e73 7572 6520 616c 6c20 6669 ., ensure all fi\n+00018380: 6e61 6c6c 7920 7472 616e 7366 6f72 6d65 nally transforme\n+00018390: 6420 6f62 6a65 6374 730a 2020 2020 2073 d objects. s\n+000183a0: 6174 6973 6679 207a 3c30 2061 6e64 2c20 atisfy z<0 and, \n+000183b0: 696e 2066 6163 742c 2064 6f20 6e6f 7420 in fact, do not \n+000183c0: 636f 6d65 2076 6572 7920 636c 6f73 6520 come very close \n+000183d0: 746f 2074 6865 206f 7269 6769 6e20 6174 to the origin at\n+000183e0: 0a20 2020 2020 616c 6c2e 0a20 2020 e280 . all.. ..\n+000183f0: a220 5465 6d70 6f72 6172 696c 7920 7573 . Temporarily us\n+00018400: 6520 e280 9863 756c 6c3d 6661 6c73 65e2 e ...cull=false.\n+00018410: 8099 2074 6f20 7365 6520 7768 6572 6520 .. to see where \n+00018420: 7665 7274 6578 206f 7264 6572 696e 6720 vertex ordering \n+00018430: 7072 6f62 6c65 6d73 0a20 2020 2020 6c69 problems. li\n+00018440: 652e 0a20 2020 e280 a220 5573 6520 7465 e.. ... Use te\n+00018450: 6d70 6f72 6172 7920 6368 616e 6765 7320 mporary changes \n+00018460: 6f66 2063 6f6c 6f72 206f 6620 6f6e 6520 of color of one \n+00018470: 6f72 206d 6f72 6520 6f62 6a65 6374 7320 or more objects \n+00018480: 746f 2065 6e73 7572 650a 2020 2020 2074 to ensure. t\n+00018490: 6861 7420 796f 7572 2075 6e64 6572 7374 hat your underst\n+000184a0: 616e 6469 6e67 206f 6620 7468 6520 7363 anding of the sc\n+000184b0: 656e 6520 6765 6f6d 6574 7279 2069 7320 ene geometry is \n+000184c0: 636f 7272 6563 742e 0a20 2020 e280 a220 correct.. ... \n+000184d0: 4966 20e2 8098 5053 5472 6963 6b73 e280 If ...PSTricks..\n+000184e0: 9920 636f 6d70 6c61 696e 7320 6162 6f75 . complains abou\n+000184f0: 7420 736f 6d65 7468 696e 672c 2069 6e73 t something, ins\n+00018500: 7065 6374 2074 6865 206f 7574 7075 740a pect the output.\n+00018510: 2020 2020 2064 6972 6563 746c 7920 666f directly fo\n+00018520: 7220 636c 7565 732e 0a0a 1f0a 4669 6c65 r clues.....File\n+00018530: 3a20 736b 6574 6368 2e69 6e66 6f2c 2020 : sketch.info, \n+00018540: 4e6f 6465 3a20 436c 6970 7069 6e67 2c20 Node: Clipping, \n+00018550: 204e 6578 743a 2048 6964 6465 6e20 7375 Next: Hidden su\n+00018560: 7266 6163 6520 7265 6d6f 7661 6c2c 2020 rface removal, \n+00018570: 5072 6576 3a20 4c69 6d69 7473 206f 6e20 Prev: Limits on \n+00018580: 6572 726f 7220 6465 7465 6374 696f 6e2c error detection,\n+00018590: 2020 5570 3a20 4361 7665 6174 730a 0a34 Up: Caveats..4\n+000185a0: 2e34 2e32 2043 6c69 7070 696e 670a 2d2d .4.2 Clipping.--\n+000185b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th\n+000185c0: 6520 6375 7272 656e 7420 7665 7273 696f e current versio\n+000185d0: 6e20 6f66 20e2 8098 736b 6574 6368 e280 n of ...sketch..\n+000185e0: 9920 6861 7320 6e6f 2063 6c69 7070 696e . has no clippin\n+000185f0: 6720 6f70 6572 6174 696f 6e73 2e20 2054 g operations. T\n+00018600: 6865 2065 6e74 6972 650a 7363 656e 6520 he entire.scene \n+00018610: 6973 2061 6c77 6179 7320 6472 6177 6e2e is always drawn.\n+00018620: 2020 5468 6973 206d 6561 6e73 2074 6861 This means tha\n+00018630: 7420 7768 656e 2061 2070 6572 7370 6563 t when a perspec\n+00018640: 7469 7665 2074 7261 6e73 666f 726d 2069 tive transform i\n+00018650: 730a 656d 706c 6f79 6564 2c20 6974 2069 s.employed, it i\n+00018660: 7320 7468 6520 7573 6572 2773 2072 6573 s the user's res\n+00018670: 706f 6e73 6962 696c 6974 7920 746f 206d ponsibility to m\n+00018680: 616b 6520 7375 7265 2074 6865 2065 6e74 ake sure the ent\n+00018690: 6972 6520 7363 656e 650a 7265 6d61 696e ire scene.remain\n+000186a0: 7320 696e 2066 726f 6e74 206f 6620 7468 s in front of th\n+000186b0: 6520 7669 6577 6572 2c20 7468 6520 7265 e viewer, the re\n+000186c0: 6769 6f6e 207a 3c30 2e0a 0a1f 0a46 696c gion z<0.....Fil\n+000186d0: 653a 2073 6b65 7463 682e 696e 666f 2c20 e: sketch.info, \n+000186e0: 204e 6f64 653a 2048 6964 6465 6e20 7375 Node: Hidden su\n+000186f0: 7266 6163 6520 7265 6d6f 7661 6c2c 2020 rface removal, \n+00018700: 5072 6576 3a20 436c 6970 7069 6e67 2c20 Prev: Clipping, \n+00018710: 2055 703a 2043 6176 6561 7473 0a0a 342e Up: Caveats..4.\n+00018720: 342e 3320 4869 6464 656e 2073 7572 6661 4.3 Hidden surfa\n+00018730: 6365 2072 656d 6f76 616c 2061 6e64 2070 ce removal and p\n+00018740: 6f6c 7967 6f6e 2073 706c 6974 7469 6e67 olygon splitting\n+00018750: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .---------------\n 00018760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ----------------\n-00018770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------.\n-00018780: 0ae2 8098 536b 6574 6368 e280 9920 7573 ....Sketch... us\n-00018790: 6573 2074 6865 20e2 809c 6465 7074 6820 es the ...depth \n-000187a0: 736f 7274 2061 6c67 6f72 6974 686d e280 sort algorithm..\n-000187b0: 9d20 666f 7220 6869 6464 656e 2073 7572 . for hidden sur\n-000187c0: 6661 6365 2072 656d 6f76 616c 2e0a 5468 face removal..Th\n-000187d0: 6973 2069 7320 6120 7665 7279 206f 6c64 is is a very old\n-000187e0: 2074 6563 686e 6971 7565 2064 7565 2074 technique due t\n-000187f0: 6f20 4e65 7765 6c6c 2e28 3129 2020 4974 o Newell.(1) It\n-00018800: 2069 7320 6765 6e65 7261 6c6c 7920 7265 is generally re\n-00018810: 6761 7264 6564 0a61 7320 746f 6f20 736c garded.as too sl\n-00018820: 6f77 2066 6f72 2072 6561 6c20 7469 6d65 ow for real time\n-00018830: 2067 7261 7068 6963 732c 2062 7574 2069 graphics, but i\n-00018840: 7420 6973 2069 6465 616c 2066 6f72 206f t is ideal for o\n-00018850: 7572 2070 7572 706f 7365 0a77 6865 7265 ur purpose.where\n-00018860: 2073 7065 6564 2069 7320 6e6f 7420 7665 speed is not ve\n-00018870: 7279 2069 6d70 6f72 7461 6e74 2e28 3229 ry important.(2)\n-00018880: 0a0a 2020 2054 6865 2064 6570 7468 2073 .. The depth s\n-00018890: 6f72 7420 616c 676f 7269 7468 6d20 6d65 ort algorithm me\n-000188a0: 7265 6c79 2073 6f72 7473 206f 626a 6563 rely sorts objec\n-000188b0: 7473 206f 6e20 6120 6b65 7920 6f66 2069 ts on a key of i\n-000188c0: 6e63 7265 6173 696e 670a 7a2d 636f 6f72 ncreasing.z-coor\n-000188d0: 6469 6e61 7465 2c20 6571 7569 7661 6c65 dinate, equivale\n-000188e0: 6e74 2074 6f20 6465 6372 6561 7369 6e67 nt to decreasing\n-000188f0: 2064 6570 7468 2e20 204f 626a 6563 7473 depth. Objects\n-00018900: 2061 7265 2074 6865 6e20 6472 6177 6e20 are then drawn \n-00018910: 696e 0a74 6865 2073 6f72 7465 6420 7365 in.the sorted se\n-00018920: 7175 656e 6365 2073 6f20 7468 6174 2074 quence so that t\n-00018930: 686f 7365 2061 7420 7468 6520 7265 6172 hose at the rear\n-00018940: 206f 6620 7468 6520 7363 656e 6520 6172 of the scene ar\n-00018950: 650a 6f76 6572 7772 6974 7465 6e20 6279 e.overwritten by\n-00018960: 2074 686f 7365 2063 6c6f 7365 7220 746f those closer to\n-00018970: 2074 6865 2076 6965 7765 722e 2020 5369 the viewer. Si\n-00018980: 6e63 6520 7468 6973 2069 7320 616c 736f nce this is also\n-00018990: 2068 6f77 206f 696c 0a70 6169 6e74 6572 how oil.painter\n-000189a0: 7320 7072 6163 7469 6365 2074 6865 6972 s practice their\n-000189b0: 2061 7274 2c20 6465 7074 6820 736f 7274 art, depth sort\n-000189c0: 2069 7320 736f 6d65 7469 6d65 7320 6361 is sometimes ca\n-000189d0: 6c6c 6564 2022 7468 650a 7061 696e 7465 lled \"the.painte\n-000189e0: 7227 7320 616c 676f 7269 7468 6d2e 220a r's algorithm.\".\n-000189f0: 0a20 2020 496e 2073 6f6d 6520 6361 7365 . In some case\n-00018a00: 7320 6974 2069 7320 696d 706f 7373 6962 s it is impossib\n-00018a10: 6c65 2074 6f20 7374 7269 6374 6c79 206f le to strictly o\n-00018a20: 7264 6572 2070 6f6c 7967 6f6e 7320 6163 rder polygons ac\n-00018a30: 636f 7264 696e 670a 746f 2064 6570 7468 cording.to depth\n-00018a40: 2e20 204d 6f72 656f 7665 722c 2065 7665 . Moreover, eve\n-00018a50: 6e20 6966 2061 2063 6f72 7265 6374 2064 n if a correct d\n-00018a60: 6570 7468 206f 7264 6572 696e 6720 6578 epth ordering ex\n-00018a70: 6973 7473 2c20 7468 650a 636f 6d70 7574 ists, the.comput\n-00018a80: 6174 696f 6e20 6e65 6564 6564 2074 6f20 ation needed to \n-00018a90: 6669 6e64 2069 7420 6d61 7920 6265 2074 find it may be t\n-00018aa0: 6f6f 2063 6f6d 706c 6578 2061 6e64 2073 oo complex and s\n-00018ab0: 6c6f 772e 2020 496e 2074 6865 7365 0a63 low. In these.c\n-00018ac0: 6173 6573 2c20 e280 9873 6b65 7463 68e2 ases, ...sketch.\n-00018ad0: 8099 2073 706c 6974 7320 6f6e 6520 6f72 .. splits one or\n-00018ae0: 206d 6f72 6520 706f 6c79 676f 6e73 2069 more polygons i\n-00018af0: 6e74 6f20 7069 6563 6573 2e20 2054 6865 nto pieces. The\n-00018b00: 0a65 7870 6563 7461 7469 6f6e 2069 7320 .expectation is \n-00018b10: 7468 6174 2074 6865 206e 6577 2c20 736d that the new, sm\n-00018b20: 616c 6c65 7220 706f 6c79 676f 6e73 2077 aller polygons w\n-00018b30: 696c 6c20 6265 2073 696d 706c 6572 2074 ill be simpler t\n-00018b40: 6f20 6f72 6465 722e 0ae2 8098 536b 6574 o order.....Sket\n-00018b50: 6368 e280 9920 7573 6573 2061 2042 5350 ch... uses a BSP\n-00018b60: 2028 6269 6e61 7279 2073 7061 6365 2070 (binary space p\n-00018b70: 6172 7469 7469 6f6e 2920 746f 2068 616e artition) to han\n-00018b80: 646c 6520 7468 6520 7370 6c69 7474 696e dle the splittin\n-00018b90: 670a 6f70 6572 6174 696f 6e2e 0a0a 2a20 g.operation...* \n-00018ba0: 4d65 6e75 3a0a 0a2a 2053 7461 7469 7374 Menu:..* Statist\n-00018bb0: 6963 733a 3a20 2020 2020 2020 2020 2020 ics:: \n-00018bc0: 2020 2020 2020 2050 6572 666f 726d 616e Performan\n-00018bd0: 6365 206e 756d 6265 7273 206f 6e20 6465 ce numbers on de\n-00018be0: 7074 6820 736f 7274 2e0a 2a20 4275 6773 pth sort..* Bugs\n-00018bf0: 2061 6e64 2061 6e6f 6d61 6c69 6573 3a3a and anomalies::\n-00018c00: 2020 2020 2020 2020 2020 496d 7065 7266 Imperf\n-00018c10: 6563 7469 6f6e 7320 696e 2074 6869 7320 ections in this \n-00018c20: 696d 706c 656d 656e 7461 7469 6f6e 2e0a implementation..\n-00018c30: 0a20 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2046 . ---------- F\n-00018c40: 6f6f 746e 6f74 6573 202d 2d2d 2d2d 2d2d ootnotes -------\n-00018c50: 2d2d 2d0a 0a20 2020 2831 2920 4e65 7765 ---.. (1) Newe\n-00018c60: 6c6c 2c20 4d2e 452e 2c20 522e 472e 204e ll, M.E., R.G. N\n-00018c70: 6577 656c 6c2c 2061 6e64 2054 2e4c 2e20 ewell, and T.L. \n-00018c80: 5361 6e63 6861 2c20 4120 736f 6c75 7469 Sancha, A soluti\n-00018c90: 6f6e 2074 6f20 7468 650a 6869 6464 656e on to the.hidden\n-00018ca0: 2073 7572 6661 6365 2070 726f 626c 656d surface problem\n-00018cb0: 2e20 2050 726f 6365 6564 696e 6773 206f . Proceedings o\n-00018cc0: 6620 7468 6520 4143 4d20 616e 6e75 616c f the ACM annual\n-00018cd0: 2063 6f6e 6665 7265 6e63 6520 2d0a 566f conference -.Vo\n-00018ce0: 6c75 6d65 2031 2c20 7061 6765 2034 3433 lume 1, page 443\n-00018cf0: 2d34 3530 2c20 4143 4d20 5072 6573 732c -450, ACM Press,\n-00018d00: 2031 3937 322e 0a0a 2020 2028 3229 2057 1972... (2) W\n-00018d10: 6520 6861 7665 2072 756e 20e2 8098 736b e have run ...sk\n-00018d20: 6574 6368 e280 9920 6f6e 2074 6865 2066 etch... on the f\n-00018d30: 616d 6f75 7320 5374 616e 666f 7264 2042 amous Stanford B\n-00018d40: 756e 6e79 2c20 7768 6963 6820 636f 6e73 unny, which cons\n-00018d50: 6973 7473 0a6f 6620 6e65 6172 6c79 2037 ists.of nearly 7\n-00018d60: 302c 3030 3020 7472 6961 6e67 6c65 732e 0,000 triangles.\n-00018d70: 2020 5275 6e20 7469 6d65 2077 6173 2061 Run time was a\n-00018d80: 626f 7574 2036 2073 6563 6f6e 6473 2e20 bout 6 seconds. \n-00018d90: 204d 6f73 7420 6f66 2074 6869 730a 7761 Most of this.wa\n-00018da0: 7320 7370 656e 7420 7772 6974 696e 6720 s spent writing \n-00018db0: 7468 6520 6f75 7470 7574 2066 696c 6520 the output file \n-00018dc0: 7261 7468 6572 2074 6861 6e20 696e 2074 rather than in t\n-00018dd0: 6865 2068 6964 6465 6e20 7375 7266 6163 he hidden surfac\n-00018de0: 650a 616c 676f 7269 7468 6d2e 2020 4c61 e.algorithm. La\n-00018df0: 5465 5820 746f 6f6b 206d 7563 6820 6c6f TeX took much lo\n-00018e00: 6e67 6572 2074 6f20 7072 6f63 6573 7320 nger to process \n-00018e10: 7468 6520 7265 7375 6c74 696e 6720 e280 the resulting ..\n-00018e20: 9850 5354 7269 636b 73e2 8099 0a63 6f64 .PSTricks....cod\n-00018e30: 652e 2020 5468 6520 6f62 7669 6f75 7320 e. The obvious \n-00018e40: 636f 6e63 6c75 7369 6f6e 2069 7320 7468 conclusion is th\n-00018e50: 6174 2074 6865 2073 7065 6564 206f 6620 at the speed of \n-00018e60: 7468 6520 6465 7074 6820 736f 7274 0a61 the depth sort.a\n-00018e70: 6c67 6f72 6974 686d 2069 7320 6e6f 7420 lgorithm is not \n-00018e80: 6120 776f 7272 792e 0a0a 1f0a 4669 6c65 a worry.....File\n-00018e90: 3a20 736b 6574 6368 2e69 6e66 6f2c 2020 : sketch.info, \n-00018ea0: 4e6f 6465 3a20 5374 6174 6973 7469 6373 Node: Statistics\n-00018eb0: 2c20 204e 6578 743a 2042 7567 7320 616e , Next: Bugs an\n-00018ec0: 6420 616e 6f6d 616c 6965 732c 2020 5072 d anomalies, Pr\n-00018ed0: 6576 3a20 4869 6464 656e 2073 7572 6661 ev: Hidden surfa\n-00018ee0: 6365 2072 656d 6f76 616c 2c20 2055 703a ce removal, Up:\n-00018ef0: 2048 6964 6465 6e20 7375 7266 6163 6520 Hidden surface \n-00018f00: 7265 6d6f 7661 6c0a 0a34 2e34 2e33 2e31 removal..4.4.3.1\n-00018f10: 2053 7461 7469 7374 6963 730a 2e2e 2e2e Statistics.....\n-00018f20: 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 0a0a ................\n-00018f30: 466f 7220 7468 6520 6375 7269 6f75 732c For the curious,\n-00018f40: 20e2 8098 736b 6574 6368 e280 9920 7772 ...sketch... wr\n-00018f50: 6974 6573 206f 6e65 206c 696e 6520 6f66 ites one line of\n-00018f60: 2064 6570 7468 2073 6f72 7420 7374 6174 depth sort stat\n-00018f70: 6973 7469 6373 2e0a 4865 7265 2069 7320 istics..Here is \n-00018f80: 616e 2065 7861 6d70 6c65 2066 6f72 2061 an example for a\n-00018f90: 206c 6172 6765 2063 6f6c 6c65 6374 696f large collectio\n-00018fa0: 6e20 6f66 2074 7269 616e 676c 6573 2e0a n of triangles..\n-00018fb0: 2020 7265 6d61 726b 2c20 6e6f 6465 3d33 remark, node=3\n-00018fc0: 3438 3234 2070 726f 6265 3d35 3831 2e39 4824 probe=581.9\n-00018fd0: 2073 7761 703d 3520 7370 6c69 743d 3220 swap=5 split=2 \n-00018fe0: 2869 6e3d 3420 6f75 743d 3629 206f 6c73 (in=4 out=6) ols\n-00018ff0: 3d32 3438 3531 2f30 0a49 7420 6d65 616e =24851/0.It mean\n-00019000: 7320 7468 6174 2033 342c 3832 3420 6f62 s that 34,824 ob\n-00019010: 6a65 6374 7320 7765 7265 2064 6570 7468 jects were depth\n-00019020: 2073 6f72 7465 6420 6166 7465 7220 6375 sorted after cu\n-00019030: 6c6c 696e 672e 2020 466f 7220 6561 6368 lling. For each\n-00019040: 2c0a 616e 2061 7665 7261 6765 206f 6620 ,.an average of \n-00019050: 3538 312e 3920 6f74 6865 7273 2068 6164 581.9 others had\n-00019060: 2074 6f20 6265 2063 6865 636b 6564 2074 to be checked t\n-00019070: 6f20 656e 7375 7265 2074 6861 7420 7468 o ensure that th\n-00019080: 6520 696e 6974 6961 6c2c 0a61 7070 726f e initial,.appro\n-00019090: 7869 6d61 7465 206f 7264 6572 696e 6720 ximate ordering \n-000190a0: 7761 7320 636f 7272 6563 742e 2020 416d was correct. Am\n-000190b0: 6f6e 6720 616c 6c20 7468 6573 6520 6368 ong all these ch\n-000190c0: 6563 6b73 2c20 6f6e 6c79 2035 0a72 6573 ecks, only 5.res\n-000190d0: 756c 7465 6420 696e 2073 7761 7073 2074 ulted in swaps t\n-000190e0: 6f20 7265 6f72 6465 7220 7468 6520 696e o reorder the in\n-000190f0: 6974 6961 6c20 736f 7274 2e20 2049 6e20 itial sort. In \n-00019100: 7477 6f20 6361 7365 732c 2061 2063 6f72 two cases, a cor\n-00019110: 7265 6374 0a6f 7264 6572 696e 6720 636f rect.ordering co\n-00019120: 756c 6420 6e6f 7420 6265 2064 6574 6572 uld not be deter\n-00019130: 6d69 6e65 642c 2073 6f20 6269 6e61 7279 mined, so binary\n-00019140: 2073 7061 6365 2070 6172 7469 7469 6f6e space partition\n-00019150: 7320 7765 7265 0a63 6f6e 7374 7275 6374 s were.construct\n-00019160: 6564 2066 6f72 2073 706c 6974 7469 6e67 ed for splitting\n-00019170: 2e20 2041 2074 6f74 616c 206f 6620 3420 . A total of 4 \n-00019180: 6f62 6a65 6374 7320 2874 7269 616e 676c objects (triangl\n-00019190: 6573 2069 6e20 7468 6973 0a63 6173 6529 es in this.case)\n-000191a0: 2077 6572 6520 696e 7365 7274 6564 2069 were inserted i\n-000191b0: 6e20 7468 6520 7061 7274 6974 696f 6e73 n the partitions\n-000191c0: 2c20 616e 6420 3620 706f 6c79 676f 6e73 , and 6 polygons\n-000191d0: 2077 6572 6520 7072 6f64 7563 6564 2e0a were produced..\n-000191e0: 4669 6e61 6c6c 792c 2032 342c 3835 3120 Finally, 24,851 \n-000191f0: 226c 6173 7420 7265 736f 7274 2220 706f \"last resort\" po\n-00019200: 6c79 676f 6e20 6f76 6572 6c61 7020 6368 lygon overlap ch\n-00019210: 6563 6b73 2077 6572 6520 7065 7266 6f72 ecks were perfor\n-00019220: 6d65 640a 6166 7465 7220 7369 6d70 6c65 med.after simple\n-00019230: 722c 2066 6173 7465 7220 6368 6563 6b73 r, faster checks\n-00019240: 2066 6169 6c65 6420 746f 2079 6965 6c64 failed to yield\n-00019250: 2063 6f6e 636c 7573 6976 6520 7265 7375 conclusive resu\n-00019260: 6c74 732e 2020 5468 650a 6669 6e61 6c20 lts. The.final \n-00019270: 2f30 2069 7320 666f 7220 6c69 6e65 2d70 /0 is for line-p\n-00019280: 6f6c 7967 6f6e 206f 7665 726c 6170 2063 olygon overlap c\n-00019290: 6865 636b 732e 2020 466f 7220 636f 6d70 hecks. For comp\n-000192a0: 6172 6973 6f6e 2c20 7468 650a 7374 6174 arison, the.stat\n-000192b0: 6973 7469 6373 2066 6f72 2074 6865 206c istics for the l\n-000192c0: 6173 7420 6669 6775 7265 2069 6e20 2a6e ast figure in *n\n-000192d0: 6f74 6520 4f76 6572 7669 6577 3a3a 2066 ote Overview:: f\n-000192e0: 6f6c 6c6f 772e 0a20 2072 656d 6172 6b2c ollow.. remark,\n-000192f0: 206e 6f64 653d 3237 2070 726f 6265 3d31 node=27 probe=1\n-00019300: 342e 3620 7377 6170 3d33 3620 7370 6c69 4.6 swap=36 spli\n-00019310: 743d 3135 2028 696e 3d33 3020 6f75 743d t=15 (in=30 out=\n-00019320: 3435 2920 6f6c 733d 302f 3639 0a4e 6f74 45) ols=0/69.Not\n-00019330: 6520 7468 6174 2074 6865 7265 2077 6173 e that there was\n-00019340: 2070 726f 706f 7274 696f 6e61 6c6c 7920 proportionally \n-00019350: 6d75 6368 206d 6f72 6520 7377 6170 7069 much more swappi\n-00019360: 6e67 2061 6e64 2073 706c 6974 7469 6e67 ng and splitting\n-00019370: 0a61 6374 6976 6974 7920 696e 2074 6869 .activity in thi\n-00019380: 7320 6869 6768 6c79 2063 6f6e 6e65 6374 s highly connect\n-00019390: 6564 2073 6365 6e65 2e0a 0a1f 0a46 696c ed scene.....Fil\n-000193a0: 653a 2073 6b65 7463 682e 696e 666f 2c20 e: sketch.info, \n-000193b0: 204e 6f64 653a 2042 7567 7320 616e 6420 Node: Bugs and \n-000193c0: 616e 6f6d 616c 6965 732c 2020 5072 6576 anomalies, Prev\n-000193d0: 3a20 5374 6174 6973 7469 6373 2c20 2055 : Statistics, U\n-000193e0: 703a 2048 6964 6465 6e20 7375 7266 6163 p: Hidden surfac\n-000193f0: 6520 7265 6d6f 7661 6c0a 0a34 2e34 2e33 e removal..4.4.3\n-00019400: 2e32 2042 7567 7320 616e 6420 616e 6f6d .2 Bugs and anom\n-00019410: 616c 6965 730a 2e2e 2e2e 2e2e 2e2e 2e2e alies...........\n+00018770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ----------------\n+00018780: 2d2d 2d0a 0ae2 8098 536b 6574 6368 e280 ---.....Sketch..\n+00018790: 9920 7573 6573 2074 6865 20e2 809c 6465 . uses the ...de\n+000187a0: 7074 6820 736f 7274 2061 6c67 6f72 6974 pth sort algorit\n+000187b0: 686d e280 9d20 666f 7220 6869 6464 656e hm... for hidden\n+000187c0: 2073 7572 6661 6365 2072 656d 6f76 616c surface removal\n+000187d0: 2e0a 5468 6973 2069 7320 6120 7665 7279 ..This is a very\n+000187e0: 206f 6c64 2074 6563 686e 6971 7565 2064 old technique d\n+000187f0: 7565 2074 6f20 4e65 7765 6c6c 2e28 3129 ue to Newell.(1)\n+00018800: 2020 4974 2069 7320 6765 6e65 7261 6c6c It is generall\n+00018810: 7920 7265 6761 7264 6564 0a61 7320 746f y regarded.as to\n+00018820: 6f20 736c 6f77 2066 6f72 2072 6561 6c20 o slow for real \n+00018830: 7469 6d65 2067 7261 7068 6963 732c 2062 time graphics, b\n+00018840: 7574 2069 7420 6973 2069 6465 616c 2066 ut it is ideal f\n+00018850: 6f72 206f 7572 2070 7572 706f 7365 0a77 or our purpose.w\n+00018860: 6865 7265 2073 7065 6564 2069 7320 6e6f here speed is no\n+00018870: 7420 7665 7279 2069 6d70 6f72 7461 6e74 t very important\n+00018880: 2e28 3229 0a0a 2020 2054 6865 2064 6570 .(2).. The dep\n+00018890: 7468 2073 6f72 7420 616c 676f 7269 7468 th sort algorith\n+000188a0: 6d20 6d65 7265 6c79 2073 6f72 7473 206f m merely sorts o\n+000188b0: 626a 6563 7473 206f 6e20 6120 6b65 7920 bjects on a key \n+000188c0: 6f66 2069 6e63 7265 6173 696e 670a 7a2d of increasing.z-\n+000188d0: 636f 6f72 6469 6e61 7465 2c20 6571 7569 coordinate, equi\n+000188e0: 7661 6c65 6e74 2074 6f20 6465 6372 6561 valent to decrea\n+000188f0: 7369 6e67 2064 6570 7468 2e20 204f 626a sing depth. Obj\n+00018900: 6563 7473 2061 7265 2074 6865 6e20 6472 ects are then dr\n+00018910: 6177 6e20 696e 0a74 6865 2073 6f72 7465 awn in.the sorte\n+00018920: 6420 7365 7175 656e 6365 2073 6f20 7468 d sequence so th\n+00018930: 6174 2074 686f 7365 2061 7420 7468 6520 at those at the \n+00018940: 7265 6172 206f 6620 7468 6520 7363 656e rear of the scen\n+00018950: 6520 6172 650a 6f76 6572 7772 6974 7465 e are.overwritte\n+00018960: 6e20 6279 2074 686f 7365 2063 6c6f 7365 n by those close\n+00018970: 7220 746f 2074 6865 2076 6965 7765 722e r to the viewer.\n+00018980: 2020 5369 6e63 6520 7468 6973 2069 7320 Since this is \n+00018990: 616c 736f 2068 6f77 206f 696c 0a70 6169 also how oil.pai\n+000189a0: 6e74 6572 7320 7072 6163 7469 6365 2074 nters practice t\n+000189b0: 6865 6972 2061 7274 2c20 6465 7074 6820 heir art, depth \n+000189c0: 736f 7274 2069 7320 736f 6d65 7469 6d65 sort is sometime\n+000189d0: 7320 6361 6c6c 6564 2022 7468 650a 7061 s called \"the.pa\n+000189e0: 696e 7465 7227 7320 616c 676f 7269 7468 inter's algorith\n+000189f0: 6d2e 220a 0a20 2020 496e 2073 6f6d 6520 m.\".. In some \n+00018a00: 6361 7365 7320 6974 2069 7320 696d 706f cases it is impo\n+00018a10: 7373 6962 6c65 2074 6f20 7374 7269 6374 ssible to strict\n+00018a20: 6c79 206f 7264 6572 2070 6f6c 7967 6f6e ly order polygon\n+00018a30: 7320 6163 636f 7264 696e 670a 746f 2064 s according.to d\n+00018a40: 6570 7468 2e20 204d 6f72 656f 7665 722c epth. Moreover,\n+00018a50: 2065 7665 6e20 6966 2061 2063 6f72 7265 even if a corre\n+00018a60: 6374 2064 6570 7468 206f 7264 6572 696e ct depth orderin\n+00018a70: 6720 6578 6973 7473 2c20 7468 650a 636f g exists, the.co\n+00018a80: 6d70 7574 6174 696f 6e20 6e65 6564 6564 mputation needed\n+00018a90: 2074 6f20 6669 6e64 2069 7420 6d61 7920 to find it may \n+00018aa0: 6265 2074 6f6f 2063 6f6d 706c 6578 2061 be too complex a\n+00018ab0: 6e64 2073 6c6f 772e 2020 496e 2074 6865 nd slow. In the\n+00018ac0: 7365 0a63 6173 6573 2c20 e280 9873 6b65 se.cases, ...ske\n+00018ad0: 7463 68e2 8099 2073 706c 6974 7320 6f6e tch... splits on\n+00018ae0: 6520 6f72 206d 6f72 6520 706f 6c79 676f e or more polygo\n+00018af0: 6e73 2069 6e74 6f20 7069 6563 6573 2e20 ns into pieces. \n+00018b00: 2054 6865 0a65 7870 6563 7461 7469 6f6e The.expectation\n+00018b10: 2069 7320 7468 6174 2074 6865 206e 6577 is that the new\n+00018b20: 2c20 736d 616c 6c65 7220 706f 6c79 676f , smaller polygo\n+00018b30: 6e73 2077 696c 6c20 6265 2073 696d 706c ns will be simpl\n+00018b40: 6572 2074 6f20 6f72 6465 722e 0ae2 8098 er to order.....\n+00018b50: 536b 6574 6368 e280 9920 7573 6573 2061 Sketch... uses a\n+00018b60: 2042 5350 2028 6269 6e61 7279 2073 7061 BSP (binary spa\n+00018b70: 6365 2070 6172 7469 7469 6f6e 2920 746f ce partition) to\n+00018b80: 2068 616e 646c 6520 7468 6520 7370 6c69 handle the spli\n+00018b90: 7474 696e 670a 6f70 6572 6174 696f 6e2e tting.operation.\n+00018ba0: 0a0a 2a20 4d65 6e75 3a0a 0a2a 2053 7461 ..* Menu:..* Sta\n+00018bb0: 7469 7374 6963 733a 3a20 2020 2020 2020 tistics:: \n+00018bc0: 2020 2020 2020 2020 2020 2050 6572 666f Perfo\n+00018bd0: 726d 616e 6365 206e 756d 6265 7273 206f rmance numbers o\n+00018be0: 6e20 6465 7074 6820 736f 7274 2e0a 2a20 n depth sort..* \n+00018bf0: 4275 6773 2061 6e64 2061 6e6f 6d61 6c69 Bugs and anomali\n+00018c00: 6573 3a3a 2020 2020 2020 2020 2020 496d es:: Im\n+00018c10: 7065 7266 6563 7469 6f6e 7320 696e 2074 perfections in t\n+00018c20: 6869 7320 696d 706c 656d 656e 7461 7469 his implementati\n+00018c30: 6f6e 2e0a 0a20 2020 2d2d 2d2d 2d2d 2d2d on... --------\n+00018c40: 2d2d 2046 6f6f 746e 6f74 6573 202d 2d2d -- Footnotes ---\n+00018c50: 2d2d 2d2d 2d2d 2d0a 0a20 2020 2831 2920 -------.. (1) \n+00018c60: 4e65 7765 6c6c 2c20 4d2e 452e 2c20 522e Newell, M.E., R.\n+00018c70: 472e 204e 6577 656c 6c2c 2061 6e64 2054 G. Newell, and T\n+00018c80: 2e4c 2e20 5361 6e63 6861 2c20 4120 736f .L. Sancha, A so\n+00018c90: 6c75 7469 6f6e 2074 6f20 7468 650a 6869 lution to the.hi\n+00018ca0: 6464 656e 2073 7572 6661 6365 2070 726f dden surface pro\n+00018cb0: 626c 656d 2e20 2050 726f 6365 6564 696e blem. Proceedin\n+00018cc0: 6773 206f 6620 7468 6520 4143 4d20 616e gs of the ACM an\n+00018cd0: 6e75 616c 2063 6f6e 6665 7265 6e63 6520 nual conference \n+00018ce0: 2d0a 566f 6c75 6d65 2031 2c20 7061 6765 -.Volume 1, page\n+00018cf0: 2034 3433 2d34 3530 2c20 4143 4d20 5072 443-450, ACM Pr\n+00018d00: 6573 732c 2031 3937 322e 0a0a 2020 2028 ess, 1972... (\n+00018d10: 3229 2057 6520 6861 7665 2072 756e 20e2 2) We have run .\n+00018d20: 8098 736b 6574 6368 e280 9920 6f6e 2074 ..sketch... on t\n+00018d30: 6865 2066 616d 6f75 7320 5374 616e 666f he famous Stanfo\n+00018d40: 7264 2042 756e 6e79 2c20 7768 6963 6820 rd Bunny, which \n+00018d50: 636f 6e73 6973 7473 0a6f 6620 6e65 6172 consists.of near\n+00018d60: 6c79 2037 302c 3030 3020 7472 6961 6e67 ly 70,000 triang\n+00018d70: 6c65 732e 2020 5275 6e20 7469 6d65 2077 les. Run time w\n+00018d80: 6173 2061 626f 7574 2036 2073 6563 6f6e as about 6 secon\n+00018d90: 6473 2e20 204d 6f73 7420 6f66 2074 6869 ds. Most of thi\n+00018da0: 730a 7761 7320 7370 656e 7420 7772 6974 s.was spent writ\n+00018db0: 696e 6720 7468 6520 6f75 7470 7574 2066 ing the output f\n+00018dc0: 696c 6520 7261 7468 6572 2074 6861 6e20 ile rather than \n+00018dd0: 696e 2074 6865 2068 6964 6465 6e20 7375 in the hidden su\n+00018de0: 7266 6163 650a 616c 676f 7269 7468 6d2e rface.algorithm.\n+00018df0: 2020 4c61 5465 5820 746f 6f6b 206d 7563 LaTeX took muc\n+00018e00: 6820 6c6f 6e67 6572 2074 6f20 7072 6f63 h longer to proc\n+00018e10: 6573 7320 7468 6520 7265 7375 6c74 696e ess the resultin\n+00018e20: 6720 e280 9850 5354 7269 636b 73e2 8099 g ...PSTricks...\n+00018e30: 0a63 6f64 652e 2020 5468 6520 6f62 7669 .code. The obvi\n+00018e40: 6f75 7320 636f 6e63 6c75 7369 6f6e 2069 ous conclusion i\n+00018e50: 7320 7468 6174 2074 6865 2073 7065 6564 s that the speed\n+00018e60: 206f 6620 7468 6520 6465 7074 6820 736f of the depth so\n+00018e70: 7274 0a61 6c67 6f72 6974 686d 2069 7320 rt.algorithm is \n+00018e80: 6e6f 7420 6120 776f 7272 792e 0a0a 1f0a not a worry.....\n+00018e90: 4669 6c65 3a20 736b 6574 6368 2e69 6e66 File: sketch.inf\n+00018ea0: 6f2c 2020 4e6f 6465 3a20 5374 6174 6973 o, Node: Statis\n+00018eb0: 7469 6373 2c20 204e 6578 743a 2042 7567 tics, Next: Bug\n+00018ec0: 7320 616e 6420 616e 6f6d 616c 6965 732c s and anomalies,\n+00018ed0: 2020 5072 6576 3a20 4869 6464 656e 2073 Prev: Hidden s\n+00018ee0: 7572 6661 6365 2072 656d 6f76 616c 2c20 urface removal, \n+00018ef0: 2055 703a 2048 6964 6465 6e20 7375 7266 Up: Hidden surf\n+00018f00: 6163 6520 7265 6d6f 7661 6c0a 0a34 2e34 ace removal..4.4\n+00018f10: 2e33 2e31 2053 7461 7469 7374 6963 730a .3.1 Statistics.\n+00018f20: 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e 2e2e ................\n+00018f30: 2e2e 0a0a 466f 7220 7468 6520 6375 7269 ....For the curi\n+00018f40: 6f75 732c 20e2 8098 736b 6574 6368 e280 ous, ...sketch..\n+00018f50: 9920 7772 6974 6573 206f 6e65 206c 696e . writes one lin\n+00018f60: 6520 6f66 2064 6570 7468 2073 6f72 7420 e of depth sort \n+00018f70: 7374 6174 6973 7469 6373 2e0a 4865 7265 statistics..Here\n+00018f80: 2069 7320 616e 2065 7861 6d70 6c65 2066 is an example f\n+00018f90: 6f72 2061 206c 6172 6765 2063 6f6c 6c65 or a large colle\n+00018fa0: 6374 696f 6e20 6f66 2074 7269 616e 676c ction of triangl\n+00018fb0: 6573 2e0a 2020 7265 6d61 726b 2c20 6e6f es.. remark, no\n+00018fc0: 6465 3d33 3438 3234 2070 726f 6265 3d35 de=34824 probe=5\n+00018fd0: 3831 2e39 2073 7761 703d 3520 7370 6c69 81.9 swap=5 spli\n+00018fe0: 743d 3220 2869 6e3d 3420 6f75 743d 3629 t=2 (in=4 out=6)\n+00018ff0: 206f 6c73 3d32 3438 3531 2f30 0a49 7420 ols=24851/0.It \n+00019000: 6d65 616e 7320 7468 6174 2033 342c 3832 means that 34,82\n+00019010: 3420 6f62 6a65 6374 7320 7765 7265 2064 4 objects were d\n+00019020: 6570 7468 2073 6f72 7465 6420 6166 7465 epth sorted afte\n+00019030: 7220 6375 6c6c 696e 672e 2020 466f 7220 r culling. For \n+00019040: 6561 6368 2c0a 616e 2061 7665 7261 6765 each,.an average\n+00019050: 206f 6620 3538 312e 3920 6f74 6865 7273 of 581.9 others\n+00019060: 2068 6164 2074 6f20 6265 2063 6865 636b had to be check\n+00019070: 6564 2074 6f20 656e 7375 7265 2074 6861 ed to ensure tha\n+00019080: 7420 7468 6520 696e 6974 6961 6c2c 0a61 t the initial,.a\n+00019090: 7070 726f 7869 6d61 7465 206f 7264 6572 pproximate order\n+000190a0: 696e 6720 7761 7320 636f 7272 6563 742e ing was correct.\n+000190b0: 2020 416d 6f6e 6720 616c 6c20 7468 6573 Among all thes\n+000190c0: 6520 6368 6563 6b73 2c20 6f6e 6c79 2035 e checks, only 5\n+000190d0: 0a72 6573 756c 7465 6420 696e 2073 7761 .resulted in swa\n+000190e0: 7073 2074 6f20 7265 6f72 6465 7220 7468 ps to reorder th\n+000190f0: 6520 696e 6974 6961 6c20 736f 7274 2e20 e initial sort. \n+00019100: 2049 6e20 7477 6f20 6361 7365 732c 2061 In two cases, a\n+00019110: 2063 6f72 7265 6374 0a6f 7264 6572 696e correct.orderin\n+00019120: 6720 636f 756c 6420 6e6f 7420 6265 2064 g could not be d\n+00019130: 6574 6572 6d69 6e65 642c 2073 6f20 6269 etermined, so bi\n+00019140: 6e61 7279 2073 7061 6365 2070 6172 7469 nary space parti\n+00019150: 7469 6f6e 7320 7765 7265 0a63 6f6e 7374 tions were.const\n+00019160: 7275 6374 6564 2066 6f72 2073 706c 6974 ructed for split\n+00019170: 7469 6e67 2e20 2041 2074 6f74 616c 206f ting. 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Th\n-000199b0: 6520 6f6e 6c79 2073 7572 652d 6669 7265 e only sure-fire\n-000199c0: 2073 6f6c 7574 696f 6e20 6973 2070 7265 solution is pre\n-000199d0: 7474 790a 7465 7272 6962 6c65 3a20 6375 tty.terrible: cu\n-000199e0: 7374 6f6d 2066 6974 20e2 8098 7370 6563 stom fit ...spec\n-000199f0: 6961 6ce2 8099 206f 7665 726c 6179 206c ial... overlay l\n-00019a00: 696e 6573 2028 7769 7468 20e2 8098 5c70 ines (with ...\\p\n-00019a10: 736c 696e 65e2 8099 2920 746f 2063 6f76 sline...) to cov\n-00019a20: 6572 0a74 6865 206e 6f74 6368 6573 2e0a er.the notches..\n-00019a30: 0a20 2020 506f 6c79 676f 6e20 7370 6c69 . Polygon spli\n-00019a40: 7474 696e 6720 616c 736f 2062 7265 616b tting also break\n-00019a50: 7320 e280 9850 5354 7269 636b 73e2 8099 s ...PSTricks...\n-00019a60: 2068 6174 6368 2070 6174 7465 726e 732e hatch patterns.\n-00019a70: 2020 5468 6520 6f6e 6c79 0a6b 6e6f 776e The only.known\n-00019a80: 2077 6f72 6b61 726f 756e 6420 6973 2074 workaround is t\n-00019a90: 6f20 7375 6273 7469 7475 7465 2061 2073 o substitute a s\n-00019aa0: 6f6c 6964 2066 696c 6c20 666f 7220 7468 olid fill for th\n-00019ab0: 6520 6861 7463 682e 0a0a 2020 202d 2d2d e hatch... ---\n-00019ac0: 2d2d 2d2d 2d2d 2d20 466f 6f74 6e6f 7465 ------- Footnote\n-00019ad0: 7320 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 2020 s ----------.. \n-00019ae0: 2028 3129 2049 206b 6e6f 7720 686f 7720 (1) I know how \n-00019af0: 746f 2066 6978 2074 6869 7320 7072 6f62 to fix this prob\n-00019b00: 6c65 6d2c 2062 7574 2049 2064 6f6e 2774 lem, but I don't\n-00019b10: 206c 696b 6520 6d79 2073 6f6c 7574 696f like my solutio\n-00019b20: 6e2c 2061 6e64 0a49 276d 2069 6e74 6572 n, and.I'm inter\n-00019b30: 6573 7465 6420 696e 2079 6f75 7273 2e0a ested in yours..\n-00019b40: 0a1f 0a46 696c 653a 2073 6b65 7463 682e ...File: sketch.\n-00019b50: 696e 666f 2c20 204e 6f64 653a 2043 6f6d info, Node: Com\n-00019b60: 6d61 6e64 206c 696e 652c 2020 4e65 7874 mand line, Next\n-00019b70: 3a20 496e 7374 616c 6c69 6e67 2073 6b65 : Installing ske\n-00019b80: 7463 682c 2020 5072 6576 3a20 4275 696c tch, Prev: Buil\n-00019b90: 6469 6e67 2061 2064 7261 7769 6e67 2c20 ding a drawing, \n-00019ba0: 2055 703a 2054 6f70 0a0a 3520 436f 6d6d Up: Top..5 Comm\n-00019bb0: 616e 6420 6c69 6e65 0a2a 2a2a 2a2a 2a2a and line.*******\n-00019bc0: 2a2a 2a2a 2a2a 2a0a 0a2a 5379 6e6f 7073 *******..*Synops\n-00019bd0: 6973 3a2a 0a20 2020 2020 736b 6574 6368 is:*. sketch\n-00019be0: 205b 2d68 5d5b 2d56 2078 2e79 5d5b 2d76 [-h][-V x.y][-v\n-00019bf0: 5d5b 2d62 5d5b 2d64 5d5b 7420 646f 6374 ][-b][-d][t doct\n-00019c00: 6d70 5d5b 2d54 5b75 7c65 5d5b 705b 507c mp][-T[u|e][p[P|\n-00019c10: 545d 5b4c 7c43 5d5d 5d5b 2d6f 206f 7574 T][L|C]]][-o out\n-00019c20: 7075 742e 7465 785d 0a20 2020 2020 2020 put.tex]. \n-00019c30: 5b2d 4420 5441 4720 2e2e 2e5d 2069 6e70 [-D TAG ...] inp\n-00019c40: 7574 312e 736b 205b 2d55 2054 4147 202e ut1.sk [-U TAG .\n-00019c50: 2e2e 5d20 696e 7075 7432 2e73 6b20 2e2e ..] input2.sk ..\n-00019c60: 2e0a 0a2a 4465 7363 7269 7074 696f 6e2a ...*Description*\n-00019c70: 2050 726f 6365 7373 6573 2074 6865 20e2 Processes the .\n-00019c80: 8098 736b 6574 6368 e280 9920 696e 7075 ..sketch... inpu\n-00019c90: 7420 6669 6c65 7320 696e 206f 7264 6572 t files in order\n-00019ca0: 2074 6f20 7072 6f64 7563 650a e280 9850 to produce....P\n-00019cb0: 5354 7269 636b 73e2 8099 206f 7574 7075 STricks... outpu\n-00019cc0: 7420 636f 6465 2073 7569 7461 626c 6520 t code suitable \n-00019cd0: 666f 7220 696e 636c 7573 696f 6e20 696e for inclusion in\n-00019ce0: 2061 2054 6558 206f 7220 4c61 5465 580a a TeX or LaTeX.\n-00019cf0: 646f 6375 6d65 6e74 2e0a 0a2a 4f70 7469 document...*Opti\n-00019d00: 6f6e 733a 2a0a e280 982d 68e2 8099 0a20 ons:*....-h.... \n-00019d10: 2020 2020 5072 696e 7420 6120 7368 6f72 Print a shor\n-00019d20: 7420 6361 7461 6c6f 6720 6f66 206f 7074 t catalog of opt\n-00019d30: 696f 6e73 2e0a e280 982d 56e2 8099 0a20 ions.....-V.... \n-00019d40: 2020 2020 5365 7420 7468 6520 e280 9850 Set the ...P\n-00019d50: 5354 7269 636b 73e2 8099 2076 6572 7369 STricks... versi\n-00019d60: 6f6e 2061 7373 756d 6564 2066 6f72 206f on assumed for o\n-00019d70: 7574 7075 7420 7075 7270 6f73 6573 2074 utput purposes t\n-00019d80: 6f20 e280 9878 2e79 e280 992c 0a20 2020 o ...x.y...,. \n-00019d90: 2020 666f 7220 6578 616d 706c 6520 312e for example 1.\n-00019da0: 3139 2e20 2055 7375 616c 6c79 206e 6565 19. Usually nee\n-00019db0: 6465 6420 6f6e 6c79 2069 6620 796f 7572 ded only if your\n-00019dc0: 20e2 8098 5053 5472 6963 6b73 e280 9920 ...PSTricks... \n-00019dd0: 6973 206f 6c64 0a20 2020 2020 636f 6d70 is old. comp\n-00019de0: 6172 6564 2074 6f20 796f 7572 20e2 8098 ared to your ...\n-00019df0: 736b 6574 6368 e280 992e 2020 5573 6520 sketch.... Use \n-00019e00: e280 982d 76e2 8099 2074 6f20 7365 6520 ...-v... to see \n-00019e10: 7768 6174 20e2 8098 736b 6574 6368 e280 what ...sketch..\n-00019e20: 9920 6173 7375 6d65 730a 2020 2020 2062 . assumes. b\n-00019e30: 7920 6465 6661 756c 742e 0ae2 8098 2d76 y default.....-v\n-00019e40: e280 990a 2020 2020 2050 7269 6e74 2076 .... Print v\n-00019e50: 6572 7369 6f6e 2069 6e66 6f72 6d61 7469 ersion informati\n-00019e60: 6f6e 2074 6f20 7374 616e 6461 7264 206f on to standard o\n-00019e70: 7574 7075 742c 2069 6e63 6c75 6469 6e67 utput, including\n-00019e80: 2074 6865 2076 6572 7369 6f6e 0a20 2020 the version. \n-00019e90: 2020 6f66 20e2 8098 5053 5472 6963 6b73 of ...PSTricks\n-00019ea0: e280 9920 6173 7375 6d65 6420 666f 7220 ... assumed for \n-00019eb0: 6f75 7470 7574 2028 6361 6e20 6265 2063 output (can be c\n-00019ec0: 6861 6e67 6564 2077 6974 6820 e280 982d hanged with ...-\n-00019ed0: 56e2 8099 2061 626f 7665 292e 0ae2 8098 V... above).....\n-00019ee0: 2d62 e280 990a 2020 2020 2055 7365 2061 -b.... Use a\n-00019ef0: 2042 5350 2028 2a4e 6f74 6520 4869 6464 BSP (*Note Hidd\n-00019f00: 656e 2073 7572 6661 6365 2072 656d 6f76 en surface remov\n-00019f10: 616c 3a3a 2e29 2020 666f 7220 5f61 6c6c al::.) for _all\n-00019f20: 5f20 6869 6464 656e 0a20 2020 2020 7375 _ hidden. su\n-00019f30: 7266 6163 6520 7265 6d6f 7661 6c20 7261 rface removal ra\n-00019f40: 7468 6572 2074 6861 6e20 7468 6520 6465 ther than the de\n-00019f50: 6661 756c 742c 2077 6869 6368 2069 7320 fault, which is \n-00019f60: 7468 6520 6465 7074 6820 736f 7274 0a20 the depth sort. \n-00019f70: 2020 2020 616c 676f 7269 7468 6d20 7769 algorithm wi\n-00019f80: 7468 2042 5350 7320 7573 6564 206f 6e6c th BSPs used onl\n-00019f90: 7920 666f 7220 6379 636c 6520 7265 736f y for cycle reso\n-00019fa0: 6c75 7469 6f6e 2e20 2054 6869 7320 6d61 lution. This ma\n-00019fb0: 790a 2020 2020 2070 726f 6475 6365 2063 y. produce c\n-00019fc0: 6f72 7265 6374 206f 7574 7075 7420 696e orrect output in\n-00019fd0: 2063 6572 7461 696e 2064 6567 656e 6572 certain degener\n-00019fe0: 6174 6520 6361 7365 7320 7768 6572 6520 ate cases where \n-00019ff0: 7468 6520 6465 7074 680a 2020 2020 2073 the depth. s\n-0001a000: 6f72 7420 6361 6e6e 6f74 2c20 6275 7420 ort cannot, but \n-0001a010: 6974 2061 6c73 6f20 6c65 6164 7320 746f it also leads to\n-0001a020: 206d 616e 7920 6772 6174 7569 746f 7573 many gratuitous\n-0001a030: 2073 706c 6974 732c 2068 656e 6365 0a20 splits, hence. \n-0001a040: 2020 2020 6d6f 7265 2061 6e6f 6d61 6c69 more anomali\n-0001a050: 6573 202a 6e6f 7465 2042 7567 7320 616e es *note Bugs an\n-0001a060: 6420 616e 6f6d 616c 6965 733a 3a20 616e d anomalies:: an\n-0001a070: 6420 6269 6720 6f75 7470 7574 2066 696c d big output fil\n-0001a080: 6573 2e0a e280 982d 64e2 8099 0a20 2020 es.....-d.... \n-0001a090: 2020 5275 6e20 e280 9873 6b65 7463 68e2 Run ...sketch.\n-0001a0a0: 8099 2773 2070 6172 7365 7220 696e 2064 ..'s parser in d\n-0001a0b0: 6562 7567 6769 6e67 206d 6f64 652e 2020 ebugging mode. \n-0001a0c0: 5468 6973 2069 7320 7072 696d 6172 696c This is primaril\n-0001a0d0: 7920 666f 720a 2020 2020 2064 6576 656c y for. devel\n-0001a0e0: 6f70 6d65 6e74 2e0a e280 982d 74e2 8099 opment.....-t...\n-0001a0f0: 0a20 2020 2020 5573 6520 636f 6e74 656e . Use conten\n-0001a100: 7473 206f 6620 6669 6c65 20e2 8098 646f ts of file ...do\n-0001a110: 6374 6d70 e280 9920 6173 2061 2064 6f63 ctmp... as a doc\n-0001a120: 756d 656e 7420 7465 6d70 6c61 7465 2069 ument template i\n-0001a130: 6e20 7768 6963 6820 746f 0a20 2020 2020 n which to. \n-0001a140: 656e 636c 6f73 6520 e280 9850 5354 7269 enclose ...PSTri\n-0001a150: 636b 73e2 8099 206f 7574 7075 7420 636f cks... output co\n-0001a160: 6465 2e20 2054 6865 2063 6f64 6520 6973 de. The code is\n-0001a170: 2069 6e73 6572 7465 6420 696e 2070 6c61 inserted in pla\n-0001a180: 6365 206f 660a 2020 2020 2074 6865 2066 ce of. the f\n-0001a190: 6972 7374 2069 6e73 7461 6e63 6520 6f66 irst instance of\n-0001a1a0: 2074 6865 2065 7363 6170 6520 7374 7269 the escape stri\n-0001a1b0: 6e67 2025 2553 4b45 5443 485f 4f55 5450 ng %%SKETCH_OUTP\n-0001a1c0: 5554 2525 2e0a e280 982d 54e2 8099 0a20 UT%%.....-T.... \n-0001a1d0: 2020 2020 4361 7573 6573 20e2 8098 5053 Causes ...PS\n-0001a1e0: 5472 6963 6b73 e280 9920 6f75 7470 7574 Tricks... output\n-0001a1f0: 2074 6f20 6265 2065 6e63 6c6f 7365 6420 to be enclosed \n-0001a200: 696e 2064 6566 6175 6c74 2055 5320 646f in default US do\n-0001a210: 6375 6d65 6e74 0a20 2020 2020 7465 6d70 cument. temp\n-0001a220: 6c61 7465 2074 6578 742e 2020 4f70 7469 late text. Opti\n-0001a230: 6f6e 20e2 8098 2d54 75e2 8099 2069 7320 on ...-Tu... is \n-0001a240: 6120 7379 6e6f 6e79 6d2e 2020 4f70 7469 a synonym. Opti\n-0001a250: 6f6e 20e2 8098 2d54 65e2 8099 2063 6175 on ...-Te... cau\n-0001a260: 7365 7320 7468 650a 2020 2020 2045 7572 ses the. Eur\n-0001a270: 6f20 7374 616e 6461 7264 2064 6f63 756d o standard docum\n-0001a280: 656e 7420 7465 6d70 6c61 7465 2074 6f20 ent template to \n-0001a290: 6265 2075 7365 642e 2020 4120 e280 9870 be used. A ...p\n-0001a2a0: e280 9920 6170 7065 6e64 6564 2074 6f20 ... appended to \n-0001a2b0: 616e 790a 2020 2020 206f 6620 7468 6573 any. of thes\n-0001a2c0: 6520 6f70 7469 6f6e 7320 6361 7573 6573 e options causes\n-0001a2d0: 2074 6865 2072 6573 7065 6374 6976 6520 the respective \n-0001a2e0: 6465 6661 756c 7420 e280 9850 5354 7269 default ...PSTri\n-0001a2f0: 636b 73e2 8099 2064 6f63 756d 656e 740a cks... document.\n-0001a300: 2020 2020 2074 656d 706c 6174 6520 746f template to\n-0001a310: 2062 6520 7072 696e 7465 6420 746f 2073 be printed to s\n-0001a320: 7461 6e64 6172 6420 6f75 7470 7574 2e20 tandard output. \n-0001a330: 2041 6e20 6170 7065 6e64 6564 20e2 8098 An appended ...\n-0001a340: 50e2 8099 2069 7320 610a 2020 2020 2073 P... is a. s\n-0001a350: 796e 6f6e 796d 2e20 2041 6e20 6170 7065 ynonym. An appe\n-0001a360: 6e64 6564 20e2 8098 54e2 8099 2063 6175 nded ...T... cau\n-0001a370: 7365 7320 7468 6520 7468 6520 e280 9854 ses the the ...T\n-0001a380: 696b 5a2f 5047 46e2 8099 2074 656d 706c ikZ/PGF... templ\n-0001a390: 6174 6520 746f 2062 650a 2020 2020 2070 ate to be. p\n-0001a3a0: 7269 6e74 6564 2e20 2041 6e20 6170 7065 rinted. An appe\n-0001a3b0: 6e64 6564 20e2 8098 4ce2 8099 2070 7269 nded ...L... pri\n-0001a3c0: 6e74 7320 7468 6520 4c61 5465 5820 7665 nts the LaTeX ve\n-0001a3d0: 7273 696f 6e20 6f66 2074 6865 2064 6f63 rsion of the doc\n-0001a3e0: 756d 656e 740a 2020 2020 2074 656d 706c ument. templ\n-0001a3f0: 6174 652c 2061 2073 796e 6f6e 796d 2066 ate, a synonym f\n-0001a400: 6f72 2074 6865 2064 6566 6175 6c74 2e20 or the default. \n-0001a410: 2041 20e2 8098 43e2 8099 2070 7269 6e74 A ...C... print\n-0001a420: 7320 7468 6520 e280 9843 6f6e 5465 5874 s the ...ConTeXt\n-0001a430: e280 990a 2020 2020 2074 656d 706c 6174 .... templat\n-0001a440: 652e 0ae2 8098 2d6f e280 990a 2020 2020 e.....-o.... \n-0001a450: 2055 7365 20e2 8098 6f75 7470 7574 2e74 Use ...output.t\n-0001a460: 6578 e280 9920 6173 2074 6865 206f 7574 ex... as the out\n-0001a470: 7075 7420 6669 6c65 2e20 2054 6865 2064 put file. The d\n-0001a480: 6566 6175 6c74 2069 7320 7374 616e 6461 efault is standa\n-0001a490: 7264 0a20 2020 2020 6f75 7470 7574 2e0a rd. output..\n-0001a4a0: e280 982d 44e2 8099 0a20 2020 2020 4465 ...-D.... De\n-0001a4b0: 6669 6e65 2061 2074 6167 2066 6f72 2070 fine a tag for p\n-0001a4c0: 7572 706f 7365 7320 6f66 2073 656c 6563 urposes of selec\n-0001a4d0: 7469 6e67 2064 6566 696e 6974 696f 6e20 ting definition \n-0001a4e0: 616c 7465 726e 6174 6976 6573 2e0a 2020 alternatives.. \n-0001a4f0: 2020 202a 4e6f 7465 2044 6566 696e 6974 *Note Definit\n-0001a500: 696f 6e73 3a3a 2e20 2054 6865 2064 6566 ions::. The def\n-0001a510: 696e 6974 696f 6e20 6170 706c 6965 7320 inition applies \n-0001a520: 666f 7220 616c 6c20 696e 7075 7420 6669 for all input fi\n-0001a530: 6c65 730a 2020 2020 2074 6861 7420 666f les. that fo\n-0001a540: 6c6c 6f77 2075 6e6c 6573 7320 7468 6520 llow unless the \n-0001a550: 7461 6720 6973 2075 6e64 6566 696e 6564 tag is undefined\n-0001a560: 2077 6974 6820 e280 982d 55e2 8099 2e0a with ...-U.....\n-0001a570: e280 9869 6e70 7574 692e 736b e280 990a ...inputi.sk....\n-0001a580: 2020 2020 2049 6e70 7574 2066 696c 6573 Input files\n-0001a590: 2c20 7265 6164 2069 6e20 7468 6520 7365 , read in the se\n-0001a5a0: 7175 656e 6365 2074 6865 7920 6172 6520 quence they are \n-0001a5b0: 6769 7665 6e2e 0ae2 8098 2d55 e280 990a given.....-U....\n-0001a5c0: 2020 2020 2055 6e2d 6465 6669 6e65 2061 Un-define a\n-0001a5d0: 2074 6167 2066 6f72 2070 7572 706f 7365 tag for purpose\n-0001a5e0: 7320 6f66 2073 656c 6563 7469 6e67 2064 s of selecting d\n-0001a5f0: 6566 696e 6974 696f 6e20 616c 7465 726e efinition altern\n-0001a600: 6174 6976 6573 2e0a 0a1f 0a46 696c 653a atives.....File:\n-0001a610: 2073 6b65 7463 682e 696e 666f 2c20 204e sketch.info, N\n-0001a620: 6f64 653a 2049 6e73 7461 6c6c 696e 6720 ode: Installing \n-0001a630: 736b 6574 6368 2c20 204e 6578 743a 2049 sketch, Next: I\n-0001a640: 6e64 6578 206f 6620 7379 6e74 6178 2c20 ndex of syntax, \n-0001a650: 2050 7265 763a 2043 6f6d 6d61 6e64 206c Prev: Command l\n-0001a660: 696e 652c 2020 5570 3a20 546f 700a 0a36 ine, Up: Top..6\n-0001a670: 2042 7569 6c64 696e 6720 616e 6420 696e Building and in\n-0001a680: 7374 616c 6c69 6e67 20e2 8098 736b 6574 stalling ...sket\n-0001a690: 6368 e280 990a 2a2a 2a2a 2a2a 2a2a 2a2a ch....**********\n+00019430: 2e2e 2e2e 0a0a 506f 6c79 676f 6e20 616e ......Polygon an\n+00019440: 6420 6c69 6e65 2073 706c 6974 7469 6e67 d line splitting\n+00019450: 2063 616e 2062 6f74 6820 6361 7573 6520 can both cause \n+00019460: 616e 6f6d 616c 6965 7320 696e 2074 6865 anomalies in the\n+00019470: 206f 7574 7075 742e 0ae2 8098 5053 5472 output.....PSTr\n+00019480: 6963 6b73 e280 9920 6461 7368 2070 6174 icks... dash pat\n+00019490: 7465 726e 732c 2073 7065 6369 6669 6564 terns, specified\n+000194a0: 2077 6974 6820 e280 986c 696e 6573 7479 with ...linesty\n+000194b0: 6c65 3d64 6173 6865 64e2 8099 2c20 6361 le=dashed..., ca\n+000194c0: 6e20 6265 0a64 6973 7275 7074 6564 2062 n be.disrupted b\n+000194d0: 7920 7370 6c69 7474 696e 672e 2020 5468 y splitting. Th\n+000194e0: 6973 206f 6363 7572 7320 7768 656e 2074 is occurs when t\n+000194f0: 6865 2064 6570 7468 2073 6f72 7420 6769 he depth sort gi\n+00019500: 7665 7320 7570 2074 6f6f 0a65 6172 6c79 ves up too.early\n+00019510: 2061 6e64 2073 706c 6974 7320 6120 6c69 and splits a li\n+00019520: 6e65 2077 6865 7265 2069 7420 6973 206e ne where it is n\n+00019530: 6f74 2072 6561 6c6c 7920 6e65 6365 7373 ot really necess\n+00019540: 6172 792e 2020 4120 776f 726b 6172 6f75 ary. A workarou\n+00019550: 6e64 0a69 7320 746f 2075 7365 2067 7261 nd.is to use gra\n+00019560: 7920 6f72 2066 696e 656c 7920 646f 7474 y or finely dott\n+00019570: 6564 206c 696e 6573 2069 6e73 7465 6164 ed lines instead\n+00019580: 2e20 2049 6620 796f 7572 2064 7261 7769 . 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I\n+00019890: 6e20 6661 6374 20e2 8098 6c69 6e65 7769 n fact ...linewi\n+000198a0: 6474 683d 3070 74e2 8099 2069 7320 6775 dth=0pt... is gu\n+000198b0: 6172 616e 7465 6564 2074 6f20 656c 696d aranteed to elim\n+000198c0: 696e 6174 650a 7468 6973 2070 726f 626c inate.this probl\n+000198d0: 656d 2c20 7468 6f75 6768 2074 6869 7320 em, though this \n+000198e0: 7265 7375 6c74 7320 696e 2074 6865 2074 results in the t\n+000198f0: 6869 6e6e 6573 7420 6c69 6e65 2079 6f75 hinnest line you\n+00019900: 7220 6f75 7470 7574 0a64 6576 6963 6520 r output.device \n+00019910: 6361 6e20 6472 6177 2c20 7768 6963 6820 can draw, which \n+00019920: 6973 2075 7375 616c 6c79 2074 6f6f 2074 is usually too t\n+00019930: 6869 6e2e 2020 596f 7520 6d69 6768 7420 hin. You might \n+00019940: 6765 7420 6c75 636b 7920 6279 0a6d 6572 get lucky by.mer\n+00019950: 656c 7920 7265 6f72 6465 7269 6e67 2074 ely reordering t\n+00019960: 6869 6e67 7320 696e 2074 6865 2069 6e70 hings in the inp\n+00019970: 7574 2066 696c 652c 2077 6869 6368 2069 ut file, which i\n+00019980: 7320 6c69 6b65 6c79 2074 6f20 6d6f 7665 s likely to move\n+00019990: 2074 6865 0a73 706c 6974 7320 746f 2064 the.splits to d\n+000199a0: 6966 6665 7265 6e74 2070 6c61 6365 732e ifferent places.\n+000199b0: 2020 5468 6520 6f6e 6c79 2073 7572 652d The only sure-\n+000199c0: 6669 7265 2073 6f6c 7574 696f 6e20 6973 fire solution is\n+000199d0: 2070 7265 7474 790a 7465 7272 6962 6c65 pretty.terrible\n+000199e0: 3a20 6375 7374 6f6d 2066 6974 20e2 8098 : custom fit ...\n+000199f0: 7370 6563 6961 6ce2 8099 206f 7665 726c special... overl\n+00019a00: 6179 206c 696e 6573 2028 7769 7468 20e2 ay lines (with .\n+00019a10: 8098 5c70 736c 696e 65e2 8099 2920 746f ..\\psline...) to\n+00019a20: 2063 6f76 6572 0a74 6865 206e 6f74 6368 cover.the notch\n+00019a30: 6573 2e0a 0a20 2020 506f 6c79 676f 6e20 es... Polygon \n+00019a40: 7370 6c69 7474 696e 6720 616c 736f 2062 splitting also b\n+00019a50: 7265 616b 7320 e280 9850 5354 7269 636b reaks ...PSTrick\n+00019a60: 73e2 8099 2068 6174 6368 2070 6174 7465 s... hatch patte\n+00019a70: 726e 732e 2020 5468 6520 6f6e 6c79 0a6b rns. The only.k\n+00019a80: 6e6f 776e 2077 6f72 6b61 726f 756e 6420 nown workaround \n+00019a90: 6973 2074 6f20 7375 6273 7469 7475 7465 is to substitute\n+00019aa0: 2061 2073 6f6c 6964 2066 696c 6c20 666f a solid fill fo\n+00019ab0: 7220 7468 6520 6861 7463 682e 0a0a 2020 r the hatch... \n+00019ac0: 202d 2d2d 2d2d 2d2d 2d2d 2d20 466f 6f74 ---------- Foot\n+00019ad0: 6e6f 7465 7320 2d2d 2d2d 2d2d 2d2d 2d2d notes ----------\n+00019ae0: 0a0a 2020 2028 3129 2049 206b 6e6f 7720 .. (1) I know \n+00019af0: 686f 7720 746f 2066 6978 2074 6869 7320 how to fix this \n+00019b00: 7072 6f62 6c65 6d2c 2062 7574 2049 2064 problem, but I d\n+00019b10: 6f6e 2774 206c 696b 6520 6d79 2073 6f6c on't like my sol\n+00019b20: 7574 696f 6e2c 2061 6e64 0a49 276d 2069 ution, and.I'm i\n+00019b30: 6e74 6572 6573 7465 6420 696e 2079 6f75 nterested in you\n+00019b40: 7273 2e0a 0a1f 0a46 696c 653a 2073 6b65 rs.....File: ske\n+00019b50: 7463 682e 696e 666f 2c20 204e 6f64 653a tch.info, Node:\n+00019b60: 2043 6f6d 6d61 6e64 206c 696e 652c 2020 Command line, \n+00019b70: 4e65 7874 3a20 496e 7374 616c 6c69 6e67 Next: Installing\n+00019b80: 2073 6b65 7463 682c 2020 5072 6576 3a20 sketch, Prev: \n+00019b90: 4275 696c 6469 6e67 2061 2064 7261 7769 Building a drawi\n+00019ba0: 6e67 2c20 2055 703a 2054 6f70 0a0a 3520 ng, Up: Top..5 \n+00019bb0: 436f 6d6d 616e 6420 6c69 6e65 0a2a 2a2a Command line.***\n+00019bc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a2a 5379 ***********..*Sy\n+00019bd0: 6e6f 7073 6973 3a2a 0a20 2020 2020 736b nopsis:*. sk\n+00019be0: 6574 6368 205b 2d68 5d5b 2d56 2078 2e79 etch [-h][-V x.y\n+00019bf0: 5d5b 2d76 5d5b 2d62 5d5b 2d64 5d5b 7420 ][-v][-b][-d][t \n+00019c00: 646f 6374 6d70 5d5b 2d54 5b75 7c65 5d5b doctmp][-T[u|e][\n+00019c10: 705b 507c 545d 5b4c 7c43 5d5d 5d5b 2d6f p[P|T][L|C]]][-o\n+00019c20: 206f 7574 7075 742e 7465 785d 0a20 2020 output.tex]. \n+00019c30: 2020 2020 5b2d 4420 5441 4720 2e2e 2e5d [-D TAG ...]\n+00019c40: 2069 6e70 7574 312e 736b 205b 2d55 2054 input1.sk [-U T\n+00019c50: 4147 202e 2e2e 5d20 696e 7075 7432 2e73 AG ...] input2.s\n+00019c60: 6b20 2e2e 2e0a 0a2a 4465 7363 7269 7074 k .....*Descript\n+00019c70: 696f 6e2a 2050 726f 6365 7373 6573 2074 ion* Processes t\n+00019c80: 6865 20e2 8098 736b 6574 6368 e280 9920 he ...sketch... \n+00019c90: 696e 7075 7420 6669 6c65 7320 696e 206f input files in o\n+00019ca0: 7264 6572 2074 6f20 7072 6f64 7563 650a rder to produce.\n+00019cb0: e280 9850 5354 7269 636b 73e2 8099 206f ...PSTricks... o\n+00019cc0: 7574 7075 7420 636f 6465 2073 7569 7461 utput code suita\n+00019cd0: 626c 6520 666f 7220 696e 636c 7573 696f ble for inclusio\n+00019ce0: 6e20 696e 2061 2054 6558 206f 7220 4c61 n in a TeX or La\n+00019cf0: 5465 580a 646f 6375 6d65 6e74 2e0a 0a2a TeX.document...*\n+00019d00: 4f70 7469 6f6e 733a 2a0a e280 982d 68e2 Options:*....-h.\n+00019d10: 8099 0a20 2020 2020 5072 696e 7420 6120 ... Print a \n+00019d20: 7368 6f72 7420 6361 7461 6c6f 6720 6f66 short catalog of\n+00019d30: 206f 7074 696f 6e73 2e0a e280 982d 56e2 options.....-V.\n+00019d40: 8099 0a20 2020 2020 5365 7420 7468 6520 ... Set the \n+00019d50: e280 9850 5354 7269 636b 73e2 8099 2076 ...PSTricks... v\n+00019d60: 6572 7369 6f6e 2061 7373 756d 6564 2066 ersion assumed f\n+00019d70: 6f72 206f 7574 7075 7420 7075 7270 6f73 or output purpos\n+00019d80: 6573 2074 6f20 e280 9878 2e79 e280 992c es to ...x.y...,\n+00019d90: 0a20 2020 2020 666f 7220 6578 616d 706c . for exampl\n+00019da0: 6520 312e 3139 2e20 2055 7375 616c 6c79 e 1.19. 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Run ...ske\n+0001a0a0: 7463 68e2 8099 2773 2070 6172 7365 7220 tch...'s parser \n+0001a0b0: 696e 2064 6562 7567 6769 6e67 206d 6f64 in debugging mod\n+0001a0c0: 652e 2020 5468 6973 2069 7320 7072 696d e. This is prim\n+0001a0d0: 6172 696c 7920 666f 720a 2020 2020 2064 arily for. d\n+0001a0e0: 6576 656c 6f70 6d65 6e74 2e0a e280 982d evelopment.....-\n+0001a0f0: 74e2 8099 0a20 2020 2020 5573 6520 636f t.... Use co\n+0001a100: 6e74 656e 7473 206f 6620 6669 6c65 20e2 ntents of file .\n+0001a110: 8098 646f 6374 6d70 e280 9920 6173 2061 ..doctmp... as a\n+0001a120: 2064 6f63 756d 656e 7420 7465 6d70 6c61 document templa\n+0001a130: 7465 2069 6e20 7768 6963 6820 746f 0a20 te in which to. \n+0001a140: 2020 2020 656e 636c 6f73 6520 e280 9850 enclose ...P\n+0001a150: 5354 7269 636b 73e2 8099 206f 7574 7075 STricks... outpu\n+0001a160: 7420 636f 6465 2e20 2054 6865 2063 6f64 t code. The cod\n+0001a170: 6520 6973 2069 6e73 6572 7465 6420 696e e is inserted in\n+0001a180: 2070 6c61 6365 206f 660a 2020 2020 2074 place of. t\n+0001a190: 6865 2066 6972 7374 2069 6e73 7461 6e63 he first instanc\n+0001a1a0: 6520 6f66 2074 6865 2065 7363 6170 6520 e of the escape \n+0001a1b0: 7374 7269 6e67 2025 2553 4b45 5443 485f string %%SKETCH_\n+0001a1c0: 4f55 5450 5554 2525 2e0a e280 982d 54e2 OUTPUT%%.....-T.\n+0001a1d0: 8099 0a20 2020 2020 4361 7573 6573 20e2 ... 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(line 1\n-0001b690: 3429 0a2a 205b 5b20 5d5b 205d 5b20 5d5b 4).* [[ ][ ][ ][\n-0001b6a0: 205d 5d2c 2074 7261 6e73 666f 726d 206c ]], transform l\n-0001b6b0: 6974 6572 616c 3a20 2020 2020 5472 616e iteral: Tran\n-0001b6c0: 7366 6f72 6d20 6c69 7465 7261 6c73 2e20 sform literals. \n-0001b6d0: 2020 286c 696e 6520 3437 290a 2a20 5b5b (line 47).* [[\n-0001b6e0: 205d 5b20 5d5b 205d 5b20 5d5d 2c20 7472 ][ ][ ][ ]], tr\n-0001b6f0: 616e 7366 6f72 6d20 6c69 7465 7261 6c20 ansform literal \n-0001b700: 3c31 3e3a 2054 7261 6e73 666f 726d 206c <1>: Transform l\n-0001b710: 6974 6572 616c 732e 2020 2028 6c69 6e65 iterals. (line\n-0001b720: 2035 3329 0a2a 205b 5b66 6f6f 5d5d 2c20 53).* [[foo]], \n-0001b730: 7472 616e 7366 6f72 6d20 7265 6665 7265 transform refere\n-0001b740: 6e63 653a 2020 2020 2020 2020 2020 466f nce: Fo\n-0001b750: 726d 7320 6f66 2072 6566 6572 656e 6365 rms of reference\n-0001b760: 732e 2020 286c 696e 6520 3136 290a 2a20 s. (line 16).* \n-0001b770: 5b5b 666f 6f5d 5d2c 2074 7261 6e73 666f [[foo]], transfo\n-0001b780: 726d 2072 6566 6572 656e 6365 203c 313e rm reference <1>\n-0001b790: 3a20 2020 2020 204f 7665 7276 6965 772e : Overview.\n-0001b7a0: 2020 2020 2020 2020 2020 2020 2028 6c69 (li\n-0001b7b0: 6e65 2033 3529 0a2a 205b 5b66 6f6f 5d5d ne 35).* [[foo]]\n-0001b7c0: 2c20 7472 616e 7366 6f72 6d20 7265 6665 , transform refe\n-0001b7d0: 7265 6e63 6520 3c32 3e3a 2020 2020 2020 rence <2>: \n-0001b7e0: 4f76 6572 7669 6577 2e20 2020 2020 2020 Overview. \n-0001b7f0: 2020 2020 2020 286c 696e 6520 3335 290a (line 35).\n-0001b800: 2a20 5b66 6f6f 2c2e 2e2e 2c62 6172 5d2c * [foo,...,bar],\n-0001b810: 206d 756c 7469 706c 6520 6f70 7469 6f6e multiple option\n-0001b820: 7320 7265 6665 7265 6e63 653a 2046 6f72 s reference: For\n-0001b830: 6d73 206f 6620 7265 6665 7265 6e63 6573 ms of references\n-0001b840: 2e0a 2020 2020 2020 2020 2020 2020 2020 .. \n+0001b5f0: 2020 2020 2020 2020 2020 2028 6c69 6e65 (line\n+0001b600: 2035 3829 0a2a 2028 666f 6f29 2c20 706f 58).* (foo), po\n+0001b610: 696e 7420 7265 6665 7265 6e63 653a 2020 int reference: \n+0001b620: 2020 2020 2020 2020 2020 2020 2020 4472 Dr\n+0001b630: 6177 696e 6720 6120 736f 6c69 642e 2020 awing a solid. \n+0001b640: 2020 2020 286c 696e 6520 3234 290a 2a20 (line 24).* \n+0001b650: 2866 6f6f 292c 2070 6f69 6e74 2072 6566 (foo), point ref\n+0001b660: 6572 656e 6365 203c 313e 3a20 2020 2020 erence <1>: \n+0001b670: 2020 2020 2020 2046 6f72 6d73 206f 6620 Forms of \n+0001b680: 7265 6665 7265 6e63 6573 2e20 2028 6c69 references. (li\n+0001b690: 6e65 2031 3429 0a2a 205b 5b20 5d5b 205d ne 14).* [[ ][ ]\n+0001b6a0: 5b20 5d5b 205d 5d2c 2074 7261 6e73 666f [ ][ ]], transfo\n+0001b6b0: 726d 206c 6974 6572 616c 3a20 2020 2020 rm literal: \n+0001b6c0: 5472 616e 7366 6f72 6d20 6c69 7465 7261 Transform litera\n+0001b6d0: 6c73 2e20 2020 286c 696e 6520 3437 290a ls. (line 47).\n+0001b6e0: 2a20 5b5b 205d 5b20 5d5b 205d 5b20 5d5d * [[ ][ ][ ][ ]]\n+0001b6f0: 2c20 7472 616e 7366 6f72 6d20 6c69 7465 , transform lite\n+0001b700: 7261 6c20 3c31 3e3a 2054 7261 6e73 666f ral <1>: Transfo\n+0001b710: 726d 206c 6974 6572 616c 732e 2020 2028 rm literals. (\n+0001b720: 6c69 6e65 2035 3329 0a2a 205b 5b66 6f6f line 53).* [[foo\n+0001b730: 5d5d 2c20 7472 616e 7366 6f72 6d20 7265 ]], transform re\n+0001b740: 6665 7265 6e63 653a 2020 2020 2020 2020 ference: \n+0001b750: 2020 466f 726d 7320 6f66 2072 6566 6572 Forms of refer\n+0001b760: 656e 6365 732e 2020 286c 696e 6520 3136 ences. (line 16\n+0001b770: 290a 2a20 5b5b 666f 6f5d 5d2c 2074 7261 ).* [[foo]], tra\n+0001b780: 6e73 666f 726d 2072 6566 6572 656e 6365 nsform reference\n+0001b790: 203c 313e 3a20 2020 2020 204f 7665 7276 <1>: Overv\n+0001b7a0: 6965 772e 2020 2020 2020 2020 2020 2020 iew. \n+0001b7b0: 2028 6c69 6e65 2033 3529 0a2a 205b 5b66 (line 35).* [[f\n+0001b7c0: 6f6f 5d5d 2c20 7472 616e 7366 6f72 6d20 oo]], transform \n+0001b7d0: 7265 6665 7265 6e63 6520 3c32 3e3a 2020 reference <2>: \n+0001b7e0: 2020 2020 4f76 6572 7669 6577 2e20 2020 Overview. \n+0001b7f0: 2020 2020 2020 2020 2020 286c 696e 6520 (line \n+0001b800: 3335 290a 2a20 5b66 6f6f 2c2e 2e2e 2c62 35).* [foo,...,b\n+0001b810: 6172 5d2c 206d 756c 7469 706c 6520 6f70 ar], multiple op\n+0001b820: 7469 6f6e 7320 7265 6665 7265 6e63 653a tions reference:\n+0001b830: 2046 6f72 6d73 206f 6620 7265 6665 7265 Forms of refere\n+0001b840: 6e63 6573 2e0a 2020 2020 2020 2020 2020 nces.. \n 0001b850: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001b860: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001b870: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001b880: 2028 6c69 6e65 2031 3829 0a2a 205b 666f (line 18).* [fo\n-0001b890: 6f5d 2c20 6f70 7469 6f6e 7320 7265 6665 o], options refe\n-0001b8a0: 7265 6e63 653a 2020 2020 2020 2020 2020 rence: \n-0001b8b0: 2020 2020 506f 6c79 6c69 6e65 2073 7765 Polyline swe\n-0001b8c0: 6570 7320 7769 7468 2063 6c6f 7375 7265 eps with closure\n-0001b8d0: 2e0a 2020 2020 2020 2020 2020 2020 2020 .. \n+0001b880: 2020 2020 2028 6c69 6e65 2031 3829 0a2a (line 18).*\n+0001b890: 205b 666f 6f5d 2c20 6f70 7469 6f6e 7320 [foo], options \n+0001b8a0: 7265 6665 7265 6e63 653a 2020 2020 2020 reference: \n+0001b8b0: 2020 2020 2020 2020 506f 6c79 6c69 6e65 Polyline\n+0001b8c0: 2073 7765 6570 7320 7769 7468 2063 6c6f sweeps with clo\n+0001b8d0: 7375 7265 2e0a 2020 2020 2020 2020 2020 sure.. \n 0001b8e0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001b8f0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001b900: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001b910: 2028 6c69 6e65 2032 3429 0a2a 205b 666f (line 24).* [fo\n-0001b920: 6f5d 2c20 6f70 7469 6f6e 7320 7265 6665 o], options refe\n-0001b930: 7265 6e63 6520 3c31 3e3a 2020 2020 2020 rence <1>: \n-0001b940: 2020 2020 466f 726d 7320 6f66 2072 6566 Forms of ref\n-0001b950: 6572 656e 6365 732e 2020 286c 696e 6520 erences. (line \n-0001b960: 3138 290a 2a20 5b66 6f6f 5d2c 2076 6563 18).* [foo], vec\n-0001b970: 746f 7220 7265 6665 7265 6e63 653a 2020 tor reference: \n-0001b980: 2020 2020 2020 2020 2020 2020 2050 6f6c Pol\n-0001b990: 796c 696e 6520 7377 6565 7073 2e20 2020 yline sweeps. \n-0001b9a0: 2020 2028 6c69 6e65 2032 3029 0a2a 205b (line 20).* [\n-0001b9b0: 666f 6f5d 2c20 7665 6374 6f72 2072 6566 foo], vector ref\n-0001b9c0: 6572 656e 6365 203c 313e 3a20 2020 2020 erence <1>: \n-0001b9d0: 2020 2020 2020 466f 726d 7320 6f66 2072 Forms of r\n-0001b9e0: 6566 6572 656e 6365 732e 2020 286c 696e eferences. (lin\n-0001b9f0: 6520 3135 290a 2a20 7b20 7d2c 2062 6c6f e 15).* { }, blo\n-0001ba00: 636b 2064 7261 7761 626c 653a 2020 2020 ck drawable: \n-0001ba10: 2020 2020 2020 2020 2020 2020 2020 204f O\n-0001ba20: 626a 6563 7420 7472 616e 7366 6f72 6d73 bject transforms\n-0001ba30: 2e20 2020 2028 6c69 6e65 2032 3429 0a2a . (line 24).*\n-0001ba40: 207b 207d 2c20 626c 6f63 6b20 6472 6177 { }, block draw\n-0001ba50: 6162 6c65 203c 313e 3a20 2020 2020 2020 able <1>: \n-0001ba60: 2020 2020 2020 2020 4e65 7374 6564 2073 Nested s\n-0001ba70: 7765 6570 732e 2020 2020 2020 2020 286c weeps. (l\n-0001ba80: 696e 6520 3235 290a 2a20 7b20 7d2c 2062 ine 25).* { }, b\n-0001ba90: 6c6f 636b 2064 7261 7761 626c 6520 3c32 lock drawable <2\n-0001baa0: 3e3a 2020 2020 2020 2020 2020 2020 2020 >: \n-0001bab0: 2053 7765 7074 2062 6c6f 636b 732e 2020 Swept blocks. \n-0001bac0: 2020 2020 2020 2028 6c69 6e65 2020 3729 (line 7)\n-0001bad0: 0a2a 207b 207d 2c20 626c 6f63 6b20 6472 .* { }, block dr\n-0001bae0: 6177 6162 6c65 203c 333e 3a20 2020 2020 awable <3>: \n-0001baf0: 2020 2020 2020 2020 2020 426c 6f63 6b73 Blocks\n-0001bb00: 2e20 2020 2020 2020 2020 2020 2020 2020 . \n-0001bb10: 286c 696e 6520 2036 290a 2a20 7b20 7d2c (line 6).* { },\n-0001bb20: 2062 6c6f 636b 2064 7261 7761 626c 6520 block drawable \n-0001bb30: 3c34 3e3a 2020 2020 2020 2020 2020 2020 <4>: \n-0001bb40: 2020 204f 7665 7276 6965 772e 2020 2020 Overview. \n-0001bb50: 2020 2020 2020 2020 2028 6c69 6e65 2033 (line 3\n-0001bb60: 3729 0a2a 207b 666f 6f7d 2c20 6472 6177 7).* {foo}, draw\n-0001bb70: 6162 6c65 2072 6566 6572 656e 6365 3a20 able reference: \n-0001bb80: 2020 2020 2020 2020 2020 2020 4f62 6a65 Obje\n-0001bb90: 6374 2074 7261 6e73 666f 726d 732e 2020 ct transforms. \n-0001bba0: 2020 286c 696e 6520 3333 290a 2a20 7b66 (line 33).* {f\n-0001bbb0: 6f6f 7d2c 2064 7261 7761 626c 6520 7265 oo}, drawable re\n-0001bbc0: 6665 7265 6e63 6520 3c31 3e3a 2020 2020 ference <1>: \n-0001bbd0: 2020 2020 2046 6f72 6d73 206f 6620 7265 Forms of re\n-0001bbe0: 6665 7265 6e63 6573 2e20 2028 6c69 6e65 ferences. (line\n-0001bbf0: 2031 3729 0a2a 207b 666f 6f7d 2c20 6472 17).* {foo}, dr\n-0001bc00: 6177 6162 6c65 2072 6566 6572 656e 6365 awable reference\n-0001bc10: 203c 323e 3a20 2020 2020 2020 2020 4f76 <2>: Ov\n-0001bc20: 6572 7669 6577 2e20 2020 2020 2020 2020 erview. \n-0001bc30: 2020 2020 286c 696e 6520 3437 290a 2a20 (line 47).* \n-0001bc40: 7b66 6f6f 7d2c 2064 7261 7761 626c 6520 {foo}, drawable \n-0001bc50: 7265 6665 7265 6e63 6520 3c33 3e3a 2020 reference <3>: \n-0001bc60: 2020 2020 2020 204f 7665 7276 6965 772e Overview.\n-0001bc70: 2020 2020 2020 2020 2020 2020 2028 6c69 (li\n-0001bc80: 6e65 2035 3729 0a2a 202a 2c20 6d75 6c74 ne 57).* *, mult\n-0001bc90: 6970 6c69 6361 7469 6f6e 206f 7065 7261 iplication opera\n-0001bca0: 746f 723a 2020 2020 2020 2020 2020 2020 tor: \n-0001bcb0: 5477 6f2d 6f70 6572 616e 6420 2862 696e Two-operand (bin\n-0001bcc0: 6172 7929 2066 6f72 6d73 2e0a 2020 2020 ary) forms.. \n+0001b910: 2020 2020 2028 6c69 6e65 2032 3429 0a2a (line 24).*\n+0001b920: 205b 666f 6f5d 2c20 6f70 7469 6f6e 7320 [foo], options \n+0001b930: 7265 6665 7265 6e63 6520 3c31 3e3a 2020 reference <1>: \n+0001b940: 2020 2020 2020 2020 466f 726d 7320 6f66 Forms of\n+0001b950: 2072 6566 6572 656e 6365 732e 2020 286c references. (l\n+0001b960: 696e 6520 3138 290a 2a20 5b66 6f6f 5d2c ine 18).* [foo],\n+0001b970: 2076 6563 746f 7220 7265 6665 7265 6e63 vector referenc\n+0001b980: 653a 2020 2020 2020 2020 2020 2020 2020 e: \n+0001b990: 2050 6f6c 796c 696e 6520 7377 6565 7073 Polyline sweeps\n+0001b9a0: 2e20 2020 2020 2028 6c69 6e65 2032 3029 . (line 20)\n+0001b9b0: 0a2a 205b 666f 6f5d 2c20 7665 6374 6f72 .* [foo], vector\n+0001b9c0: 2072 6566 6572 656e 6365 203c 313e 3a20 reference <1>: \n+0001b9d0: 2020 2020 2020 2020 2020 466f 726d 7320 Forms \n+0001b9e0: 6f66 2072 6566 6572 656e 6365 732e 2020 of references. \n+0001b9f0: 286c 696e 6520 3135 290a 2a20 7b20 7d2c (line 15).* { },\n+0001ba00: 2062 6c6f 636b 2064 7261 7761 626c 653a block drawable:\n+0001ba10: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001ba20: 2020 204f 626a 6563 7420 7472 616e 7366 Object transf\n+0001ba30: 6f72 6d73 2e20 2020 2028 6c69 6e65 2032 orms. (line 2\n+0001ba40: 3429 0a2a 207b 207d 2c20 626c 6f63 6b20 4).* { }, block \n+0001ba50: 6472 6177 6162 6c65 203c 313e 3a20 2020 drawable <1>: \n+0001ba60: 2020 2020 2020 2020 2020 2020 4e65 7374 Nest\n+0001ba70: 6564 2073 7765 6570 732e 2020 2020 2020 ed sweeps. \n+0001ba80: 2020 286c 696e 6520 3235 290a 2a20 7b20 (line 25).* { \n+0001ba90: 7d2c 2062 6c6f 636b 2064 7261 7761 626c }, block drawabl\n+0001baa0: 6520 3c32 3e3a 2020 2020 2020 2020 2020 e <2>: \n+0001bab0: 2020 2020 2053 7765 7074 2062 6c6f 636b Swept block\n+0001bac0: 732e 2020 2020 2020 2020 2028 6c69 6e65 s. (line\n+0001bad0: 2020 3729 0a2a 207b 207d 2c20 626c 6f63 7).* { }, bloc\n+0001bae0: 6b20 6472 6177 6162 6c65 203c 333e 3a20 k drawable <3>: \n+0001baf0: 2020 2020 2020 2020 2020 2020 2020 426c Bl\n+0001bb00: 6f63 6b73 2e20 2020 2020 2020 2020 2020 ocks. \n+0001bb10: 2020 2020 286c 696e 6520 2036 290a 2a20 (line 6).* \n+0001bb20: 7b20 7d2c 2062 6c6f 636b 2064 7261 7761 { }, block drawa\n+0001bb30: 626c 6520 3c34 3e3a 2020 2020 2020 2020 ble <4>: \n+0001bb40: 2020 2020 2020 204f 7665 7276 6965 772e Overview.\n+0001bb50: 2020 2020 2020 2020 2020 2020 2028 6c69 (li\n+0001bb60: 6e65 2033 3729 0a2a 207b 666f 6f7d 2c20 ne 37).* {foo}, \n+0001bb70: 6472 6177 6162 6c65 2072 6566 6572 656e drawable referen\n+0001bb80: 6365 3a20 2020 2020 2020 2020 2020 2020 ce: \n+0001bb90: 4f62 6a65 6374 2074 7261 6e73 666f 726d Object transform\n+0001bba0: 732e 2020 2020 286c 696e 6520 3333 290a s. (line 33).\n+0001bbb0: 2a20 7b66 6f6f 7d2c 2064 7261 7761 626c * {foo}, drawabl\n+0001bbc0: 6520 7265 6665 7265 6e63 6520 3c31 3e3a e reference <1>:\n+0001bbd0: 2020 2020 2020 2020 2046 6f72 6d73 206f Forms o\n+0001bbe0: 6620 7265 6665 7265 6e63 6573 2e20 2028 f references. (\n+0001bbf0: 6c69 6e65 2031 3729 0a2a 207b 666f 6f7d line 17).* {foo}\n+0001bc00: 2c20 6472 6177 6162 6c65 2072 6566 6572 , drawable refer\n+0001bc10: 656e 6365 203c 323e 3a20 2020 2020 2020 ence <2>: \n+0001bc20: 2020 4f76 6572 7669 6577 2e20 2020 2020 Overview. \n+0001bc30: 2020 2020 2020 2020 286c 696e 6520 3437 (line 47\n+0001bc40: 290a 2a20 7b66 6f6f 7d2c 2064 7261 7761 ).* {foo}, drawa\n+0001bc50: 626c 6520 7265 6665 7265 6e63 6520 3c33 ble reference <3\n+0001bc60: 3e3a 2020 2020 2020 2020 204f 7665 7276 >: Overv\n+0001bc70: 6965 772e 2020 2020 2020 2020 2020 2020 iew. \n+0001bc80: 2028 6c69 6e65 2035 3729 0a2a 202a 2c20 (line 57).* *, \n+0001bc90: 6d75 6c74 6970 6c69 6361 7469 6f6e 206f multiplication o\n+0001bca0: 7065 7261 746f 723a 2020 2020 2020 2020 perator: \n+0001bcb0: 2020 2020 5477 6f2d 6f70 6572 616e 6420 Two-operand \n+0001bcc0: 2862 696e 6172 7929 2066 6f72 6d73 2e0a (binary) forms..\n 0001bcd0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001bce0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001bcf0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001bd00: 2020 2020 2020 2020 2020 2028 6c69 6e65 (line\n-0001bd10: 2031 3929 0a2a 202a 2c20 6d75 6c74 6970 19).* *, multip\n-0001bd20: 6c69 6361 7469 6f6e 206f 7065 7261 746f lication operato\n-0001bd30: 7220 3c31 3e3a 2020 2020 2020 2020 5477 r <1>: Tw\n-0001bd40: 6f2d 6f70 6572 616e 6420 2862 696e 6172 o-operand (binar\n-0001bd50: 7929 2066 6f72 6d73 2e0a 2020 2020 2020 y) forms.. \n+0001bd00: 2020 2020 2020 2020 2020 2020 2020 2028 (\n+0001bd10: 6c69 6e65 2031 3929 0a2a 202a 2c20 6d75 line 19).* *, mu\n+0001bd20: 6c74 6970 6c69 6361 7469 6f6e 206f 7065 ltiplication ope\n+0001bd30: 7261 746f 7220 3c31 3e3a 2020 2020 2020 rator <1>: \n+0001bd40: 2020 5477 6f2d 6f70 6572 616e 6420 2862 Two-operand (b\n+0001bd50: 696e 6172 7929 2066 6f72 6d73 2e0a 2020 inary) forms.. \n 0001bd60: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001bd70: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001bd80: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001bd90: 2020 2020 2020 2020 2028 6c69 6e65 2036 (line 6\n-0001bda0: 3829 0a2a 202f 2c20 6469 7669 7369 6f6e 8).* /, division\n-0001bdb0: 206f 7065 7261 746f 723a 2020 2020 2020 operator: \n-0001bdc0: 2020 2020 2020 2020 2020 2020 5477 6f2d Two-\n-0001bdd0: 6f70 6572 616e 6420 2862 696e 6172 7929 operand (binary)\n-0001bde0: 2066 6f72 6d73 2e0a 2020 2020 2020 2020 forms.. \n+0001bd90: 2020 2020 2020 2020 2020 2020 2028 6c69 (li\n+0001bda0: 6e65 2036 3829 0a2a 202f 2c20 6469 7669 ne 68).* /, divi\n+0001bdb0: 7369 6f6e 206f 7065 7261 746f 723a 2020 sion operator: \n+0001bdc0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001bdd0: 5477 6f2d 6f70 6572 616e 6420 2862 696e Two-operand (bin\n+0001bde0: 6172 7929 2066 6f72 6d73 2e0a 2020 2020 ary) forms.. \n 0001bdf0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001be00: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001be10: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001be20: 2020 2020 2020 2028 6c69 6e65 2033 3829 (line 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(line 11).\n+0001c1e0: 2a20 3c66 6f6f 3e2c 2074 6167 2072 6566 * , tag ref\n+0001c1f0: 6572 656e 6365 3a20 2020 2020 2020 2020 erence: \n+0001c200: 2020 2020 2020 2020 2046 6f72 6d73 206f Forms o\n+0001c210: 6620 6465 6669 6e69 7469 6f6e 732e 2028 f definitions. (\n+0001c220: 6c69 6e65 2032 3029 0a2a 203c 666f 6f3e line 20).* \n+0001c230: 2c20 7461 6720 7265 6665 7265 6e63 6520 , tag reference \n+0001c240: 3c31 3e3a 2020 2020 2020 2020 2020 2020 <1>: \n+0001c250: 2020 466f 726d 7320 6f66 2072 6566 6572 Forms of refer\n+0001c260: 656e 6365 732e 2020 286c 696e 6520 3139 ences. (line 19\n+0001c270: 290a 2a20 7c58 7c2c 206d 6167 6e69 7475 ).* |X|, magnitu\n+0001c280: 6465 206f 7065 7261 746f 723a 2020 2020 de operator: \n+0001c290: 2020 2020 2020 2020 2020 2055 6e61 7279 Unary\n+0001c2a0: 2066 6f72 6d73 2e20 2020 2020 2020 2020 forms. \n+0001c2b0: 2028 6c69 6e65 2031 3229 0a2a 2061 636f (line 12).* aco\n+0001c2c0: 733a 2020 2020 2020 2020 2020 2020 2020 s: \n 0001c2d0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001c2e0: 556e 6172 7920 666f 726d 732e 2020 2020 Unary forms. \n-0001c2f0: 2020 2020 2020 286c 696e 6520 3138 290a (line 18).\n-0001c300: 2a20 6172 726f 7773 3a20 2020 2020 2020 * arrows: \n+0001c2e0: 2020 2020 556e 6172 7920 666f 726d 732e Unary forms.\n+0001c2f0: 2020 2020 2020 2020 2020 286c 696e 6520 (line \n+0001c300: 3138 290a 2a20 6172 726f 7773 3a20 2020 18).* arrows: \n 0001c310: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001c320: 2020 2020 2020 2020 2041 6666 696e 6520 Affine \n-0001c330: 6172 6974 686d 6574 6963 2e20 2020 2028 arithmetic. (\n-0001c340: 6c69 6e65 2031 3129 0a2a 2061 7272 6f77 line 11).* arrow\n-0001c350: 7320 3c31 3e3a 2020 2020 2020 2020 2020 s <1>: \n+0001c320: 2020 2020 2020 2020 2020 2020 2041 6666 Aff\n+0001c330: 696e 6520 6172 6974 686d 6574 6963 2e20 ine arithmetic. \n+0001c340: 2020 2028 6c69 6e65 2031 3129 0a2a 2061 (line 11).* a\n+0001c350: 7272 6f77 7320 3c31 3e3a 2020 2020 2020 rrows <1>: \n 0001c360: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001c370: 2020 4166 6669 6e65 2061 7269 7468 6d65 Affine arithme\n-0001c380: 7469 632e 2020 2020 286c 696e 6520 3238 tic. (line 28\n-0001c390: 290a 2a20 6172 726f 7773 203c 323e 3a20 ).* arrows <2>: \n-0001c3a0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001c3b0: 2020 2020 2020 2020 2020 204f 7665 7276 Overv\n-0001c3c0: 6965 772e 2020 2020 2020 2020 2020 2020 iew. \n-0001c3d0: 2028 6c69 6e65 2036 3929 0a2a 2061 7272 (line 69).* arr\n-0001c3e0: 6f77 7320 3c33 3e3a 2020 2020 2020 2020 ows <3>: \n+0001c370: 2020 2020 2020 4166 6669 6e65 2061 7269 Affine ari\n+0001c380: 7468 6d65 7469 632e 2020 2020 286c 696e thmetic. (lin\n+0001c390: 6520 3238 290a 2a20 6172 726f 7773 203c e 28).* arrows <\n+0001c3a0: 323e 3a20 2020 2020 2020 2020 2020 2020 2>: \n+0001c3b0: 2020 2020 2020 2020 2020 2020 2020 204f O\n+0001c3c0: 7665 7276 6965 772e 2020 2020 2020 2020 verview. \n+0001c3d0: 2020 2020 2028 6c69 6e65 2036 3929 0a2a (line 69).*\n+0001c3e0: 2061 7272 6f77 7320 3c33 3e3a 2020 2020 arrows <3>: \n 0001c3f0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001c400: 2020 2020 4120 7465 6368 6e69 6361 6c20 A technical \n-0001c410: 6472 6177 696e 672e 2020 286c 696e 6520 drawing. (line \n-0001c420: 3236 290a 2a20 6172 726f 7773 203c 343e 26).* arrows <4>\n-0001c430: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n-0001c440: 2020 2020 2020 2020 2020 2020 2041 2074 A t\n-0001c450: 6563 686e 6963 616c 2064 7261 7769 6e67 echnical drawing\n-0001c460: 2e20 2028 6c69 6e65 2036 3829 0a2a 2061 . 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(lin\n-0001c4b0: 6520 3137 290a 2a20 6174 616e 323a 2020 e 17).* atan2: \n-0001c4c0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001c4d0: 2020 2020 2020 2020 2020 2020 2020 2055 U\n-0001c4e0: 6e61 7279 2066 6f72 6d73 2e20 2020 2020 nary forms. \n-0001c4f0: 2020 2020 2028 6c69 6e65 2031 3929 0a2a (line 19).*\n-0001c500: 2063 616d 6572 613a 2020 2020 2020 2020 camera: \n+0001c490: 2020 2020 2020 2020 2020 556e 6172 7920 Unary \n+0001c4a0: 666f 726d 732e 2020 2020 2020 2020 2020 forms. \n+0001c4b0: 286c 696e 6520 3137 290a 2a20 6174 616e (line 17).* atan\n+0001c4c0: 323a 2020 2020 2020 2020 2020 2020 2020 2: \n+0001c4d0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001c4e0: 2020 2055 6e61 7279 2066 6f72 6d73 2e20 Unary forms. \n+0001c4f0: 2020 2020 2020 2020 2028 6c69 6e65 2031 (line 1\n+0001c500: 3929 0a2a 2063 616d 6572 613a 2020 2020 9).* camera: \n 0001c510: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001c520: 2020 2020 2020 2020 4361 6d65 7261 2e20 Camera. \n-0001c530: 2020 2020 2020 2020 2020 2020 2020 286c (l\n-0001c540: 696e 6520 2036 290a 2a20 636f 6e74 6578 ine 6).* contex\n-0001c550: 743a 2020 2020 2020 2020 2020 2020 2020 t: \n+0001c520: 2020 2020 2020 2020 2020 2020 4361 6d65 Came\n+0001c530: 7261 2e20 2020 2020 2020 2020 2020 2020 ra. \n+0001c540: 2020 286c 696e 6520 2036 290a 2a20 636f (line 6).* co\n+0001c550: 6e74 6578 743a 2020 2020 2020 2020 2020 ntext: \n 0001c560: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001c570: 204c 616e 6775 6167 652e 2020 2020 2020 Language. \n-0001c580: 2020 2020 2020 2028 6c69 6e65 2031 3729 (line 17)\n-0001c590: 0a2a 2063 6f73 3a20 2020 2020 2020 2020 .* cos: \n+0001c570: 2020 2020 204c 616e 6775 6167 652e 2020 Language. \n+0001c580: 2020 2020 2020 2020 2020 2028 6c69 6e65 (line\n+0001c590: 2031 3729 0a2a 2063 6f73 3a20 2020 2020 17).* cos: \n 0001c5a0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001c5b0: 2020 2020 2020 2020 2020 556e 6172 7920 Unary \n-0001c5c0: 666f 726d 732e 2020 2020 2020 2020 2020 forms. \n-0001c5d0: 286c 696e 6520 3136 290a 2a20 6375 6c6c (line 16).* cull\n-0001c5e0: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+0001c5b0: 2020 2020 2020 2020 2020 2020 2020 556e Un\n+0001c5c0: 6172 7920 666f 726d 732e 2020 2020 2020 ary forms. \n+0001c5d0: 2020 2020 286c 696e 6520 3136 290a 2a20 (line 16).* \n+0001c5e0: 6375 6c6c 3a20 2020 2020 2020 2020 2020 cull: \n 0001c5f0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001c600: 2020 2044 7261 7769 6e67 2061 2073 6f6c Drawing a sol\n-0001c610: 6964 2e20 2020 2020 2028 6c69 6e65 2034 id. (line 4\n-0001c620: 3529 0a2a 2063 756c 6c20 3c31 3e3a 2020 5).* cull <1>: \n-0001c630: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001c640: 2020 2020 2020 2020 2020 2020 506f 6c79 Poly\n-0001c650: 6c69 6e65 2073 7765 6570 732e 2020 2020 line sweeps. \n-0001c660: 2020 286c 696e 6520 2039 290a 2a20 6375 (line 9).* cu\n-0001c670: 6c6c 203c 323e 3a20 2020 2020 2020 2020 ll <2>: \n+0001c600: 2020 2020 2020 2044 7261 7769 6e67 2061 Drawing a\n+0001c610: 2073 6f6c 6964 2e20 2020 2020 2028 6c69 solid. (li\n+0001c620: 6e65 2034 3529 0a2a 2063 756c 6c20 3c31 ne 45).* cull <1\n+0001c630: 3e3a 2020 2020 2020 2020 2020 2020 2020 >: \n+0001c640: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001c650: 506f 6c79 6c69 6e65 2073 7765 6570 732e Polyline sweeps.\n+0001c660: 2020 2020 2020 286c 696e 6520 2039 290a (line 9).\n+0001c670: 2a20 6375 6c6c 203c 323e 3a20 2020 2020 * cull <2>: \n 0001c680: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001c690: 2020 2020 2054 7261 6e73 7061 7265 6e63 Transparenc\n-0001c6a0: 792e 2020 2020 2020 2020 2028 6c69 6e65 y. (line\n-0001c6b0: 2031 3529 0a2a 2063 756c 6c20 3c33 3e3a 15).* cull <3>:\n-0001c6c0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001c6d0: 2020 2020 2020 2020 2020 2020 2020 4120 A \n-0001c6e0: 7465 6368 6e69 6361 6c20 6472 6177 696e technical drawin\n-0001c6f0: 672e 2020 286c 696e 6520 3230 290a 2a20 g. (line 20).* \n-0001c700: 6375 6c6c 203c 343e 3a20 2020 2020 2020 cull <4>: \n+0001c690: 2020 2020 2020 2020 2054 7261 6e73 7061 Transpa\n+0001c6a0: 7265 6e63 792e 2020 2020 2020 2020 2028 rency. (\n+0001c6b0: 6c69 6e65 2031 3529 0a2a 2063 756c 6c20 line 15).* cull \n+0001c6c0: 3c33 3e3a 2020 2020 2020 2020 2020 2020 <3>: \n+0001c6d0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001c6e0: 2020 4120 7465 6368 6e69 6361 6c20 6472 A technical dr\n+0001c6f0: 6177 696e 672e 2020 286c 696e 6520 3230 awing. (line 20\n+0001c700: 290a 2a20 6375 6c6c 203c 343e 3a20 2020 ).* cull <4>: \n 0001c710: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001c720: 2020 2020 2020 2049 6e74 6572 6e61 6c20 Internal \n-0001c730: 6f70 7469 6f6e 732e 2020 2020 2028 6c69 options. (li\n-0001c740: 6e65 2020 3729 0a2a 2063 756c 6c20 3c35 ne 7).* cull <5\n-0001c750: 3e3a 2020 2020 2020 2020 2020 2020 2020 >: \n+0001c720: 2020 2020 2020 2020 2020 2049 6e74 6572 Inter\n+0001c730: 6e61 6c20 6f70 7469 6f6e 732e 2020 2020 nal options. \n+0001c740: 2028 6c69 6e65 2020 3729 0a2a 2063 756c (line 7).* cul\n+0001c750: 6c20 3c35 3e3a 2020 2020 2020 2020 2020 l <5>: \n 0001c760: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001c770: 476c 6f62 616c 206f 7074 696f 6e73 2e20 Global options. \n-0001c780: 2020 2020 2020 286c 696e 6520 3133 290a (line 13).\n-0001c790: 2a20 6375 7276 653a 2020 2020 2020 2020 * curve: \n+0001c770: 2020 2020 476c 6f62 616c 206f 7074 696f Global optio\n+0001c780: 6e73 2e20 2020 2020 2020 286c 696e 6520 ns. (line \n+0001c790: 3133 290a 2a20 6375 7276 653a 2020 2020 13).* curve: \n 0001c7a0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001c7b0: 2020 2020 2020 2020 2043 7572 7665 732e Curves.\n-0001c7c0: 2020 2020 2020 2020 2020 2020 2020 2028 (\n-0001c7d0: 6c69 6e65 2020 3629 0a2a 2064 6566 3a20 line 6).* def: \n-0001c7e0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001c7b0: 2020 2020 2020 2020 2020 2020 2043 7572 Cur\n+0001c7c0: 7665 732e 2020 2020 2020 2020 2020 2020 ves. \n+0001c7d0: 2020 2028 6c69 6e65 2020 3629 0a2a 2064 (line 6).* d\n+0001c7e0: 6566 3a20 2020 2020 2020 2020 2020 2020 ef: \n 0001c7f0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001c800: 2020 4472 6177 696e 6720 6120 736f 6c69 Drawing a soli\n-0001c810: 642e 2020 2020 2020 286c 696e 6520 2039 d. (line 9\n-0001c820: 290a 2a20 6465 6620 3c31 3e3a 2020 2020 ).* def <1>: \n+0001c800: 2020 2020 2020 4472 6177 696e 6720 6120 Drawing a \n+0001c810: 736f 6c69 642e 2020 2020 2020 286c 696e solid. (lin\n+0001c820: 6520 2039 290a 2a20 6465 6620 3c31 3e3a e 9).* def <1>:\n 0001c830: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001c840: 2020 2020 2020 2020 2020 2044 7261 7769 Drawi\n-0001c850: 6e67 2061 2073 6f6c 6964 2e20 2020 2020 ng a solid. \n-0001c860: 2028 6c69 6e65 2032 3229 0a2a 2064 6566 (line 22).* def\n-0001c870: 203c 323e 3a20 2020 2020 2020 2020 2020 <2>: \n+0001c840: 2020 2020 2020 2020 2020 2020 2020 2044 D\n+0001c850: 7261 7769 6e67 2061 2073 6f6c 6964 2e20 rawing a solid. \n+0001c860: 2020 2020 2028 6c69 6e65 2032 3229 0a2a (line 22).*\n+0001c870: 2064 6566 203c 323e 3a20 2020 2020 2020 def <2>: \n 0001c880: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001c890: 2020 2020 4f62 6a65 6374 2074 7261 6e73 Object trans\n-0001c8a0: 666f 726d 732e 2020 2020 286c 696e 6520 forms. (line \n-0001c8b0: 3130 290a 2a20 6465 6620 3c33 3e3a 2020 10).* def <3>: \n-0001c8c0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001c8d0: 2020 2020 2020 2020 2020 2020 2050 6f69 Poi\n-0001c8e0: 6e74 2073 7765 6570 732e 2020 2020 2020 nt sweeps. \n-0001c8f0: 2020 2028 6c69 6e65 2031 3929 0a2a 2064 (line 19).* d\n-0001c900: 6566 203c 343e 3a20 2020 2020 2020 2020 ef <4>: \n+0001c890: 2020 2020 2020 2020 4f62 6a65 6374 2074 Object t\n+0001c8a0: 7261 6e73 666f 726d 732e 2020 2020 286c ransforms. (l\n+0001c8b0: 696e 6520 3130 290a 2a20 6465 6620 3c33 ine 10).* def <3\n+0001c8c0: 3e3a 2020 2020 2020 2020 2020 2020 2020 >: \n+0001c8d0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001c8e0: 2050 6f69 6e74 2073 7765 6570 732e 2020 Point sweeps. \n+0001c8f0: 2020 2020 2020 2028 6c69 6e65 2031 3929 (line 19)\n+0001c900: 0a2a 2064 6566 203c 343e 3a20 2020 2020 .* def <4>: \n 0001c910: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001c920: 2020 2020 2020 506f 6c79 6c69 6e65 2073 Polyline s\n-0001c930: 7765 6570 732e 2020 2020 2020 286c 696e weeps. (lin\n-0001c940: 6520 2039 290a 2a20 6465 6620 3c35 3e3a e 9).* def <5>:\n-0001c950: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001c960: 2020 2020 2020 2020 2020 2020 2020 204e N\n-0001c970: 6573 7465 6420 7377 6565 7073 2e20 2020 ested sweeps. \n-0001c980: 2020 2020 2028 6c69 6e65 2020 3829 0a2a (line 8).*\n-0001c990: 2064 6566 203c 363e 3a20 2020 2020 2020 def <6>: \n+0001c920: 2020 2020 2020 2020 2020 506f 6c79 6c69 Polyli\n+0001c930: 6e65 2073 7765 6570 732e 2020 2020 2020 ne sweeps. \n+0001c940: 286c 696e 6520 2039 290a 2a20 6465 6620 (line 9).* def \n+0001c950: 3c35 3e3a 2020 2020 2020 2020 2020 2020 <5>: \n+0001c960: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001c970: 2020 204e 6573 7465 6420 7377 6565 7073 Nested sweeps\n+0001c980: 2e20 2020 2020 2020 2028 6c69 6e65 2020 . (line \n+0001c990: 3829 0a2a 2064 6566 203c 363e 3a20 2020 8).* def <6>: \n 0001c9a0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001c9b0: 2020 2020 2020 2020 506f 6c79 6c69 6e65 Polyline\n-0001c9c0: 2073 7765 6570 7320 7769 7468 2063 6c6f sweeps with clo\n-0001c9d0: 7375 7265 2e0a 2020 2020 2020 2020 2020 sure.. \n+0001c9b0: 2020 2020 2020 2020 2020 2020 506f 6c79 Poly\n+0001c9c0: 6c69 6e65 2073 7765 6570 7320 7769 7468 line sweeps with\n+0001c9d0: 2063 6c6f 7375 7265 2e0a 2020 2020 2020 closure.. \n 0001c9e0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001c9f0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001ca00: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001ca10: 2020 2020 2028 6c69 6e65 2031 3029 0a2a (line 10).*\n-0001ca20: 2064 6566 203c 373e 3a20 2020 2020 2020 def <7>: \n+0001ca10: 2020 2020 2020 2020 2028 6c69 6e65 2031 (line 1\n+0001ca20: 3029 0a2a 2064 6566 203c 373e 3a20 2020 0).* def <7>: \n 0001ca30: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001ca40: 2020 2020 2020 2020 4166 6669 6e65 2061 Affine a\n-0001ca50: 7269 7468 6d65 7469 632e 2020 2020 286c rithmetic. (l\n-0001ca60: 696e 6520 3131 290a 2a20 6465 6620 3c38 ine 11).* def <8\n-0001ca70: 3e3a 2020 2020 2020 2020 2020 2020 2020 >: \n+0001ca40: 2020 2020 2020 2020 2020 2020 4166 6669 Affi\n+0001ca50: 6e65 2061 7269 7468 6d65 7469 632e 2020 ne arithmetic. \n+0001ca60: 2020 286c 696e 6520 3131 290a 2a20 6465 (line 11).* de\n+0001ca70: 6620 3c38 3e3a 2020 2020 2020 2020 2020 f <8>: \n 0001ca80: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001ca90: 204f 7665 7276 6965 772e 2020 2020 2020 Overview. \n-0001caa0: 2020 2020 2020 2028 6c69 6e65 2031 3229 (line 12)\n-0001cab0: 0a2a 2064 6566 203c 393e 3a20 2020 2020 .* def <9>: \n+0001ca90: 2020 2020 204f 7665 7276 6965 772e 2020 Overview. \n+0001caa0: 2020 2020 2020 2020 2020 2028 6c69 6e65 (line\n+0001cab0: 2031 3229 0a2a 2064 6566 203c 393e 3a20 12).* def <9>: \n 0001cac0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001cad0: 2020 2020 2020 2020 2020 4f76 6572 7669 Overvi\n-0001cae0: 6577 2e20 2020 2020 2020 2020 2020 2020 ew. \n-0001caf0: 286c 696e 6520 3337 290a 2a20 6465 6620 (line 37).* def \n-0001cb00: 3c31 303e 3a20 2020 2020 2020 2020 2020 <10>: \n+0001cad0: 2020 2020 2020 2020 2020 2020 2020 4f76 Ov\n+0001cae0: 6572 7669 6577 2e20 2020 2020 2020 2020 erview. \n+0001caf0: 2020 2020 286c 696e 6520 3337 290a 2a20 (line 37).* \n+0001cb00: 6465 6620 3c31 303e 3a20 2020 2020 2020 def <10>: \n 0001cb10: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001cb20: 2020 204f 7665 7276 6965 772e 2020 2020 Overview. \n-0001cb30: 2020 2020 2020 2020 2028 6c69 6e65 2036 (line 6\n-0001cb40: 3929 0a2a 2064 6566 203c 3131 3e3a 2020 9).* def <11>: \n-0001cb50: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001cb60: 2020 2020 2020 2020 2020 2020 4120 7465 A te\n-0001cb70: 6368 6e69 6361 6c20 6472 6177 696e 672e chnical drawing.\n-0001cb80: 2020 286c 696e 6520 3134 290a 2a20 6465 (line 14).* de\n-0001cb90: 6620 3c31 323e 3a20 2020 2020 2020 2020 f <12>: \n+0001cb20: 2020 2020 2020 204f 7665 7276 6965 772e Overview.\n+0001cb30: 2020 2020 2020 2020 2020 2020 2028 6c69 (li\n+0001cb40: 6e65 2036 3929 0a2a 2064 6566 203c 3131 ne 69).* def <11\n+0001cb50: 3e3a 2020 2020 2020 2020 2020 2020 2020 >: \n+0001cb60: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001cb70: 4120 7465 6368 6e69 6361 6c20 6472 6177 A technical draw\n+0001cb80: 696e 672e 2020 286c 696e 6520 3134 290a ing. (line 14).\n+0001cb90: 2a20 6465 6620 3c31 323e 3a20 2020 2020 * def <12>: \n 0001cba0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001cbb0: 2020 2020 2041 2074 6563 686e 6963 616c A technical\n-0001cbc0: 2064 7261 7769 6e67 2e20 2028 6c69 6e65 drawing. (line\n-0001cbd0: 2032 3629 0a2a 2064 6566 203c 3133 3e3a 26).* def <13>:\n-0001cbe0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001cbf0: 2020 2020 2020 2020 2020 2020 2020 4120 A \n-0001cc00: 7465 6368 6e69 6361 6c20 6472 6177 696e technical drawin\n-0001cc10: 672e 2020 286c 696e 6520 3431 290a 2a20 g. (line 41).* \n-0001cc20: 6465 6620 3c31 343e 3a20 2020 2020 2020 def <14>: \n+0001cbb0: 2020 2020 2020 2020 2041 2074 6563 686e A techn\n+0001cbc0: 6963 616c 2064 7261 7769 6e67 2e20 2028 ical drawing. (\n+0001cbd0: 6c69 6e65 2032 3629 0a2a 2064 6566 203c line 26).* def <\n+0001cbe0: 3133 3e3a 2020 2020 2020 2020 2020 2020 13>: \n+0001cbf0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001cc00: 2020 4120 7465 6368 6e69 6361 6c20 6472 A technical dr\n+0001cc10: 6177 696e 672e 2020 286c 696e 6520 3431 awing. (line 41\n+0001cc20: 290a 2a20 6465 6620 3c31 343e 3a20 2020 ).* def <14>: \n 0001cc30: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001cc40: 2020 2020 2020 2041 2074 6563 686e 6963 A technic\n-0001cc50: 616c 2064 7261 7769 6e67 2e20 2028 6c69 al drawing. (li\n-0001cc60: 6e65 2036 3829 0a2a 2064 6566 203c 3135 ne 68).* def <15\n-0001cc70: 3e3a 2020 2020 2020 2020 2020 2020 2020 >: \n+0001cc40: 2020 2020 2020 2020 2020 2041 2074 6563 A tec\n+0001cc50: 686e 6963 616c 2064 7261 7769 6e67 2e20 hnical drawing. \n+0001cc60: 2028 6c69 6e65 2036 3829 0a2a 2064 6566 (line 68).* def\n+0001cc70: 203c 3135 3e3a 2020 2020 2020 2020 2020 <15>: \n 0001cc80: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001cc90: 4120 7465 6368 6e69 6361 6c20 6472 6177 A technical draw\n-0001cca0: 696e 672e 2020 286c 696e 6520 3935 290a ing. (line 95).\n-0001ccb0: 2a20 6465 6620 3c31 363e 3a20 2020 2020 * def <16>: \n+0001cc90: 2020 2020 4120 7465 6368 6e69 6361 6c20 A technical \n+0001cca0: 6472 6177 696e 672e 2020 286c 696e 6520 drawing. (line \n+0001ccb0: 3935 290a 2a20 6465 6620 3c31 363e 3a20 95).* def <16>: \n 0001ccc0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001ccd0: 2020 2020 2020 2020 2041 2068 6965 7261 A hiera\n-0001cce0: 7263 6869 6361 6c20 6d6f 6465 6c2e 2028 rchical model. (\n-0001ccf0: 6c69 6e65 2031 3029 0a2a 2064 6566 203c line 10).* def <\n-0001cd00: 3137 3e3a 2020 2020 2020 2020 2020 2020 17>: \n+0001ccd0: 2020 2020 2020 2020 2020 2020 2041 2068 A h\n+0001cce0: 6965 7261 7263 6869 6361 6c20 6d6f 6465 ierarchical mode\n+0001ccf0: 6c2e 2028 6c69 6e65 2031 3029 0a2a 2064 l. (line 10).* d\n+0001cd00: 6566 203c 3137 3e3a 2020 2020 2020 2020 ef <17>: \n 0001cd10: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001cd20: 2020 4120 6869 6572 6172 6368 6963 616c A hierarchical\n-0001cd30: 206d 6f64 656c 2e20 286c 696e 6520 3233 model. (line 23\n-0001cd40: 290a 2a20 6465 6620 3c31 383e 3a20 2020 ).* def <18>: \n-0001cd50: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001cd60: 2020 2020 2020 2020 2020 2041 2068 6965 A hie\n-0001cd70: 7261 7263 6869 6361 6c20 6d6f 6465 6c2e rarchical model.\n-0001cd80: 2028 6c69 6e65 2033 3229 0a2a 2064 6f74 (line 32).* dot\n-0001cd90: 733a 2020 2020 2020 2020 2020 2020 2020 s: \n+0001cd20: 2020 2020 2020 4120 6869 6572 6172 6368 A hierarch\n+0001cd30: 6963 616c 206d 6f64 656c 2e20 286c 696e ical model. (lin\n+0001cd40: 6520 3233 290a 2a20 6465 6620 3c31 383e e 23).* def <18>\n+0001cd50: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+0001cd60: 2020 2020 2020 2020 2020 2020 2020 2041 A\n+0001cd70: 2068 6965 7261 7263 6869 6361 6c20 6d6f hierarchical mo\n+0001cd80: 6465 6c2e 2028 6c69 6e65 2033 3229 0a2a del. (line 32).*\n+0001cd90: 2064 6f74 733a 2020 2020 2020 2020 2020 dots: \n 0001cda0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001cdb0: 2020 2020 446f 7473 2e20 2020 2020 2020 Dots. \n-0001cdc0: 2020 2020 2020 2020 2020 286c 696e 6520 (line \n-0001cdd0: 2036 290a 2a20 6669 6c6c 206f 7061 6369 6).* fill opaci\n-0001cde0: 7479 3a20 2020 2020 2020 2020 2020 2020 ty: \n-0001cdf0: 2020 2020 2020 2020 2020 2020 2054 7261 Tra\n-0001ce00: 6e73 7061 7265 6e63 792e 2020 2020 2020 nsparency. \n-0001ce10: 2020 2028 6c69 6e65 2031 3029 0a2a 2066 (line 10).* f\n-0001ce20: 696c 6c20 7374 796c 653a 2020 2020 2020 ill style: \n+0001cdb0: 2020 2020 2020 2020 446f 7473 2e20 2020 Dots. \n+0001cdc0: 2020 2020 2020 2020 2020 2020 2020 286c (l\n+0001cdd0: 696e 6520 2036 290a 2a20 6669 6c6c 206f ine 6).* fill o\n+0001cde0: 7061 6369 7479 3a20 2020 2020 2020 2020 pacity: \n+0001cdf0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001ce00: 2054 7261 6e73 7061 7265 6e63 792e 2020 Transparency. \n+0001ce10: 2020 2020 2020 2028 6c69 6e65 2031 3029 (line 10)\n+0001ce20: 0a2a 2066 696c 6c20 7374 796c 653a 2020 .* fill style: \n 0001ce30: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001ce40: 2020 2020 2020 5469 6b5a 2f50 4746 2075 TikZ/PGF u\n-0001ce50: 7365 722d 6465 6669 6e65 6420 7374 796c ser-defined styl\n-0001ce60: 6573 2e0a 2020 2020 2020 2020 2020 2020 es.. \n+0001ce40: 2020 2020 2020 2020 2020 5469 6b5a 2f50 TikZ/P\n+0001ce50: 4746 2075 7365 722d 6465 6669 6e65 6420 GF user-defined \n+0001ce60: 7374 796c 6573 2e0a 2020 2020 2020 2020 styles.. \n 0001ce70: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001ce80: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001ce90: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001cea0: 2020 2028 6c69 6e65 2032 3529 0a2a 2066 (line 25).* f\n-0001ceb0: 696c 6c63 6f6c 6f72 3a20 2020 2020 2020 illcolor: \n+0001cea0: 2020 2020 2020 2028 6c69 6e65 2032 3529 (line 25)\n+0001ceb0: 0a2a 2066 696c 6c63 6f6c 6f72 3a20 2020 .* fillcolor: \n 0001cec0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001ced0: 2020 2020 2020 4f62 6a65 6374 2074 7261 Object tra\n-0001cee0: 6e73 666f 726d 732e 2020 2020 286c 696e nsforms. (lin\n-0001cef0: 6520 3130 290a 2a20 6669 6c6c 636f 6c6f e 10).* fillcolo\n-0001cf00: 7220 3c31 3e3a 2020 2020 2020 2020 2020 r <1>: \n-0001cf10: 2020 2020 2020 2020 2020 2020 2020 2050 P\n-0001cf20: 6f6c 796c 696e 6520 7377 6565 7073 2077 olyline sweeps w\n-0001cf30: 6974 6820 636c 6f73 7572 652e 0a20 2020 ith closure.. \n-0001cf40: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001ced0: 2020 2020 2020 2020 2020 4f62 6a65 6374 Object\n+0001cee0: 2074 7261 6e73 666f 726d 732e 2020 2020 transforms. \n+0001cef0: 286c 696e 6520 3130 290a 2a20 6669 6c6c (line 10).* fill\n+0001cf00: 636f 6c6f 7220 3c31 3e3a 2020 2020 2020 color <1>: \n+0001cf10: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001cf20: 2020 2050 6f6c 796c 696e 6520 7377 6565 Polyline swee\n+0001cf30: 7073 2077 6974 6820 636c 6f73 7572 652e ps with closure.\n+0001cf40: 0a20 2020 2020 2020 2020 2020 2020 2020 . \n 0001cf50: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001cf60: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001cf70: 2020 2020 2020 2020 2020 2020 286c 696e (lin\n-0001cf80: 6520 3130 290a 2a20 6669 6c6c 636f 6c6f e 10).* fillcolo\n-0001cf90: 7220 3c32 3e3a 2020 2020 2020 2020 2020 r <2>: \n-0001cfa0: 2020 2020 2020 2020 2020 2020 2020 204f O\n-0001cfb0: 7665 7276 6965 772e 2020 2020 2020 2020 verview. \n-0001cfc0: 2020 2020 2028 6c69 6e65 2033 3729 0a2a (line 37).*\n-0001cfd0: 2066 7261 6d65 3a20 2020 2020 2020 2020 frame: \n+0001cf70: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001cf80: 286c 696e 6520 3130 290a 2a20 6669 6c6c (line 10).* fill\n+0001cf90: 636f 6c6f 7220 3c32 3e3a 2020 2020 2020 color <2>: \n+0001cfa0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001cfb0: 2020 204f 7665 7276 6965 772e 2020 2020 Overview. \n+0001cfc0: 2020 2020 2020 2020 2028 6c69 6e65 2033 (line 3\n+0001cfd0: 3729 0a2a 2066 7261 6d65 3a20 2020 2020 7).* frame: \n 0001cfe0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001cff0: 2020 2020 2020 2020 4672 616d 652e 2020 Frame. \n-0001d000: 2020 2020 2020 2020 2020 2020 2020 286c (l\n-0001d010: 696e 6520 2036 290a 2a20 6672 616d 6573 ine 6).* frames\n-0001d020: 6570 3a20 2020 2020 2020 2020 2020 2020 ep: \n+0001cff0: 2020 2020 2020 2020 2020 2020 4672 616d Fram\n+0001d000: 652e 2020 2020 2020 2020 2020 2020 2020 e. \n+0001d010: 2020 286c 696e 6520 2036 290a 2a20 6672 (line 6).* fr\n+0001d020: 616d 6573 6570 3a20 2020 2020 2020 2020 amesep: \n 0001d030: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001d040: 2046 7261 6d65 2e20 2020 2020 2020 2020 Frame. \n-0001d050: 2020 2020 2020 2028 6c69 6e65 2031 3129 (line 11)\n-0001d060: 0a2a 2067 6c6f 6261 6c3a 2020 2020 2020 .* global: \n+0001d040: 2020 2020 2046 7261 6d65 2e20 2020 2020 Frame. \n+0001d050: 2020 2020 2020 2020 2020 2028 6c69 6e65 (line\n+0001d060: 2031 3129 0a2a 2067 6c6f 6261 6c3a 2020 11).* global: \n 0001d070: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001d080: 2020 2020 2020 2020 2020 476c 6f62 616c Global\n-0001d090: 2065 6e76 6972 6f6e 6d65 6e74 2e20 2020 environment. \n-0001d0a0: 286c 696e 6520 3132 290a 2a20 696e 7075 (line 12).* inpu\n-0001d0b0: 743a 2020 2020 2020 2020 2020 2020 2020 t: \n+0001d080: 2020 2020 2020 2020 2020 2020 2020 476c Gl\n+0001d090: 6f62 616c 2065 6e76 6972 6f6e 6d65 6e74 obal environment\n+0001d0a0: 2e20 2020 286c 696e 6520 3132 290a 2a20 . (line 12).* \n+0001d0b0: 696e 7075 743a 2020 2020 2020 2020 2020 input: \n 0001d0c0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001d0d0: 2020 204c 616e 6775 6167 6520 6261 7369 Language basi\n-0001d0e0: 6373 2e20 2020 2020 2028 6c69 6e65 2031 cs. (line 1\n-0001d0f0: 3929 0a2a 2069 6e76 6572 7365 3a20 2020 9).* inverse: \n-0001d100: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001d110: 2020 2020 2020 2020 2020 2020 556e 6172 Unar\n-0001d120: 7920 666f 726d 732e 2020 2020 2020 2020 y forms. \n-0001d130: 2020 286c 696e 6520 3231 290a 2a20 6c61 (line 21).* la\n-0001d140: 6e67 7561 6765 3a20 2020 2020 2020 2020 nguage: \n+0001d0d0: 2020 2020 2020 204c 616e 6775 6167 6520 Language \n+0001d0e0: 6261 7369 6373 2e20 2020 2020 2028 6c69 basics. (li\n+0001d0f0: 6e65 2031 3929 0a2a 2069 6e76 6572 7365 ne 19).* inverse\n+0001d100: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+0001d110: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001d120: 556e 6172 7920 666f 726d 732e 2020 2020 Unary forms. \n+0001d130: 2020 2020 2020 286c 696e 6520 3231 290a (line 21).\n+0001d140: 2a20 6c61 6e67 7561 6765 3a20 2020 2020 * language: \n 0001d150: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001d160: 2020 2020 204c 616e 6775 6167 652e 2020 Language. \n-0001d170: 2020 2020 2020 2020 2020 2028 6c69 6e65 (line\n-0001d180: 2020 3629 0a2a 206c 6174 6578 3a20 2020 6).* latex: \n-0001d190: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001d1a0: 2020 2020 2020 2020 2020 2020 2020 4c61 La\n-0001d1b0: 6e67 7561 6765 2e20 2020 2020 2020 2020 nguage. \n-0001d1c0: 2020 2020 286c 696e 6520 3137 290a 2a20 (line 17).* \n-0001d1d0: 6c61 793a 2020 2020 2020 2020 2020 2020 lay: \n+0001d160: 2020 2020 2020 2020 204c 616e 6775 6167 Languag\n+0001d170: 652e 2020 2020 2020 2020 2020 2020 2028 e. (\n+0001d180: 6c69 6e65 2020 3629 0a2a 206c 6174 6578 line 6).* latex\n+0001d190: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+0001d1a0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001d1b0: 2020 4c61 6e67 7561 6765 2e20 2020 2020 Language. \n+0001d1c0: 2020 2020 2020 2020 286c 696e 6520 3137 (line 17\n+0001d1d0: 290a 2a20 6c61 793a 2020 2020 2020 2020 ).* lay: \n 0001d1e0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001d1f0: 2020 2020 2020 2053 7065 6369 616c 206f Special o\n-0001d200: 626a 6563 7473 2e20 2020 2020 2028 6c69 bjects. 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(line 26).\n-0001d2f0: 2a20 6c61 7920 3c33 3e3a 2020 2020 2020 * lay <3>: \n+0001d2d0: 2020 2020 4120 7465 6368 6e69 6361 6c20 A technical \n+0001d2e0: 6472 6177 696e 672e 2020 286c 696e 6520 drawing. (line \n+0001d2f0: 3236 290a 2a20 6c61 7920 3c33 3e3a 2020 26).* lay <3>: \n 0001d300: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001d310: 2020 2020 2020 2020 2041 2074 6563 686e A techn\n-0001d320: 6963 616c 2064 7261 7769 6e67 2e20 2028 ical drawing. (\n-0001d330: 6c69 6e65 2032 3629 0a2a 206c 6179 203c line 26).* lay <\n-0001d340: 343e 3a20 2020 2020 2020 2020 2020 2020 4>: \n+0001d310: 2020 2020 2020 2020 2020 2020 2041 2074 A t\n+0001d320: 6563 686e 6963 616c 2064 7261 7769 6e67 echnical drawing\n+0001d330: 2e20 2028 6c69 6e65 2032 3629 0a2a 206c . (line 26).* l\n+0001d340: 6179 203c 343e 3a20 2020 2020 2020 2020 ay <4>: \n 0001d350: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001d360: 2020 4120 7465 6368 6e69 6361 6c20 6472 A technical dr\n-0001d370: 6177 696e 672e 2020 286c 696e 6520 3935 awing. (line 95\n-0001d380: 290a 2a20 6c61 7920 3c35 3e3a 2020 2020 ).* lay <5>: \n+0001d360: 2020 2020 2020 4120 7465 6368 6e69 6361 A technica\n+0001d370: 6c20 6472 6177 696e 672e 2020 286c 696e l drawing. (lin\n+0001d380: 6520 3935 290a 2a20 6c61 7920 3c35 3e3a e 95).* lay <5>:\n 0001d390: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001d3a0: 2020 2020 2020 2020 2020 2049 6e74 6572 Inter\n-0001d3b0: 6e61 6c20 6f70 7469 6f6e 732e 2020 2020 nal options. \n-0001d3c0: 2028 6c69 6e65 2020 3729 0a2a 206c 6179 (line 7).* lay\n-0001d3d0: 203c 363e 3a20 2020 2020 2020 2020 2020 <6>: \n+0001d3a0: 2020 2020 2020 2020 2020 2020 2020 2049 I\n+0001d3b0: 6e74 6572 6e61 6c20 6f70 7469 6f6e 732e nternal options.\n+0001d3c0: 2020 2020 2028 6c69 6e65 2020 3729 0a2a (line 7).*\n+0001d3d0: 206c 6179 203c 363e 3a20 2020 2020 2020 lay <6>: \n 0001d3e0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001d3f0: 2020 2020 5370 6563 6961 6c73 2e20 2020 Specials. \n-0001d400: 2020 2020 2020 2020 2020 286c 696e 6520 (line \n-0001d410: 3432 290a 2a20 6c61 7920 3c37 3e3a 2020 42).* lay <7>: \n-0001d420: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001d430: 2020 2020 2020 2020 2020 2020 2047 6c6f Glo\n-0001d440: 6261 6c20 6f70 7469 6f6e 732e 2020 2020 bal options. \n-0001d450: 2020 2028 6c69 6e65 2031 3329 0a2a 206c (line 13).* l\n-0001d460: 696e 653a 2020 2020 2020 2020 2020 2020 ine: \n+0001d3f0: 2020 2020 2020 2020 5370 6563 6961 6c73 Specials\n+0001d400: 2e20 2020 2020 2020 2020 2020 2020 286c . (l\n+0001d410: 696e 6520 3432 290a 2a20 6c61 7920 3c37 ine 42).* lay <7\n+0001d420: 3e3a 2020 2020 2020 2020 2020 2020 2020 >: \n+0001d430: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001d440: 2047 6c6f 6261 6c20 6f70 7469 6f6e 732e Global options.\n+0001d450: 2020 2020 2020 2028 6c69 6e65 2031 3329 (line 13)\n+0001d460: 0a2a 206c 696e 653a 2020 2020 2020 2020 .* line: \n 0001d470: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001d480: 2020 2020 2020 4472 6177 696e 6720 6f70 Drawing op\n-0001d490: 7469 6f6e 732e 2020 2020 2020 286c 696e tions. (lin\n-0001d4a0: 6520 2037 290a 2a20 6c69 6e65 203c 313e e 7).* line <1>\n-0001d4b0: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n-0001d4c0: 2020 2020 2020 2020 2020 2020 2020 204f O\n-0001d4d0: 626a 6563 7420 7472 616e 7366 6f72 6d73 bject transforms\n-0001d4e0: 2e20 2020 2028 6c69 6e65 2031 3029 0a2a . (line 10).*\n-0001d4f0: 206c 696e 6520 3c32 3e3a 2020 2020 2020 line <2>: \n+0001d480: 2020 2020 2020 2020 2020 4472 6177 696e Drawin\n+0001d490: 6720 6f70 7469 6f6e 732e 2020 2020 2020 g options. \n+0001d4a0: 286c 696e 6520 2037 290a 2a20 6c69 6e65 (line 7).* line\n+0001d4b0: 203c 313e 3a20 2020 2020 2020 2020 2020 <1>: \n+0001d4c0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001d4d0: 2020 204f 626a 6563 7420 7472 616e 7366 Object transf\n+0001d4e0: 6f72 6d73 2e20 2020 2028 6c69 6e65 2031 orms. 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(l\n-0001d5c0: 696e 6520 3131 290a 2a20 6c69 6e65 203c ine 11).* line <\n-0001d5d0: 343e 3a20 2020 2020 2020 2020 2020 2020 4>: \n+0001d5a0: 2020 2020 2020 2020 2020 2020 4166 6669 Affi\n+0001d5b0: 6e65 2061 7269 7468 6d65 7469 632e 2020 ne arithmetic. \n+0001d5c0: 2020 286c 696e 6520 3131 290a 2a20 6c69 (line 11).* li\n+0001d5d0: 6e65 203c 343e 3a20 2020 2020 2020 2020 ne <4>: \n 0001d5e0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001d5f0: 204c 696e 6573 2e20 2020 2020 2020 2020 Lines. \n-0001d600: 2020 2020 2020 2028 6c69 6e65 2020 3629 (line 6)\n-0001d610: 0a2a 206c 696e 6520 3c35 3e3a 2020 2020 .* line <5>: \n+0001d5f0: 2020 2020 204c 696e 6573 2e20 2020 2020 Lines. \n+0001d600: 2020 2020 2020 2020 2020 2028 6c69 6e65 (line\n+0001d610: 2020 3629 0a2a 206c 696e 6520 3c35 3e3a 6).* line <5>:\n 0001d620: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001d630: 2020 2020 2020 2020 2020 5377 6570 7420 Swept \n-0001d640: 626c 6f63 6b73 2e20 2020 2020 2020 2020 blocks. \n-0001d650: 286c 696e 6520 2037 290a 2a20 6c69 6e65 (line 7).* line\n-0001d660: 203c 363e 3a20 2020 2020 2020 2020 2020 <6>: \n+0001d630: 2020 2020 2020 2020 2020 2020 2020 5377 Sw\n+0001d640: 6570 7420 626c 6f63 6b73 2e20 2020 2020 ept blocks. \n+0001d650: 2020 2020 286c 696e 6520 2037 290a 2a20 (line 7).* \n+0001d660: 6c69 6e65 203c 363e 3a20 2020 2020 2020 line <6>: \n 0001d670: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001d680: 2020 204f 7665 7276 6965 772e 2020 2020 Overview. \n-0001d690: 2020 2020 2020 2020 2028 6c69 6e65 2036 (line 6\n-0001d6a0: 3929 0a2a 206c 696e 6520 3c37 3e3a 2020 9).* line <7>: \n-0001d6b0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001d6c0: 2020 2020 2020 2020 2020 2020 4120 7465 A te\n-0001d6d0: 6368 6e69 6361 6c20 6472 6177 696e 672e chnical drawing.\n-0001d6e0: 2020 286c 696e 6520 3230 290a 2a20 6c69 (line 20).* li\n-0001d6f0: 6e65 203c 383e 3a20 2020 2020 2020 2020 ne <8>: \n+0001d680: 2020 2020 2020 204f 7665 7276 6965 772e Overview.\n+0001d690: 2020 2020 2020 2020 2020 2020 2028 6c69 (li\n+0001d6a0: 6e65 2036 3929 0a2a 206c 696e 6520 3c37 ne 69).* line <7\n+0001d6b0: 3e3a 2020 2020 2020 2020 2020 2020 2020 >: \n+0001d6c0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001d6d0: 4120 7465 6368 6e69 6361 6c20 6472 6177 A technical draw\n+0001d6e0: 696e 672e 2020 286c 696e 6520 3230 290a ing. 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(line \n+0001d810: 3130 290a 2a20 6c69 6e65 2073 7479 6c65 10).* line style\n+0001d820: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+0001d830: 2020 2020 2020 2020 2020 2020 2054 696b Tik\n+0001d840: 5a2f 5047 4620 7573 6572 2d64 6566 696e Z/PGF user-defin\n+0001d850: 6564 2073 7479 6c65 732e 0a20 2020 2020 ed styles.. \n 0001d860: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001d870: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001d880: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001d890: 2020 2020 2020 286c 696e 6520 3235 290a (line 25).\n-0001d8a0: 2a20 6c69 6e65 636f 6c6f 723a 2020 2020 * linecolor: \n+0001d890: 2020 2020 2020 2020 2020 286c 696e 6520 (line \n+0001d8a0: 3235 290a 2a20 6c69 6e65 636f 6c6f 723a 25).* linecolor:\n 0001d8b0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001d8c0: 2020 2020 2020 2020 204f 626a 6563 7420 Object \n-0001d8d0: 7472 616e 7366 6f72 6d73 2e20 2020 2028 transforms. (\n-0001d8e0: 6c69 6e65 2031 3029 0a2a 206c 696e 6563 line 10).* linec\n-0001d8f0: 6f6c 6f72 203c 313e 3a20 2020 2020 2020 olor <1>: \n+0001d8c0: 2020 2020 2020 2020 2020 2020 204f 626a Obj\n+0001d8d0: 6563 7420 7472 616e 7366 6f72 6d73 2e20 ect transforms. \n+0001d8e0: 2020 2028 6c69 6e65 2031 3029 0a2a 206c (line 10).* l\n+0001d8f0: 696e 6563 6f6c 6f72 203c 313e 3a20 2020 inecolor <1>: \n 0001d900: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001d910: 2020 4672 616d 652e 2020 2020 2020 2020 Frame. \n-0001d920: 2020 2020 2020 2020 286c 696e 6520 3130 (line 10\n-0001d930: 290a 2a20 6c69 6e65 636f 6c6f 7220 3c32 ).* linecolor <2\n-0001d940: 3e3a 2020 2020 2020 2020 2020 2020 2020 >: \n-0001d950: 2020 2020 2020 2020 2020 204f 7665 7276 Overv\n-0001d960: 6965 772e 2020 2020 2020 2020 2020 2020 iew. \n-0001d970: 2028 6c69 6e65 2036 3929 0a2a 206c 696e (line 69).* lin\n-0001d980: 6573 7479 6c65 3a20 2020 2020 2020 2020 estyle: \n+0001d910: 2020 2020 2020 4672 616d 652e 2020 2020 Frame. \n+0001d920: 2020 2020 2020 2020 2020 2020 286c 696e (lin\n+0001d930: 6520 3130 290a 2a20 6c69 6e65 636f 6c6f e 10).* linecolo\n+0001d940: 7220 3c32 3e3a 2020 2020 2020 2020 2020 r <2>: \n+0001d950: 2020 2020 2020 2020 2020 2020 2020 204f O\n+0001d960: 7665 7276 6965 772e 2020 2020 2020 2020 verview. \n+0001d970: 2020 2020 2028 6c69 6e65 2036 3929 0a2a (line 69).*\n+0001d980: 206c 696e 6573 7479 6c65 3a20 2020 2020 linestyle: \n 0001d990: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001d9a0: 2020 2020 4672 616d 652e 2020 2020 2020 Frame. \n-0001d9b0: 2020 2020 2020 2020 2020 286c 696e 6520 (line \n-0001d9c0: 3130 290a 2a20 6c69 6e65 7374 796c 6520 10).* linestyle \n-0001d9d0: 3c31 3e3a 2020 2020 2020 2020 2020 2020 <1>: \n-0001d9e0: 2020 2020 2020 2020 2020 2020 204f 7665 Ove\n-0001d9f0: 7276 6965 772e 2020 2020 2020 2020 2020 rview. \n-0001da00: 2020 2028 6c69 6e65 2036 3929 0a2a 206c (line 69).* l\n-0001da10: 696e 6573 7479 6c65 203c 323e 3a20 2020 inestyle <2>: \n-0001da20: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001da30: 2020 2020 2020 4120 7465 6368 6e69 6361 A technica\n-0001da40: 6c20 6472 6177 696e 672e 2020 286c 696e l drawing. (lin\n-0001da50: 6520 3236 290a 2a20 6c69 6e65 7374 796c e 26).* linestyl\n-0001da60: 6520 3c33 3e3a 2020 2020 2020 2020 2020 e <3>: \n-0001da70: 2020 2020 2020 2020 2020 2020 2020 2042 B\n-0001da80: 7567 7320 616e 6420 616e 6f6d 616c 6965 ugs and anomalie\n-0001da90: 732e 2020 2028 6c69 6e65 2020 3729 0a2a s. 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(line \n+0001daa0: 3729 0a2a 206c 696e 6577 6964 7468 3a20 7).* linewidth: \n 0001dab0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001dac0: 2020 2020 2020 2020 506f 6c79 6c69 6e65 Polyline\n-0001dad0: 2073 7765 6570 732e 2020 2020 2020 286c sweeps. (l\n-0001dae0: 696e 6520 2039 290a 2a20 6c69 6e65 7769 ine 9).* linewi\n-0001daf0: 6474 6820 3c31 3e3a 2020 2020 2020 2020 dth <1>: \n+0001dac0: 2020 2020 2020 2020 2020 2020 506f 6c79 Poly\n+0001dad0: 6c69 6e65 2073 7765 6570 732e 2020 2020 line sweeps. \n+0001dae0: 2020 286c 696e 6520 2039 290a 2a20 6c69 (line 9).* li\n+0001daf0: 6e65 7769 6474 6820 3c31 3e3a 2020 2020 newidth <1>: \n 0001db00: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001db10: 2046 7261 6d65 2e20 2020 2020 2020 2020 Frame. \n-0001db20: 2020 2020 2020 2028 6c69 6e65 2031 3029 (line 10)\n-0001db30: 0a2a 206c 696e 6577 6964 7468 203c 323e .* linewidth <2>\n-0001db40: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n-0001db50: 2020 2020 2020 2020 2020 4f76 6572 7669 Overvi\n-0001db60: 6577 2e20 2020 2020 2020 2020 2020 2020 ew. \n-0001db70: 286c 696e 6520 3639 290a 2a20 6c69 6e65 (line 69).* line\n-0001db80: 7769 6474 6820 3c33 3e3a 2020 2020 2020 width <3>: \n+0001db10: 2020 2020 2046 7261 6d65 2e20 2020 2020 Frame. \n+0001db20: 2020 2020 2020 2020 2020 2028 6c69 6e65 (line\n+0001db30: 2031 3029 0a2a 206c 696e 6577 6964 7468 10).* linewidth\n+0001db40: 203c 323e 3a20 2020 2020 2020 2020 2020 <2>: \n+0001db50: 2020 2020 2020 2020 2020 2020 2020 4f76 Ov\n+0001db60: 6572 7669 6577 2e20 2020 2020 2020 2020 erview. \n+0001db70: 2020 2020 286c 696e 6520 3639 290a 2a20 (line 69).* \n+0001db80: 6c69 6e65 7769 6474 6820 3c33 3e3a 2020 linewidth <3>: \n 0001db90: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001dba0: 2020 2041 2074 6563 686e 6963 616c 2064 A technical d\n-0001dbb0: 7261 7769 6e67 2e20 2028 6c69 6e65 2032 rawing. (line 2\n-0001dbc0: 3629 0a2a 206f 7061 6369 7479 3a20 2020 6).* opacity: \n-0001dbd0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001dbe0: 2020 2020 2020 2020 2020 2020 5472 616e Tran\n-0001dbf0: 7370 6172 656e 6379 2e20 2020 2020 2020 sparency. \n-0001dc00: 2020 286c 696e 6520 3130 290a 2a20 7065 (line 10).* pe\n-0001dc10: 7273 7065 6374 6976 653a 2020 2020 2020 rspective: \n+0001dba0: 2020 2020 2020 2041 2074 6563 686e 6963 A technic\n+0001dbb0: 616c 2064 7261 7769 6e67 2e20 2028 6c69 al drawing. (li\n+0001dbc0: 6e65 2032 3629 0a2a 206f 7061 6369 7479 ne 26).* opacity\n+0001dbd0: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+0001dbe0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001dbf0: 5472 616e 7370 6172 656e 6379 2e20 2020 Transparency. \n+0001dc00: 2020 2020 2020 286c 696e 6520 3130 290a (line 10).\n+0001dc10: 2a20 7065 7273 7065 6374 6976 653a 2020 * perspective: \n 0001dc20: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001dc30: 2020 2020 2054 7261 6e73 666f 726d 206c Transform l\n-0001dc40: 6974 6572 616c 732e 2020 2028 6c69 6e65 iterals. (line\n-0001dc50: 2032 3129 0a2a 2070 6963 7475 7265 626f 21).* picturebo\n-0001dc60: 783a 2020 2020 2020 2020 2020 2020 2020 x: \n-0001dc70: 2020 2020 2020 2020 2020 2020 2020 5069 Pi\n-0001dc80: 6374 7572 6520 626f 782e 2020 2020 2020 cture box. \n-0001dc90: 2020 2020 286c 696e 6520 2036 290a 2a20 (line 6).* \n-0001dca0: 706f 696e 7420 6c69 7374 3a20 2020 2020 point list: \n+0001dc30: 2020 2020 2020 2020 2054 7261 6e73 666f Transfo\n+0001dc40: 726d 206c 6974 6572 616c 732e 2020 2028 rm literals. (\n+0001dc50: 6c69 6e65 2032 3129 0a2a 2070 6963 7475 line 21).* pictu\n+0001dc60: 7265 626f 783a 2020 2020 2020 2020 2020 rebox: \n+0001dc70: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001dc80: 2020 5069 6374 7572 6520 626f 782e 2020 Picture box. \n+0001dc90: 2020 2020 2020 2020 286c 696e 6520 2036 (line 6\n+0001dca0: 290a 2a20 706f 696e 7420 6c69 7374 3a20 ).* point list: \n 0001dcb0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001dcc0: 2020 2020 2020 2049 6e74 6572 6e61 6c20 Internal \n-0001dcd0: 6f70 7469 6f6e 732e 2020 2020 2028 6c69 options. (li\n-0001dce0: 6e65 2032 3429 0a2a 2070 6f6c 7967 6f6e ne 24).* polygon\n-0001dcf0: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+0001dcc0: 2020 2020 2020 2020 2020 2049 6e74 6572 Inter\n+0001dcd0: 6e61 6c20 6f70 7469 6f6e 732e 2020 2020 nal options. \n+0001dce0: 2028 6c69 6e65 2032 3429 0a2a 2070 6f6c (line 24).* pol\n+0001dcf0: 7967 6f6e 3a20 2020 2020 2020 2020 2020 ygon: \n 0001dd00: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001dd10: 4865 6c6c 6f20 776f 726c 642e 2020 2020 Hello world. \n-0001dd20: 2020 2020 2020 286c 696e 6520 3338 290a (line 38).\n-0001dd30: 2a20 706f 6c79 676f 6e20 3c31 3e3a 2020 * polygon <1>: \n-0001dd40: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001dd50: 2020 2020 2020 2020 2044 7261 7769 6e67 Drawing\n-0001dd60: 2061 2073 6f6c 6964 2e20 2020 2020 2028 a solid. (\n-0001dd70: 6c69 6e65 2020 3929 0a2a 2070 6f6c 7967 line 9).* polyg\n-0001dd80: 6f6e 203c 323e 3a20 2020 2020 2020 2020 on <2>: \n+0001dd10: 2020 2020 4865 6c6c 6f20 776f 726c 642e Hello world.\n+0001dd20: 2020 2020 2020 2020 2020 286c 696e 6520 (line \n+0001dd30: 3338 290a 2a20 706f 6c79 676f 6e20 3c31 38).* polygon <1\n+0001dd40: 3e3a 2020 2020 2020 2020 2020 2020 2020 >: \n+0001dd50: 2020 2020 2020 2020 2020 2020 2044 7261 Dra\n+0001dd60: 7769 6e67 2061 2073 6f6c 6964 2e20 2020 wing a solid. \n+0001dd70: 2020 2028 6c69 6e65 2020 3929 0a2a 2070 (line 9).* p\n+0001dd80: 6f6c 7967 6f6e 203c 323e 3a20 2020 2020 olygon <2>: \n 0001dd90: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001dda0: 2020 4f62 6a65 6374 2074 7261 6e73 666f Object transfo\n-0001ddb0: 726d 732e 2020 2020 286c 696e 6520 3130 rms. (line 10\n-0001ddc0: 290a 2a20 706f 6c79 676f 6e20 3c33 3e3a ).* polygon <3>:\n-0001ddd0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001dde0: 2020 2020 2020 2020 2020 2041 6666 696e Affin\n-0001ddf0: 6520 6172 6974 686d 6574 6963 2e20 2020 e arithmetic. \n-0001de00: 2028 6c69 6e65 2031 3129 0a2a 2070 6f6c (line 11).* pol\n-0001de10: 7967 6f6e 203c 343e 3a20 2020 2020 2020 ygon <4>: \n+0001dda0: 2020 2020 2020 4f62 6a65 6374 2074 7261 Object tra\n+0001ddb0: 6e73 666f 726d 732e 2020 2020 286c 696e nsforms. (lin\n+0001ddc0: 6520 3130 290a 2a20 706f 6c79 676f 6e20 e 10).* polygon \n+0001ddd0: 3c33 3e3a 2020 2020 2020 2020 2020 2020 <3>: \n+0001dde0: 2020 2020 2020 2020 2020 2020 2020 2041 A\n+0001ddf0: 6666 696e 6520 6172 6974 686d 6574 6963 ffine arithmetic\n+0001de00: 2e20 2020 2028 6c69 6e65 2031 3129 0a2a . (line 11).*\n+0001de10: 2070 6f6c 7967 6f6e 203c 343e 3a20 2020 polygon <4>: \n 0001de20: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001de30: 2020 2020 506f 6c79 676f 6e73 2e20 2020 Polygons. \n-0001de40: 2020 2020 2020 2020 2020 286c 696e 6520 (line \n-0001de50: 2036 290a 2a20 706f 6c79 676f 6e20 3c35 6).* polygon <5\n-0001de60: 3e3a 2020 2020 2020 2020 2020 2020 2020 >: \n-0001de70: 2020 2020 2020 2020 2020 2020 2053 7765 Swe\n-0001de80: 7074 2062 6c6f 636b 732e 2020 2020 2020 pt blocks. \n-0001de90: 2020 2028 6c69 6e65 2020 3729 0a2a 2070 (line 7).* p\n-0001dea0: 6f6c 7967 6f6e 203c 363e 3a20 2020 2020 olygon <6>: \n+0001de30: 2020 2020 2020 2020 506f 6c79 676f 6e73 Polygons\n+0001de40: 2e20 2020 2020 2020 2020 2020 2020 286c . (l\n+0001de50: 696e 6520 2036 290a 2a20 706f 6c79 676f ine 6).* polygo\n+0001de60: 6e20 3c35 3e3a 2020 2020 2020 2020 2020 n <5>: \n+0001de70: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001de80: 2053 7765 7074 2062 6c6f 636b 732e 2020 Swept blocks. \n+0001de90: 2020 2020 2020 2028 6c69 6e65 2020 3729 (line 7)\n+0001dea0: 0a2a 2070 6f6c 7967 6f6e 203c 363e 3a20 .* polygon <6>: \n 0001deb0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001dec0: 2020 2020 2020 4f76 6572 7669 6577 2e20 Overview. \n-0001ded0: 2020 2020 2020 2020 2020 2020 286c 696e (lin\n-0001dee0: 6520 3337 290a 2a20 7072 6f6a 6563 743a e 37).* project:\n-0001def0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001df00: 2020 2020 2020 2020 2020 2020 2020 2054 T\n-0001df10: 7261 6e73 666f 726d 206c 6974 6572 616c ransform literal\n-0001df20: 732e 2020 2028 6c69 6e65 2031 3829 0a2a s. (line 18).*\n-0001df30: 2070 726f 6a65 6374 203c 313e 3a20 2020 project <1>: \n-0001df40: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001df50: 2020 2020 2020 2020 5472 616e 7366 6f72 Transfor\n-0001df60: 6d20 6c69 7465 7261 6c73 2e20 2020 286c m literals. (l\n-0001df70: 696e 6520 3139 290a 2a20 7072 6f6a 6563 ine 19).* projec\n-0001df80: 7420 3c32 3e3a 2020 2020 2020 2020 2020 t <2>: \n+0001dec0: 2020 2020 2020 2020 2020 4f76 6572 7669 Overvi\n+0001ded0: 6577 2e20 2020 2020 2020 2020 2020 2020 ew. \n+0001dee0: 286c 696e 6520 3337 290a 2a20 7072 6f6a (line 37).* proj\n+0001def0: 6563 743a 2020 2020 2020 2020 2020 2020 ect: \n+0001df00: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001df10: 2020 2054 7261 6e73 666f 726d 206c 6974 Transform lit\n+0001df20: 6572 616c 732e 2020 2028 6c69 6e65 2031 erals. (line 1\n+0001df30: 3829 0a2a 2070 726f 6a65 6374 203c 313e 8).* project <1>\n+0001df40: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+0001df50: 2020 2020 2020 2020 2020 2020 5472 616e Tran\n+0001df60: 7366 6f72 6d20 6c69 7465 7261 6c73 2e20 sform literals. \n+0001df70: 2020 286c 696e 6520 3139 290a 2a20 7072 (line 19).* pr\n+0001df80: 6f6a 6563 7420 3c32 3e3a 2020 2020 2020 oject <2>: \n 0001df90: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001dfa0: 2054 7261 6e73 666f 726d 206c 6974 6572 Transform liter\n-0001dfb0: 616c 732e 2020 2028 6c69 6e65 2034 3729 als. (line 47)\n-0001dfc0: 0a2a 2070 7374 7269 636b 733a 2020 2020 .* pstricks: \n+0001dfa0: 2020 2020 2054 7261 6e73 666f 726d 206c Transform l\n+0001dfb0: 6974 6572 616c 732e 2020 2028 6c69 6e65 iterals. (line\n+0001dfc0: 2034 3729 0a2a 2070 7374 7269 636b 733a 47).* pstricks:\n 0001dfd0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001dfe0: 2020 2020 2020 2020 2020 4c61 6e67 7561 Langua\n-0001dff0: 6765 2e20 2020 2020 2020 2020 2020 2020 ge. \n-0001e000: 286c 696e 6520 3130 290a 2a20 7075 743a (line 10).* put:\n-0001e010: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001dfe0: 2020 2020 2020 2020 2020 2020 2020 4c61 La\n+0001dff0: 6e67 7561 6765 2e20 2020 2020 2020 2020 nguage. \n+0001e000: 2020 2020 286c 696e 6520 3130 290a 2a20 (line 10).* \n+0001e010: 7075 743a 2020 2020 2020 2020 2020 2020 put: \n 0001e020: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001e030: 2020 204f 626a 6563 7420 7472 616e 7366 Object transf\n-0001e040: 6f72 6d73 2e20 2020 2028 6c69 6e65 2031 orms. (line 1\n-0001e050: 3029 0a2a 2070 7574 203c 313e 3a20 2020 0).* put <1>: \n-0001e060: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001e070: 2020 2020 2020 2020 2020 2020 4e65 7374 Nest\n-0001e080: 6564 2073 7765 6570 732e 2020 2020 2020 ed sweeps. \n-0001e090: 2020 286c 696e 6520 3235 290a 2a20 7075 (line 25).* pu\n-0001e0a0: 7420 3c32 3e3a 2020 2020 2020 2020 2020 t <2>: \n+0001e030: 2020 2020 2020 204f 626a 6563 7420 7472 Object tr\n+0001e040: 616e 7366 6f72 6d73 2e20 2020 2028 6c69 ansforms. (li\n+0001e050: 6e65 2031 3029 0a2a 2070 7574 203c 313e ne 10).* put <1>\n+0001e060: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+0001e070: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001e080: 4e65 7374 6564 2073 7765 6570 732e 2020 Nested sweeps. \n+0001e090: 2020 2020 2020 286c 696e 6520 3235 290a (line 25).\n+0001e0a0: 2a20 7075 7420 3c32 3e3a 2020 2020 2020 * put <2>: \n 0001e0b0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001e0c0: 2020 2020 2050 7574 732e 2020 2020 2020 Puts. \n-0001e0d0: 2020 2020 2020 2020 2020 2028 6c69 6e65 (line\n-0001e0e0: 2020 3629 0a2a 2070 7574 203c 333e 3a20 6).* put <3>: \n-0001e0f0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001e100: 2020 2020 2020 2020 2020 2020 2020 4120 A \n-0001e110: 6869 6572 6172 6368 6963 616c 206d 6f64 hierarchical mod\n-0001e120: 656c 2e20 286c 696e 6520 3332 290a 2a20 el. (line 32).* \n-0001e130: 7265 7065 6174 3a20 2020 2020 2020 2020 repeat: \n+0001e0c0: 2020 2020 2020 2020 2050 7574 732e 2020 Puts. \n+0001e0d0: 2020 2020 2020 2020 2020 2020 2020 2028 (\n+0001e0e0: 6c69 6e65 2020 3629 0a2a 2070 7574 203c line 6).* put <\n+0001e0f0: 333e 3a20 2020 2020 2020 2020 2020 2020 3>: \n+0001e100: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001e110: 2020 4120 6869 6572 6172 6368 6963 616c A hierarchical\n+0001e120: 206d 6f64 656c 2e20 286c 696e 6520 3332 model. (line 32\n+0001e130: 290a 2a20 7265 7065 6174 3a20 2020 2020 ).* repeat: \n 0001e140: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001e150: 2020 2020 2020 2052 6570 6561 7465 6420 Repeated \n-0001e160: 6f62 6a65 6374 732e 2020 2020 2028 6c69 objects. (li\n-0001e170: 6e65 2020 3829 0a2a 2072 6570 6561 7420 ne 8).* repeat \n-0001e180: 3c31 3e3a 2020 2020 2020 2020 2020 2020 <1>: \n+0001e150: 2020 2020 2020 2020 2020 2052 6570 6561 Repea\n+0001e160: 7465 6420 6f62 6a65 6374 732e 2020 2020 ted objects. \n+0001e170: 2028 6c69 6e65 2020 3829 0a2a 2072 6570 (line 8).* rep\n+0001e180: 6561 7420 3c31 3e3a 2020 2020 2020 2020 eat <1>: \n 0001e190: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001e1a0: 4e65 7374 6564 2073 7765 6570 732e 2020 Nested sweeps. \n-0001e1b0: 2020 2020 2020 286c 696e 6520 3235 290a (line 25).\n-0001e1c0: 2a20 7265 7065 6174 203c 323e 3a20 2020 * repeat <2>: \n-0001e1d0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001e1e0: 2020 2020 2020 2020 2050 6f6c 796c 696e Polylin\n-0001e1f0: 6520 7377 6565 7073 2077 6974 6820 636c e sweeps with cl\n-0001e200: 6f73 7572 652e 0a20 2020 2020 2020 2020 osure.. \n+0001e1a0: 2020 2020 4e65 7374 6564 2073 7765 6570 Nested sweep\n+0001e1b0: 732e 2020 2020 2020 2020 286c 696e 6520 s. (line \n+0001e1c0: 3235 290a 2a20 7265 7065 6174 203c 323e 25).* repeat <2>\n+0001e1d0: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+0001e1e0: 2020 2020 2020 2020 2020 2020 2050 6f6c Pol\n+0001e1f0: 796c 696e 6520 7377 6565 7073 2077 6974 yline sweeps wit\n+0001e200: 6820 636c 6f73 7572 652e 0a20 2020 2020 h closure.. \n 0001e210: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001e220: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001e230: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001e240: 2020 2020 2020 286c 696e 6520 3130 290a (line 10).\n-0001e250: 2a20 7265 7065 6174 203c 333e 3a20 2020 * repeat <3>: \n-0001e260: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001e270: 2020 2020 2020 2020 2052 6570 6561 7473 Repeats\n-0001e280: 2e20 2020 2020 2020 2020 2020 2020 2028 . (\n-0001e290: 6c69 6e65 2020 3629 0a2a 2072 6570 6561 line 6).* repea\n-0001e2a0: 7420 3c34 3e3a 2020 2020 2020 2020 2020 t <4>: \n+0001e240: 2020 2020 2020 2020 2020 286c 696e 6520 (line \n+0001e250: 3130 290a 2a20 7265 7065 6174 203c 333e 10).* repeat <3>\n+0001e260: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+0001e270: 2020 2020 2020 2020 2020 2020 2052 6570 Rep\n+0001e280: 6561 7473 2e20 2020 2020 2020 2020 2020 eats. \n+0001e290: 2020 2028 6c69 6e65 2020 3629 0a2a 2072 (line 6).* r\n+0001e2a0: 6570 6561 7420 3c34 3e3a 2020 2020 2020 epeat <4>: \n 0001e2b0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001e2c0: 2020 4f76 6572 7669 6577 2e20 2020 2020 Overview. \n-0001e2d0: 2020 2020 2020 2020 286c 696e 6520 3537 (line 57\n-0001e2e0: 290a 2a20 726f 7461 7465 3a20 2020 2020 ).* rotate: \n+0001e2c0: 2020 2020 2020 4f76 6572 7669 6577 2e20 Overview. \n+0001e2d0: 2020 2020 2020 2020 2020 2020 286c 696e (lin\n+0001e2e0: 6520 3537 290a 2a20 726f 7461 7465 3a20 e 57).* rotate: \n 0001e2f0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001e300: 2020 2020 2020 2020 2020 204f 626a 6563 Objec\n-0001e310: 7420 7472 616e 7366 6f72 6d73 2e20 2020 t transforms. \n-0001e320: 2028 6c69 6e65 2031 3029 0a2a 2072 6f74 (line 10).* rot\n-0001e330: 6174 6520 3c31 3e3a 2020 2020 2020 2020 ate <1>: \n+0001e300: 2020 2020 2020 2020 2020 2020 2020 204f O\n+0001e310: 626a 6563 7420 7472 616e 7366 6f72 6d73 bject transforms\n+0001e320: 2e20 2020 2028 6c69 6e65 2031 3029 0a2a . (line 10).*\n+0001e330: 2072 6f74 6174 6520 3c31 3e3a 2020 2020 rotate <1>: \n 0001e340: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001e350: 2020 2020 4f62 6a65 6374 2074 7261 6e73 Object trans\n-0001e360: 666f 726d 732e 2020 2020 286c 696e 6520 forms. (line \n-0001e370: 3339 290a 2a20 726f 7461 7465 203c 323e 39).* rotate <2>\n-0001e380: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n-0001e390: 2020 2020 2020 2020 2020 2020 2052 6570 Rep\n-0001e3a0: 6561 7465 6420 6f62 6a65 6374 732e 2020 eated objects. \n-0001e3b0: 2020 2028 6c69 6e65 2020 3829 0a2a 2072 (line 8).* r\n-0001e3c0: 6f74 6174 6520 3c33 3e3a 2020 2020 2020 otate <3>: \n+0001e350: 2020 2020 2020 2020 4f62 6a65 6374 2074 Object t\n+0001e360: 7261 6e73 666f 726d 732e 2020 2020 286c ransforms. (l\n+0001e370: 696e 6520 3339 290a 2a20 726f 7461 7465 ine 39).* rotate\n+0001e380: 203c 323e 3a20 2020 2020 2020 2020 2020 <2>: \n+0001e390: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001e3a0: 2052 6570 6561 7465 6420 6f62 6a65 6374 Repeated object\n+0001e3b0: 732e 2020 2020 2028 6c69 6e65 2020 3829 s. (line 8)\n+0001e3c0: 0a2a 2072 6f74 6174 6520 3c33 3e3a 2020 .* rotate <3>: \n 0001e3d0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001e3e0: 2020 2020 2020 5377 6570 7420 6f62 6a65 Swept obje\n-0001e3f0: 6374 732e 2020 2020 2020 2020 286c 696e cts. (lin\n-0001e400: 6520 2038 290a 2a20 726f 7461 7465 203c e 8).* rotate <\n-0001e410: 343e 3a20 2020 2020 2020 2020 2020 2020 4>: \n-0001e420: 2020 2020 2020 2020 2020 2020 2020 2050 P\n-0001e430: 6f69 6e74 2073 7765 6570 732e 2020 2020 oint sweeps. \n-0001e440: 2020 2020 2028 6c69 6e65 2031 3929 0a2a (line 19).*\n-0001e450: 2072 6f74 6174 6520 3c35 3e3a 2020 2020 rotate <5>: \n+0001e3e0: 2020 2020 2020 2020 2020 5377 6570 7420 Swept \n+0001e3f0: 6f62 6a65 6374 732e 2020 2020 2020 2020 objects. \n+0001e400: 286c 696e 6520 2038 290a 2a20 726f 7461 (line 8).* rota\n+0001e410: 7465 203c 343e 3a20 2020 2020 2020 2020 te <4>: \n+0001e420: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001e430: 2020 2050 6f69 6e74 2073 7765 6570 732e Point sweeps.\n+0001e440: 2020 2020 2020 2020 2028 6c69 6e65 2031 (line 1\n+0001e450: 3929 0a2a 2072 6f74 6174 6520 3c35 3e3a 9).* rotate <5>:\n 0001e460: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001e470: 2020 2020 2020 2020 506f 6c79 6c69 6e65 Polyline\n-0001e480: 2073 7765 6570 732e 2020 2020 2020 286c sweeps. (l\n-0001e490: 696e 6520 2039 290a 2a20 726f 7461 7465 ine 9).* rotate\n-0001e4a0: 203c 363e 3a20 2020 2020 2020 2020 2020 <6>: \n+0001e470: 2020 2020 2020 2020 2020 2020 506f 6c79 Poly\n+0001e480: 6c69 6e65 2073 7765 6570 732e 2020 2020 line sweeps. \n+0001e490: 2020 286c 696e 6520 2039 290a 2a20 726f (line 9).* ro\n+0001e4a0: 7461 7465 203c 363e 3a20 2020 2020 2020 tate <6>: \n 0001e4b0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001e4c0: 204e 6573 7465 6420 7377 6565 7073 2e20 Nested sweeps. \n-0001e4d0: 2020 2020 2020 2028 6c69 6e65 2020 3829 (line 8)\n-0001e4e0: 0a2a 2072 6f74 6174 6520 3c37 3e3a 2020 .* rotate <7>: \n-0001e4f0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001e500: 2020 2020 2020 2020 2020 506f 6c79 6c69 Polyli\n-0001e510: 6e65 2073 7765 6570 7320 7769 7468 2063 ne sweeps with c\n-0001e520: 6c6f 7375 7265 2e0a 2020 2020 2020 2020 losure.. \n+0001e4c0: 2020 2020 204e 6573 7465 6420 7377 6565 Nested swee\n+0001e4d0: 7073 2e20 2020 2020 2020 2028 6c69 6e65 ps. (line\n+0001e4e0: 2020 3829 0a2a 2072 6f74 6174 6520 3c37 8).* rotate <7\n+0001e4f0: 3e3a 2020 2020 2020 2020 2020 2020 2020 >: \n+0001e500: 2020 2020 2020 2020 2020 2020 2020 506f Po\n+0001e510: 6c79 6c69 6e65 2073 7765 6570 7320 7769 lyline sweeps wi\n+0001e520: 7468 2063 6c6f 7375 7265 2e0a 2020 2020 th closure.. \n 0001e530: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001e540: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001e550: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001e560: 2020 2020 2020 2028 6c69 6e65 2031 3029 (line 10)\n-0001e570: 0a2a 2072 6f74 6174 6520 3c38 3e3a 2020 .* rotate <8>: \n-0001e580: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001e590: 2020 2020 2020 2020 2020 5472 616e 7366 Transf\n-0001e5a0: 6f72 6d20 6c69 7465 7261 6c73 2e20 2020 orm literals. \n-0001e5b0: 286c 696e 6520 3130 290a 2a20 726f 7461 (line 10).* rota\n-0001e5c0: 7465 203c 393e 3a20 2020 2020 2020 2020 te <9>: \n+0001e560: 2020 2020 2020 2020 2020 2028 6c69 6e65 (line\n+0001e570: 2031 3029 0a2a 2072 6f74 6174 6520 3c38 10).* rotate <8\n+0001e580: 3e3a 2020 2020 2020 2020 2020 2020 2020 >: \n+0001e590: 2020 2020 2020 2020 2020 2020 2020 5472 Tr\n+0001e5a0: 616e 7366 6f72 6d20 6c69 7465 7261 6c73 ansform literals\n+0001e5b0: 2e20 2020 286c 696e 6520 3130 290a 2a20 . (line 10).* \n+0001e5c0: 726f 7461 7465 203c 393e 3a20 2020 2020 rotate <9>: \n 0001e5d0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001e5e0: 2020 204f 7665 7276 6965 772e 2020 2020 Overview. \n-0001e5f0: 2020 2020 2020 2020 2028 6c69 6e65 2035 (line 5\n-0001e600: 3729 0a2a 2072 6f74 6174 6520 3c31 303e 7).* rotate <10>\n-0001e610: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n-0001e620: 2020 2020 2020 2020 2020 2020 4f76 6572 Over\n-0001e630: 7669 6577 2e20 2020 2020 2020 2020 2020 view. \n-0001e640: 2020 286c 696e 6520 3639 290a 2a20 726f (line 69).* ro\n-0001e650: 7461 7465 203c 3131 3e3a 2020 2020 2020 tate <11>: \n+0001e5e0: 2020 2020 2020 204f 7665 7276 6965 772e Overview.\n+0001e5f0: 2020 2020 2020 2020 2020 2020 2028 6c69 (li\n+0001e600: 6e65 2035 3729 0a2a 2072 6f74 6174 6520 ne 57).* rotate \n+0001e610: 3c31 303e 3a20 2020 2020 2020 2020 2020 <10>: \n+0001e620: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001e630: 4f76 6572 7669 6577 2e20 2020 2020 2020 Overview. \n+0001e640: 2020 2020 2020 286c 696e 6520 3639 290a (line 69).\n+0001e650: 2a20 726f 7461 7465 203c 3131 3e3a 2020 * rotate <11>: \n 0001e660: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001e670: 2020 2020 2041 2074 6563 686e 6963 616c A technical\n-0001e680: 2064 7261 7769 6e67 2e20 2028 6c69 6e65 drawing. (line\n-0001e690: 2031 3429 0a2a 2072 6f74 6174 6520 3c31 14).* rotate <1\n-0001e6a0: 323e 3a20 2020 2020 2020 2020 2020 2020 2>: \n-0001e6b0: 2020 2020 2020 2020 2020 2020 2020 4120 A \n-0001e6c0: 6869 6572 6172 6368 6963 616c 206d 6f64 hierarchical mod\n-0001e6d0: 656c 2e20 286c 696e 6520 3130 290a 2a20 el. (line 10).* \n-0001e6e0: 726f 7461 7465 203c 3133 3e3a 2020 2020 rotate <13>: \n+0001e670: 2020 2020 2020 2020 2041 2074 6563 686e A techn\n+0001e680: 6963 616c 2064 7261 7769 6e67 2e20 2028 ical drawing. (\n+0001e690: 6c69 6e65 2031 3429 0a2a 2072 6f74 6174 line 14).* rotat\n+0001e6a0: 6520 3c31 323e 3a20 2020 2020 2020 2020 e <12>: \n+0001e6b0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001e6c0: 2020 4120 6869 6572 6172 6368 6963 616c A hierarchical\n+0001e6d0: 206d 6f64 656c 2e20 286c 696e 6520 3130 model. (line 10\n+0001e6e0: 290a 2a20 726f 7461 7465 203c 3133 3e3a ).* rotate <13>:\n 0001e6f0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001e700: 2020 2020 2020 2041 2068 6965 7261 7263 A hierarc\n-0001e710: 6869 6361 6c20 6d6f 6465 6c2e 2028 6c69 hical model. (li\n-0001e720: 6e65 2032 3329 0a2a 2072 6f74 6174 6520 ne 23).* rotate \n-0001e730: 3c31 343e 3a20 2020 2020 2020 2020 2020 <14>: \n+0001e700: 2020 2020 2020 2020 2020 2041 2068 6965 A hie\n+0001e710: 7261 7263 6869 6361 6c20 6d6f 6465 6c2e rarchical model.\n+0001e720: 2028 6c69 6e65 2032 3329 0a2a 2072 6f74 (line 23).* rot\n+0001e730: 6174 6520 3c31 343e 3a20 2020 2020 2020 ate <14>: \n 0001e740: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001e750: 4120 6869 6572 6172 6368 6963 616c 206d A hierarchical m\n-0001e760: 6f64 656c 2e20 286c 696e 6520 3332 290a odel. (line 32).\n-0001e770: 2a20 7363 616c 653a 2020 2020 2020 2020 * scale: \n+0001e750: 2020 2020 4120 6869 6572 6172 6368 6963 A hierarchic\n+0001e760: 616c 206d 6f64 656c 2e20 286c 696e 6520 al model. (line \n+0001e770: 3332 290a 2a20 7363 616c 653a 2020 2020 32).* scale: \n 0001e780: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001e790: 2020 2020 2020 2020 2054 7261 6e73 666f Transfo\n-0001e7a0: 726d 206c 6974 6572 616c 732e 2020 2028 rm literals. (\n-0001e7b0: 6c69 6e65 2031 3629 0a2a 2073 6361 6c65 line 16).* scale\n-0001e7c0: 203c 313e 3a20 2020 2020 2020 2020 2020 <1>: \n+0001e790: 2020 2020 2020 2020 2020 2020 2054 7261 Tra\n+0001e7a0: 6e73 666f 726d 206c 6974 6572 616c 732e nsform literals.\n+0001e7b0: 2020 2028 6c69 6e65 2031 3629 0a2a 2073 (line 16).* s\n+0001e7c0: 6361 6c65 203c 313e 3a20 2020 2020 2020 cale <1>: \n 0001e7d0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001e7e0: 2020 5472 616e 7366 6f72 6d20 6c69 7465 Transform lite\n-0001e7f0: 7261 6c73 2e20 2020 286c 696e 6520 3137 rals. (line 17\n-0001e800: 290a 2a20 7363 616c 6520 3c32 3e3a 2020 ).* scale <2>: \n-0001e810: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001e820: 2020 2020 2020 2020 2020 2041 2074 6563 A tec\n-0001e830: 686e 6963 616c 2064 7261 7769 6e67 2e20 hnical drawing. \n-0001e840: 2028 6c69 6e65 2034 3129 0a2a 2073 6361 (line 41).* sca\n-0001e850: 6c65 203c 333e 3a20 2020 2020 2020 2020 le <3>: \n+0001e7e0: 2020 2020 2020 5472 616e 7366 6f72 6d20 Transform \n+0001e7f0: 6c69 7465 7261 6c73 2e20 2020 286c 696e literals. (lin\n+0001e800: 6520 3137 290a 2a20 7363 616c 6520 3c32 e 17).* scale <2\n+0001e810: 3e3a 2020 2020 2020 2020 2020 2020 2020 >: \n+0001e820: 2020 2020 2020 2020 2020 2020 2020 2041 A\n+0001e830: 2074 6563 686e 6963 616c 2064 7261 7769 technical drawi\n+0001e840: 6e67 2e20 2028 6c69 6e65 2034 3129 0a2a ng. (line 41).*\n+0001e850: 2073 6361 6c65 203c 333e 3a20 2020 2020 scale <3>: \n 0001e860: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001e870: 2020 2020 4120 7465 6368 6e69 6361 6c20 A technical \n-0001e880: 6472 6177 696e 672e 2020 286c 696e 6520 drawing. (line \n-0001e890: 3638 290a 2a20 7363 616c 6520 3c34 3e3a 68).* scale <4>:\n-0001e8a0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001e8b0: 2020 2020 2020 2020 2020 2020 2041 2068 A h\n-0001e8c0: 6965 7261 7263 6869 6361 6c20 6d6f 6465 ierarchical mode\n-0001e8d0: 6c2e 2028 6c69 6e65 2033 3229 0a2a 2073 l. (line 32).* s\n-0001e8e0: 6574 3a20 2020 2020 2020 2020 2020 2020 et: \n+0001e870: 2020 2020 2020 2020 4120 7465 6368 6e69 A techni\n+0001e880: 6361 6c20 6472 6177 696e 672e 2020 286c cal drawing. (l\n+0001e890: 696e 6520 3638 290a 2a20 7363 616c 6520 ine 68).* scale \n+0001e8a0: 3c34 3e3a 2020 2020 2020 2020 2020 2020 <4>: \n+0001e8b0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001e8c0: 2041 2068 6965 7261 7263 6869 6361 6c20 A hierarchical \n+0001e8d0: 6d6f 6465 6c2e 2028 6c69 6e65 2033 3229 model. (line 32)\n+0001e8e0: 0a2a 2073 6574 3a20 2020 2020 2020 2020 .* set: \n 0001e8f0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001e900: 2020 2020 2020 476c 6f62 616c 206f 7074 Global opt\n-0001e910: 696f 6e73 2e20 2020 2020 2020 286c 696e ions. (lin\n-0001e920: 6520 2036 290a 2a20 7369 6e3a 2020 2020 e 6).* sin: \n+0001e900: 2020 2020 2020 2020 2020 476c 6f62 616c Global\n+0001e910: 206f 7074 696f 6e73 2e20 2020 2020 2020 options. \n+0001e920: 286c 696e 6520 2036 290a 2a20 7369 6e3a (line 6).* sin:\n 0001e930: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001e940: 2020 2020 2020 2020 2020 2020 2020 2055 U\n-0001e950: 6e61 7279 2066 6f72 6d73 2e20 2020 2020 nary forms. \n-0001e960: 2020 2020 2028 6c69 6e65 2031 3529 0a2a (line 15).*\n-0001e970: 2073 7065 6369 616c 3a20 2020 2020 2020 special: \n+0001e940: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001e950: 2020 2055 6e61 7279 2066 6f72 6d73 2e20 Unary forms. \n+0001e960: 2020 2020 2020 2020 2028 6c69 6e65 2031 (line 1\n+0001e970: 3529 0a2a 2073 7065 6369 616c 3a20 2020 5).* special: \n 0001e980: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001e990: 2020 2020 2020 2020 5370 6563 6961 6c20 Special \n-0001e9a0: 6f62 6a65 6374 732e 2020 2020 2020 286c objects. (l\n-0001e9b0: 696e 6520 2036 290a 2a20 7370 6563 6961 ine 6).* specia\n-0001e9c0: 6c20 3c31 3e3a 2020 2020 2020 2020 2020 l <1>: \n+0001e990: 2020 2020 2020 2020 2020 2020 5370 6563 Spec\n+0001e9a0: 6961 6c20 6f62 6a65 6374 732e 2020 2020 ial objects. \n+0001e9b0: 2020 286c 696e 6520 2036 290a 2a20 7370 (line 6).* sp\n+0001e9c0: 6563 6961 6c20 3c31 3e3a 2020 2020 2020 ecial <1>: \n 0001e9d0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001e9e0: 2053 7065 6369 616c 732e 2020 2020 2020 Specials. \n-0001e9f0: 2020 2020 2020 2028 6c69 6e65 2020 3629 (line 6)\n-0001ea00: 0a2a 2073 7065 6369 616c 203c 323e 3a20 .* special <2>: \n-0001ea10: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001ea20: 2020 2020 2020 2020 2020 4f76 6572 7669 Overvi\n-0001ea30: 6577 2e20 2020 2020 2020 2020 2020 2020 ew. \n-0001ea40: 286c 696e 6520 3639 290a 2a20 7370 6563 (line 69).* spec\n-0001ea50: 6961 6c20 3c33 3e3a 2020 2020 2020 2020 ial <3>: \n+0001e9e0: 2020 2020 2053 7065 6369 616c 732e 2020 Specials. \n+0001e9f0: 2020 2020 2020 2020 2020 2028 6c69 6e65 (line\n+0001ea00: 2020 3629 0a2a 2073 7065 6369 616c 203c 6).* special <\n+0001ea10: 323e 3a20 2020 2020 2020 2020 2020 2020 2>: \n+0001ea20: 2020 2020 2020 2020 2020 2020 2020 4f76 Ov\n+0001ea30: 6572 7669 6577 2e20 2020 2020 2020 2020 erview. \n+0001ea40: 2020 2020 286c 696e 6520 3639 290a 2a20 (line 69).* \n+0001ea50: 7370 6563 6961 6c20 3c33 3e3a 2020 2020 special <3>: \n 0001ea60: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001ea70: 2020 2041 2074 6563 686e 6963 616c 2064 A technical d\n-0001ea80: 7261 7769 6e67 2e20 2028 6c69 6e65 2032 rawing. (line 2\n-0001ea90: 3629 0a2a 2073 7065 6369 616c 203c 343e 6).* special <4>\n-0001eaa0: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n-0001eab0: 2020 2020 2020 2020 2020 2020 4120 7465 A te\n-0001eac0: 6368 6e69 6361 6c20 6472 6177 696e 672e chnical drawing.\n-0001ead0: 2020 286c 696e 6520 3431 290a 2a20 7370 (line 41).* sp\n-0001eae0: 6563 6961 6c20 3c35 3e3a 2020 2020 2020 ecial <5>: \n+0001ea70: 2020 2020 2020 2041 2074 6563 686e 6963 A technic\n+0001ea80: 616c 2064 7261 7769 6e67 2e20 2028 6c69 al drawing. (li\n+0001ea90: 6e65 2032 3629 0a2a 2073 7065 6369 616c ne 26).* special\n+0001eaa0: 203c 343e 3a20 2020 2020 2020 2020 2020 <4>: \n+0001eab0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001eac0: 4120 7465 6368 6e69 6361 6c20 6472 6177 A technical draw\n+0001ead0: 696e 672e 2020 286c 696e 6520 3431 290a ing. (line 41).\n+0001eae0: 2a20 7370 6563 6961 6c20 3c35 3e3a 2020 * special <5>: \n 0001eaf0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001eb00: 2020 2020 2041 2074 6563 686e 6963 616c A technical\n-0001eb10: 2064 7261 7769 6e67 2e20 2028 6c69 6e65 drawing. (line\n-0001eb20: 2036 3829 0a2a 2073 7065 6369 616c 203c 68).* special <\n-0001eb30: 363e 3a20 2020 2020 2020 2020 2020 2020 6>: \n-0001eb40: 2020 2020 2020 2020 2020 2020 2020 4120 A \n-0001eb50: 7465 6368 6e69 6361 6c20 6472 6177 696e technical drawin\n-0001eb60: 672e 2020 286c 696e 6520 3935 290a 2a20 g. (line 95).* \n-0001eb70: 7370 6c69 743a 2020 2020 2020 2020 2020 split: \n+0001eb00: 2020 2020 2020 2020 2041 2074 6563 686e A techn\n+0001eb10: 6963 616c 2064 7261 7769 6e67 2e20 2028 ical drawing. (\n+0001eb20: 6c69 6e65 2036 3829 0a2a 2073 7065 6369 line 68).* speci\n+0001eb30: 616c 203c 363e 3a20 2020 2020 2020 2020 al <6>: \n+0001eb40: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001eb50: 2020 4120 7465 6368 6e69 6361 6c20 6472 A technical dr\n+0001eb60: 6177 696e 672e 2020 286c 696e 6520 3935 awing. (line 95\n+0001eb70: 290a 2a20 7370 6c69 743a 2020 2020 2020 ).* split: \n 0001eb80: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001eb90: 2020 2020 2020 2049 6e74 6572 6e61 6c20 Internal \n-0001eba0: 6f70 7469 6f6e 732e 2020 2020 2028 6c69 options. (li\n-0001ebb0: 6e65 2020 3729 0a2a 2073 706c 6974 203c ne 7).* split <\n-0001ebc0: 313e 3a20 2020 2020 2020 2020 2020 2020 1>: \n+0001eb90: 2020 2020 2020 2020 2020 2049 6e74 6572 Inter\n+0001eba0: 6e61 6c20 6f70 7469 6f6e 732e 2020 2020 nal options. \n+0001ebb0: 2028 6c69 6e65 2020 3729 0a2a 2073 706c (line 7).* spl\n+0001ebc0: 6974 203c 313e 3a20 2020 2020 2020 2020 it <1>: \n 0001ebd0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001ebe0: 476c 6f62 616c 206f 7074 696f 6e73 2e20 Global options. \n-0001ebf0: 2020 2020 2020 286c 696e 6520 3133 290a (line 13).\n-0001ec00: 2a20 7371 7274 3a20 2020 2020 2020 2020 * sqrt: \n+0001ebe0: 2020 2020 476c 6f62 616c 206f 7074 696f Global optio\n+0001ebf0: 6e73 2e20 2020 2020 2020 286c 696e 6520 ns. (line \n+0001ec00: 3133 290a 2a20 7371 7274 3a20 2020 2020 13).* sqrt: \n 0001ec10: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001ec20: 2020 2020 2020 2020 2055 6e61 7279 2066 Unary f\n-0001ec30: 6f72 6d73 2e20 2020 2020 2020 2020 2028 orms. (\n-0001ec40: 6c69 6e65 2031 3429 0a2a 2073 7765 6570 line 14).* sweep\n-0001ec50: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+0001ec20: 2020 2020 2020 2020 2020 2020 2055 6e61 Una\n+0001ec30: 7279 2066 6f72 6d73 2e20 2020 2020 2020 ry forms. \n+0001ec40: 2020 2028 6c69 6e65 2031 3429 0a2a 2073 (line 14).* s\n+0001ec50: 7765 6570 3a20 2020 2020 2020 2020 2020 weep: \n 0001ec60: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001ec70: 2020 5377 6570 7420 6f62 6a65 6374 732e Swept objects.\n-0001ec80: 2020 2020 2020 2020 286c 696e 6520 2038 (line 8\n-0001ec90: 290a 2a20 7377 6565 7020 3c31 3e3a 2020 ).* sweep <1>: \n-0001eca0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001ecb0: 2020 2020 2020 2020 2020 2050 6f69 6e74 Point\n-0001ecc0: 2073 7765 6570 732e 2020 2020 2020 2020 sweeps. \n-0001ecd0: 2028 6c69 6e65 2031 3929 0a2a 2073 7765 (line 19).* swe\n-0001ece0: 6570 203c 323e 3a20 2020 2020 2020 2020 ep <2>: \n+0001ec70: 2020 2020 2020 5377 6570 7420 6f62 6a65 Swept obje\n+0001ec80: 6374 732e 2020 2020 2020 2020 286c 696e cts. (lin\n+0001ec90: 6520 2038 290a 2a20 7377 6565 7020 3c31 e 8).* sweep <1\n+0001eca0: 3e3a 2020 2020 2020 2020 2020 2020 2020 >: \n+0001ecb0: 2020 2020 2020 2020 2020 2020 2020 2050 P\n+0001ecc0: 6f69 6e74 2073 7765 6570 732e 2020 2020 oint sweeps. \n+0001ecd0: 2020 2020 2028 6c69 6e65 2031 3929 0a2a (line 19).*\n+0001ece0: 2073 7765 6570 203c 323e 3a20 2020 2020 sweep <2>: \n 0001ecf0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001ed00: 2020 2020 506f 6c79 6c69 6e65 2073 7765 Polyline swe\n-0001ed10: 6570 732e 2020 2020 2020 286c 696e 6520 eps. (line \n-0001ed20: 2039 290a 2a20 7377 6565 7020 3c33 3e3a 9).* sweep <3>:\n-0001ed30: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001ed40: 2020 2020 2020 2020 2020 2020 204e 6573 Nes\n-0001ed50: 7465 6420 7377 6565 7073 2e20 2020 2020 ted sweeps. \n-0001ed60: 2020 2028 6c69 6e65 2020 3829 0a2a 2073 (line 8).* s\n-0001ed70: 7765 6570 203c 343e 3a20 2020 2020 2020 weep <4>: \n+0001ed00: 2020 2020 2020 2020 506f 6c79 6c69 6e65 Polyline\n+0001ed10: 2073 7765 6570 732e 2020 2020 2020 286c sweeps. (l\n+0001ed20: 696e 6520 2039 290a 2a20 7377 6565 7020 ine 9).* sweep \n+0001ed30: 3c33 3e3a 2020 2020 2020 2020 2020 2020 <3>: \n+0001ed40: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001ed50: 204e 6573 7465 6420 7377 6565 7073 2e20 Nested sweeps. \n+0001ed60: 2020 2020 2020 2028 6c69 6e65 2020 3829 (line 8)\n+0001ed70: 0a2a 2073 7765 6570 203c 343e 3a20 2020 .* sweep <4>: \n 0001ed80: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001ed90: 2020 2020 2020 4e65 7374 6564 2073 7765 Nested swe\n-0001eda0: 6570 732e 2020 2020 2020 2020 286c 696e eps. (lin\n-0001edb0: 6520 3232 290a 2a20 7377 6565 7020 3c35 e 22).* sweep <5\n-0001edc0: 3e3a 2020 2020 2020 2020 2020 2020 2020 >: \n-0001edd0: 2020 2020 2020 2020 2020 2020 2020 2050 P\n-0001ede0: 6f6c 796c 696e 6520 7377 6565 7073 2077 olyline sweeps w\n-0001edf0: 6974 6820 636c 6f73 7572 652e 0a20 2020 ith closure.. \n-0001ee00: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001ed90: 2020 2020 2020 2020 2020 4e65 7374 6564 Nested\n+0001eda0: 2073 7765 6570 732e 2020 2020 2020 2020 sweeps. \n+0001edb0: 286c 696e 6520 3232 290a 2a20 7377 6565 (line 22).* swee\n+0001edc0: 7020 3c35 3e3a 2020 2020 2020 2020 2020 p <5>: \n+0001edd0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001ede0: 2020 2050 6f6c 796c 696e 6520 7377 6565 Polyline swee\n+0001edf0: 7073 2077 6974 6820 636c 6f73 7572 652e ps with closure.\n+0001ee00: 0a20 2020 2020 2020 2020 2020 2020 2020 . \n 0001ee10: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001ee20: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001ee30: 2020 2020 2020 2020 2020 2020 286c 696e (lin\n-0001ee40: 6520 3130 290a 2a20 7377 6565 7020 3c36 e 10).* sweep <6\n-0001ee50: 3e3a 2020 2020 2020 2020 2020 2020 2020 >: \n-0001ee60: 2020 2020 2020 2020 2020 2020 2020 2053 S\n-0001ee70: 7765 6570 732e 2020 2020 2020 2020 2020 weeps. \n-0001ee80: 2020 2020 2028 6c69 6e65 2020 3629 0a2a (line 6).*\n-0001ee90: 2073 7765 6570 203c 373e 3a20 2020 2020 sweep <7>: \n+0001ee30: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001ee40: 286c 696e 6520 3130 290a 2a20 7377 6565 (line 10).* swee\n+0001ee50: 7020 3c36 3e3a 2020 2020 2020 2020 2020 p <6>: \n+0001ee60: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001ee70: 2020 2053 7765 6570 732e 2020 2020 2020 Sweeps. \n+0001ee80: 2020 2020 2020 2020 2028 6c69 6e65 2020 (line \n+0001ee90: 3629 0a2a 2073 7765 6570 203c 373e 3a20 6).* sweep <7>: \n 0001eea0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001eeb0: 2020 2020 2020 2020 4120 7465 6368 6e69 A techni\n-0001eec0: 6361 6c20 6472 6177 696e 672e 2020 286c cal drawing. (l\n-0001eed0: 696e 6520 3230 290a 2a20 7377 6565 7020 ine 20).* sweep \n-0001eee0: 3c38 3e3a 2020 2020 2020 2020 2020 2020 <8>: \n+0001eeb0: 2020 2020 2020 2020 2020 2020 4120 7465 A te\n+0001eec0: 6368 6e69 6361 6c20 6472 6177 696e 672e chnical drawing.\n+0001eed0: 2020 286c 696e 6520 3230 290a 2a20 7377 (line 20).* sw\n+0001eee0: 6565 7020 3c38 3e3a 2020 2020 2020 2020 eep <8>: \n 0001eef0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001ef00: 2041 2068 6965 7261 7263 6869 6361 6c20 A hierarchical \n-0001ef10: 6d6f 6465 6c2e 2028 6c69 6e65 2031 3029 model. (line 10)\n-0001ef20: 0a2a 2073 7765 6570 203c 393e 3a20 2020 .* sweep <9>: \n-0001ef30: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001ef40: 2020 2020 2020 2020 2020 4120 6869 6572 A hier\n-0001ef50: 6172 6368 6963 616c 206d 6f64 656c 2e20 archical model. \n-0001ef60: 286c 696e 6520 3233 290a 2a20 7468 656e (line 23).* then\n-0001ef70: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+0001ef00: 2020 2020 2041 2068 6965 7261 7263 6869 A hierarchi\n+0001ef10: 6361 6c20 6d6f 6465 6c2e 2028 6c69 6e65 cal model. (line\n+0001ef20: 2031 3029 0a2a 2073 7765 6570 203c 393e 10).* sweep <9>\n+0001ef30: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+0001ef40: 2020 2020 2020 2020 2020 2020 2020 4120 A \n+0001ef50: 6869 6572 6172 6368 6963 616c 206d 6f64 hierarchical mod\n+0001ef60: 656c 2e20 286c 696e 6520 3233 290a 2a20 el. (line 23).* \n+0001ef70: 7468 656e 3a20 2020 2020 2020 2020 2020 then: \n 0001ef80: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001ef90: 2020 204f 626a 6563 7420 7472 616e 7366 Object transf\n-0001efa0: 6f72 6d73 2e20 2020 2028 6c69 6e65 2031 orms. (line 1\n-0001efb0: 3029 0a2a 2074 6865 6e20 3c31 3e3a 2020 0).* then <1>: \n-0001efc0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001efd0: 2020 2020 2020 2020 2020 2020 506f 6c79 Poly\n-0001efe0: 6c69 6e65 2073 7765 6570 7320 7769 7468 line sweeps with\n-0001eff0: 2063 6c6f 7375 7265 2e0a 2020 2020 2020 closure.. \n+0001ef90: 2020 2020 2020 204f 626a 6563 7420 7472 Object tr\n+0001efa0: 616e 7366 6f72 6d73 2e20 2020 2028 6c69 ansforms. (li\n+0001efb0: 6e65 2031 3029 0a2a 2074 6865 6e20 3c31 ne 10).* then <1\n+0001efc0: 3e3a 2020 2020 2020 2020 2020 2020 2020 >: \n+0001efd0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001efe0: 506f 6c79 6c69 6e65 2073 7765 6570 7320 Polyline sweeps \n+0001eff0: 7769 7468 2063 6c6f 7375 7265 2e0a 2020 with closure.. \n 0001f000: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001f010: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001f020: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001f030: 2020 2020 2020 2020 2028 6c69 6e65 2031 (line 1\n-0001f040: 3029 0a2a 2074 6865 6e20 3c32 3e3a 2020 0).* then <2>: \n-0001f050: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001f060: 2020 2020 2020 2020 2020 2020 5477 6f2d Two-\n-0001f070: 6f70 6572 616e 6420 2862 696e 6172 7929 operand (binary)\n-0001f080: 2066 6f72 6d73 2e0a 2020 2020 2020 2020 forms.. \n+0001f030: 2020 2020 2020 2020 2020 2020 2028 6c69 (li\n+0001f040: 6e65 2031 3029 0a2a 2074 6865 6e20 3c32 ne 10).* then <2\n+0001f050: 3e3a 2020 2020 2020 2020 2020 2020 2020 >: \n+0001f060: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001f070: 5477 6f2d 6f70 6572 616e 6420 2862 696e Two-operand (bin\n+0001f080: 6172 7929 2066 6f72 6d73 2e0a 2020 2020 ary) forms.. \n 0001f090: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001f0a0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001f0b0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001f0c0: 2020 2020 2020 2028 6c69 6e65 2033 3629 (line 36)\n-0001f0d0: 0a2a 2074 6865 6e20 3c33 3e3a 2020 2020 .* then <3>: \n+0001f0c0: 2020 2020 2020 2020 2020 2028 6c69 6e65 (line\n+0001f0d0: 2033 3629 0a2a 2074 6865 6e20 3c33 3e3a 36).* then <3>:\n 0001f0e0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001f0f0: 2020 2020 2020 2020 2020 5477 6f2d 6f70 Two-op\n-0001f100: 6572 616e 6420 2862 696e 6172 7929 2066 erand (binary) f\n-0001f110: 6f72 6d73 2e0a 2020 2020 2020 2020 2020 orms.. \n+0001f0f0: 2020 2020 2020 2020 2020 2020 2020 5477 Tw\n+0001f100: 6f2d 6f70 6572 616e 6420 2862 696e 6172 o-operand (binar\n+0001f110: 7929 2066 6f72 6d73 2e0a 2020 2020 2020 y) forms.. \n 0001f120: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001f130: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001f140: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001f150: 2020 2020 2028 6c69 6e65 2036 3329 0a2a (line 63).*\n-0001f160: 2074 696b 7a3a 2020 2020 2020 2020 2020 tikz: \n+0001f150: 2020 2020 2020 2020 2028 6c69 6e65 2036 (line 6\n+0001f160: 3329 0a2a 2074 696b 7a3a 2020 2020 2020 3).* tikz: \n 0001f170: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001f180: 2020 2020 2020 2020 4c61 6e67 7561 6765 Language\n-0001f190: 2e20 2020 2020 2020 2020 2020 2020 286c . (l\n-0001f1a0: 696e 6520 3130 290a 2a20 7472 616e 736c ine 10).* transl\n-0001f1b0: 6174 653a 2020 2020 2020 2020 2020 2020 ate: \n+0001f180: 2020 2020 2020 2020 2020 2020 4c61 6e67 Lang\n+0001f190: 7561 6765 2e20 2020 2020 2020 2020 2020 uage. \n+0001f1a0: 2020 286c 696e 6520 3130 290a 2a20 7472 (line 10).* tr\n+0001f1b0: 616e 736c 6174 653a 2020 2020 2020 2020 anslate: \n 0001f1c0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001f1d0: 204f 626a 6563 7420 7472 616e 7366 6f72 Object transfor\n-0001f1e0: 6d73 2e20 2020 2028 6c69 6e65 2031 3029 ms. (line 10)\n-0001f1f0: 0a2a 2074 7261 6e73 6c61 7465 203c 313e .* translate <1>\n-0001f200: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n-0001f210: 2020 2020 2020 2020 2020 4f62 6a65 6374 Object\n-0001f220: 2074 7261 6e73 666f 726d 732e 2020 2020 transforms. \n-0001f230: 286c 696e 6520 3437 290a 2a20 7472 616e (line 47).* tran\n-0001f240: 736c 6174 6520 3c32 3e3a 2020 2020 2020 slate <2>: \n+0001f1d0: 2020 2020 204f 626a 6563 7420 7472 616e Object tran\n+0001f1e0: 7366 6f72 6d73 2e20 2020 2028 6c69 6e65 sforms. (line\n+0001f1f0: 2031 3029 0a2a 2074 7261 6e73 6c61 7465 10).* translate\n+0001f200: 203c 313e 3a20 2020 2020 2020 2020 2020 <1>: \n+0001f210: 2020 2020 2020 2020 2020 2020 2020 4f62 Ob\n+0001f220: 6a65 6374 2074 7261 6e73 666f 726d 732e ject transforms.\n+0001f230: 2020 2020 286c 696e 6520 3437 290a 2a20 (line 47).* \n+0001f240: 7472 616e 736c 6174 6520 3c32 3e3a 2020 translate <2>: \n 0001f250: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001f260: 2020 2052 6570 6561 7465 6420 6f62 6a65 Repeated obje\n-0001f270: 6374 732e 2020 2020 2028 6c69 6e65 2020 cts. (line \n-0001f280: 3829 0a2a 2074 7261 6e73 6c61 7465 203c 8).* translate <\n-0001f290: 333e 3a20 2020 2020 2020 2020 2020 2020 3>: \n-0001f2a0: 2020 2020 2020 2020 2020 2020 506f 6c79 Poly\n-0001f2b0: 6c69 6e65 2073 7765 6570 732e 2020 2020 line sweeps. \n-0001f2c0: 2020 286c 696e 6520 2039 290a 2a20 7472 (line 9).* tr\n-0001f2d0: 616e 736c 6174 6520 3c34 3e3a 2020 2020 anslate <4>: \n+0001f260: 2020 2020 2020 2052 6570 6561 7465 6420 Repeated \n+0001f270: 6f62 6a65 6374 732e 2020 2020 2028 6c69 objects. (li\n+0001f280: 6e65 2020 3829 0a2a 2074 7261 6e73 6c61 ne 8).* transla\n+0001f290: 7465 203c 333e 3a20 2020 2020 2020 2020 te <3>: \n+0001f2a0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001f2b0: 506f 6c79 6c69 6e65 2073 7765 6570 732e Polyline sweeps.\n+0001f2c0: 2020 2020 2020 286c 696e 6520 2039 290a (line 9).\n+0001f2d0: 2a20 7472 616e 736c 6174 6520 3c34 3e3a * translate <4>:\n 0001f2e0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001f2f0: 2020 2020 2050 6f6c 796c 696e 6520 7377 Polyline sw\n-0001f300: 6565 7073 2077 6974 6820 636c 6f73 7572 eeps with closur\n-0001f310: 652e 0a20 2020 2020 2020 2020 2020 2020 e.. \n+0001f2f0: 2020 2020 2020 2020 2050 6f6c 796c 696e Polylin\n+0001f300: 6520 7377 6565 7073 2077 6974 6820 636c e sweeps with cl\n+0001f310: 6f73 7572 652e 0a20 2020 2020 2020 2020 osure.. \n 0001f320: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001f330: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001f340: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001f350: 2020 286c 696e 6520 3130 290a 2a20 7472 (line 10).* tr\n-0001f360: 616e 736c 6174 6520 3c35 3e3a 2020 2020 anslate <5>: \n+0001f350: 2020 2020 2020 286c 696e 6520 3130 290a (line 10).\n+0001f360: 2a20 7472 616e 736c 6174 6520 3c35 3e3a * translate <5>:\n 0001f370: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001f380: 2020 2020 2054 7261 6e73 666f 726d 206c Transform l\n-0001f390: 6974 6572 616c 732e 2020 2028 6c69 6e65 iterals. (line\n-0001f3a0: 2031 3529 0a2a 2074 7261 6e73 6c61 7465 15).* translate\n-0001f3b0: 203c 363e 3a20 2020 2020 2020 2020 2020 <6>: \n-0001f3c0: 2020 2020 2020 2020 2020 2020 2020 4f76 Ov\n-0001f3d0: 6572 7669 6577 2e20 2020 2020 2020 2020 erview. \n-0001f3e0: 2020 2020 286c 696e 6520 3537 290a 2a20 (line 57).* \n-0001f3f0: 7472 616e 736c 6174 6520 3c37 3e3a 2020 translate <7>: \n-0001f400: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001f410: 2020 2020 2020 204f 7665 7276 6965 772e Overview.\n-0001f420: 2020 2020 2020 2020 2020 2020 2028 6c69 (li\n-0001f430: 6e65 2036 3929 0a2a 2074 7261 6e73 6c61 ne 69).* transla\n-0001f440: 7465 203c 383e 3a20 2020 2020 2020 2020 te <8>: \n+0001f380: 2020 2020 2020 2020 2054 7261 6e73 666f Transfo\n+0001f390: 726d 206c 6974 6572 616c 732e 2020 2028 rm literals. (\n+0001f3a0: 6c69 6e65 2031 3529 0a2a 2074 7261 6e73 line 15).* trans\n+0001f3b0: 6c61 7465 203c 363e 3a20 2020 2020 2020 late <6>: \n+0001f3c0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001f3d0: 2020 4f76 6572 7669 6577 2e20 2020 2020 Overview. \n+0001f3e0: 2020 2020 2020 2020 286c 696e 6520 3537 (line 57\n+0001f3f0: 290a 2a20 7472 616e 736c 6174 6520 3c37 ).* translate <7\n+0001f400: 3e3a 2020 2020 2020 2020 2020 2020 2020 >: \n+0001f410: 2020 2020 2020 2020 2020 204f 7665 7276 Overv\n+0001f420: 6965 772e 2020 2020 2020 2020 2020 2020 iew. \n+0001f430: 2028 6c69 6e65 2036 3929 0a2a 2074 7261 (line 69).* tra\n+0001f440: 6e73 6c61 7465 203c 383e 3a20 2020 2020 nslate <8>: \n 0001f450: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001f460: 4120 6869 6572 6172 6368 6963 616c 206d A hierarchical m\n-0001f470: 6f64 656c 2e20 286c 696e 6520 3332 290a odel. (line 32).\n-0001f480: 2a20 7472 616e 7370 616c 7068 613a 2020 * transpalpha: \n-0001f490: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001f4a0: 2020 2020 2020 2020 2054 7261 6e73 7061 Transpa\n-0001f4b0: 7265 6e63 792e 2020 2020 2020 2020 2028 rency. (\n-0001f4c0: 6c69 6e65 2031 3029 0a2a 2075 6e69 743a line 10).* unit:\n-0001f4d0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001f460: 2020 2020 4120 6869 6572 6172 6368 6963 A hierarchic\n+0001f470: 616c 206d 6f64 656c 2e20 286c 696e 6520 al model. (line \n+0001f480: 3332 290a 2a20 7472 616e 7370 616c 7068 32).* transpalph\n+0001f490: 613a 2020 2020 2020 2020 2020 2020 2020 a: \n+0001f4a0: 2020 2020 2020 2020 2020 2020 2054 7261 Tra\n+0001f4b0: 6e73 7061 7265 6e63 792e 2020 2020 2020 nsparency. \n+0001f4c0: 2020 2028 6c69 6e65 2031 3029 0a2a 2075 (line 10).* u\n+0001f4d0: 6e69 743a 2020 2020 2020 2020 2020 2020 nit: \n 0001f4e0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001f4f0: 2020 556e 6172 7920 666f 726d 732e 2020 Unary forms. \n-0001f500: 2020 2020 2020 2020 286c 696e 6520 3133 (line 13\n-0001f510: 290a 2a20 756e 6974 203c 313e 3a20 2020 ).* unit <1>: \n-0001f520: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001f530: 2020 2020 2020 2020 2020 2041 2074 6563 A tec\n-0001f540: 686e 6963 616c 2064 7261 7769 6e67 2e20 hnical drawing. \n-0001f550: 2028 6c69 6e65 2034 3129 0a2a 2076 6965 (line 41).* vie\n-0001f560: 773a 2020 2020 2020 2020 2020 2020 2020 w: \n+0001f4f0: 2020 2020 2020 556e 6172 7920 666f 726d Unary form\n+0001f500: 732e 2020 2020 2020 2020 2020 286c 696e s. (lin\n+0001f510: 6520 3133 290a 2a20 756e 6974 203c 313e e 13).* unit <1>\n+0001f520: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+0001f530: 2020 2020 2020 2020 2020 2020 2020 2041 A\n+0001f540: 2074 6563 686e 6963 616c 2064 7261 7769 technical drawi\n+0001f550: 6e67 2e20 2028 6c69 6e65 2034 3129 0a2a ng. (line 41).*\n+0001f560: 2076 6965 773a 2020 2020 2020 2020 2020 view: \n 0001f570: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001f580: 2020 2020 5472 616e 7366 6f72 6d20 6c69 Transform li\n-0001f590: 7465 7261 6c73 2e20 2020 286c 696e 6520 terals. (line \n-0001f5a0: 3235 290a 2a20 7669 6577 203c 313e 3a20 25).* view <1>: \n-0001f5b0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001f5c0: 2020 2020 2020 2020 2020 2020 2054 7261 Tra\n-0001f5d0: 6e73 666f 726d 206c 6974 6572 616c 732e nsform literals.\n-0001f5e0: 2020 2028 6c69 6e65 2033 3529 0a0a 1f0a (line 35)....\n-0001f5f0: 4669 6c65 3a20 736b 6574 6368 2e69 6e66 File: sketch.inf\n-0001f600: 6f2c 2020 4e6f 6465 3a20 496e 6465 782c o, Node: Index,\n-0001f610: 2020 5072 6576 3a20 496e 6465 7820 6f66 Prev: Index of\n-0001f620: 2073 796e 7461 782c 2020 5570 3a20 546f syntax, Up: To\n-0001f630: 700a 0a49 6e64 6578 206f 6620 636f 6e63 p..Index of conc\n-0001f640: 6570 7473 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a epts.***********\n-0001f650: 2a2a 2a2a 2a2a 0a0a 0008 5b69 6e64 6578 ******....[index\n-0001f660: 0008 5d0a 2a20 4d65 6e75 3a0a 0a2a 2061 ..].* Menu:..* a\n-0001f670: 6666 696e 6520 6172 6974 686d 6574 6963 ffine arithmetic\n-0001f680: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n-0001f690: 2020 2020 2020 4166 6669 6e65 2061 7269 Affine ari\n-0001f6a0: 7468 6d65 7469 632e 2020 2028 6c69 6e65 thmetic. (line\n-0001f6b0: 2020 2038 290a 2a20 6166 6669 6e65 2061 8).* affine a\n-0001f6c0: 7269 7468 6d65 7469 6320 3c31 3e3a 2020 rithmetic <1>: \n-0001f6d0: 2020 2020 2020 2020 2020 2020 2020 2041 A\n-0001f6e0: 2074 6563 686e 6963 616c 2064 7261 7769 technical drawi\n-0001f6f0: 6e67 2e20 286c 696e 6520 2034 3029 0a2a ng. (line 40).*\n-0001f700: 2061 6c74 6572 6e61 7469 7665 732c 2064 alternatives, d\n-0001f710: 6566 696e 6974 696f 6e3a 2020 2020 2020 efinition: \n-0001f720: 2020 2020 2020 2020 466f 726d 7320 6f66 Forms of\n-0001f730: 2064 6566 696e 6974 696f 6e73 2e0a 2020 definitions.. \n-0001f740: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001f580: 2020 2020 2020 2020 5472 616e 7366 6f72 Transfor\n+0001f590: 6d20 6c69 7465 7261 6c73 2e20 2020 286c m literals. (l\n+0001f5a0: 696e 6520 3235 290a 2a20 7669 6577 203c ine 25).* view <\n+0001f5b0: 313e 3a20 2020 2020 2020 2020 2020 2020 1>: \n+0001f5c0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001f5d0: 2054 7261 6e73 666f 726d 206c 6974 6572 Transform liter\n+0001f5e0: 616c 732e 2020 2028 6c69 6e65 2033 3529 als. (line 35)\n+0001f5f0: 0a0a 1f0a 4669 6c65 3a20 736b 6574 6368 ....File: sketch\n+0001f600: 2e69 6e66 6f2c 2020 4e6f 6465 3a20 496e .info, Node: In\n+0001f610: 6465 782c 2020 5072 6576 3a20 496e 6465 dex, Prev: Inde\n+0001f620: 7820 6f66 2073 796e 7461 782c 2020 5570 x of syntax, Up\n+0001f630: 3a20 546f 700a 0a49 6e64 6578 206f 6620 : Top..Index of \n+0001f640: 636f 6e63 6570 7473 0a2a 2a2a 2a2a 2a2a concepts.*******\n+0001f650: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 0008 5b69 **********....[i\n+0001f660: 6e64 6578 0008 5d0a 2a20 4d65 6e75 3a0a ndex..].* Menu:.\n+0001f670: 0a2a 2061 6666 696e 6520 6172 6974 686d .* affine arithm\n+0001f680: 6574 6963 3a20 2020 2020 2020 2020 2020 etic: \n+0001f690: 2020 2020 2020 2020 2020 4166 6669 6e65 Affine\n+0001f6a0: 2061 7269 7468 6d65 7469 632e 2020 2028 arithmetic. (\n+0001f6b0: 6c69 6e65 2020 2038 290a 2a20 6166 6669 line 8).* affi\n+0001f6c0: 6e65 2061 7269 7468 6d65 7469 6320 3c31 ne arithmetic <1\n+0001f6d0: 3e3a 2020 2020 2020 2020 2020 2020 2020 >: \n+0001f6e0: 2020 2041 2074 6563 686e 6963 616c 2064 A technical d\n+0001f6f0: 7261 7769 6e67 2e20 286c 696e 6520 2034 rawing. (line 4\n+0001f700: 3029 0a2a 2061 6c74 6572 6e61 7469 7665 0).* alternative\n+0001f710: 732c 2064 6566 696e 6974 696f 6e3a 2020 s, definition: \n+0001f720: 2020 2020 2020 2020 2020 2020 466f 726d Form\n+0001f730: 7320 6f66 2064 6566 696e 6974 696f 6e73 s of definitions\n+0001f740: 2e0a 2020 2020 2020 2020 2020 2020 2020 .. \n 0001f750: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001f760: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001f770: 2020 2020 2020 2020 2020 2020 286c 696e (lin\n-0001f780: 6520 2020 3629 0a2a 2061 7267 756d 656e e 6).* argumen\n-0001f790: 742c 2073 7065 6369 616c 3a20 2020 2020 t, special: \n+0001f770: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001f780: 286c 696e 6520 2020 3629 0a2a 2061 7267 (line 6).* arg\n+0001f790: 756d 656e 742c 2073 7065 6369 616c 3a20 ument, special: \n 0001f7a0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001f7b0: 5370 6563 6961 6c20 6f62 6a65 6374 732e Special objects.\n-0001f7c0: 2020 2020 2028 6c69 6e65 2020 3235 290a (line 25).\n-0001f7d0: 2a20 6172 6775 6d65 6e74 2c20 7370 6563 * argument, spec\n-0001f7e0: 6961 6c20 3c31 3e3a 2020 2020 2020 2020 ial <1>: \n-0001f7f0: 2020 2020 2020 2020 2053 7065 6369 616c Special\n-0001f800: 732e 2020 2020 2020 2020 2020 2020 286c s. (l\n-0001f810: 696e 6520 2031 3029 0a2a 2061 7373 6f63 ine 10).* assoc\n-0001f820: 6961 7469 7669 7479 2c20 6f70 6572 6174 iativity, operat\n-0001f830: 6f72 3a20 2020 2020 2020 2020 2020 2020 or: \n-0001f840: 2020 5477 6f2d 6f70 6572 616e 6420 2862 Two-operand (b\n-0001f850: 696e 6172 7929 2066 6f72 6d73 2e0a 2020 inary) forms.. \n-0001f860: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001f7b0: 2020 2020 5370 6563 6961 6c20 6f62 6a65 Special obje\n+0001f7c0: 6374 732e 2020 2020 2028 6c69 6e65 2020 cts. (line \n+0001f7d0: 3235 290a 2a20 6172 6775 6d65 6e74 2c20 25).* argument, \n+0001f7e0: 7370 6563 6961 6c20 3c31 3e3a 2020 2020 special <1>: \n+0001f7f0: 2020 2020 2020 2020 2020 2020 2053 7065 Spe\n+0001f800: 6369 616c 732e 2020 2020 2020 2020 2020 cials. \n+0001f810: 2020 286c 696e 6520 2031 3029 0a2a 2061 (line 10).* a\n+0001f820: 7373 6f63 6961 7469 7669 7479 2c20 6f70 ssociativity, op\n+0001f830: 6572 6174 6f72 3a20 2020 2020 2020 2020 erator: \n+0001f840: 2020 2020 2020 5477 6f2d 6f70 6572 616e Two-operan\n+0001f850: 6420 2862 696e 6172 7929 2066 6f72 6d73 d (binary) forms\n+0001f860: 2e0a 2020 2020 2020 2020 2020 2020 2020 .. \n 0001f870: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001f880: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001f890: 2020 2020 2020 2020 2020 2020 286c 696e (lin\n-0001f8a0: 6520 2035 3829 0a2a 2061 7869 732c 2072 e 58).* axis, r\n-0001f8b0: 6f74 6174 696f 6e3a 2020 2020 2020 2020 otation: \n+0001f890: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001f8a0: 286c 696e 6520 2035 3829 0a2a 2061 7869 (line 58).* axi\n+0001f8b0: 732c 2072 6f74 6174 696f 6e3a 2020 2020 s, rotation: \n 0001f8c0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001f8d0: 4f62 6a65 6374 2074 7261 6e73 666f 726d Object transform\n-0001f8e0: 732e 2020 2028 6c69 6e65 2020 3430 290a s. (line 40).\n-0001f8f0: 2a20 6261 636b 2066 6163 653a 2020 2020 * back face: \n+0001f8d0: 2020 2020 4f62 6a65 6374 2074 7261 6e73 Object trans\n+0001f8e0: 666f 726d 732e 2020 2028 6c69 6e65 2020 forms. (line \n+0001f8f0: 3430 290a 2a20 6261 636b 2066 6163 653a 40).* back face:\n 0001f900: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001f910: 2020 2020 2020 2020 2044 7261 7769 6e67 Drawing\n-0001f920: 2061 2073 6f6c 6964 2e20 2020 2020 286c a solid. (l\n-0001f930: 696e 6520 2033 3529 0a2a 2062 6173 656c ine 35).* basel\n-0001f940: 696e 6520 6672 6163 7469 6f6e 3a20 2020 ine fraction: \n-0001f950: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001f960: 2020 5069 6374 7572 6520 626f 782e 2020 Picture box. \n-0001f970: 2020 2020 2020 2028 6c69 6e65 2020 3130 (line 10\n-0001f980: 290a 2a20 6269 6e61 7279 2066 6f72 6d3a ).* binary form:\n-0001f990: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001f9a0: 2020 2020 2020 2020 2020 2054 776f 2d6f Two-o\n-0001f9b0: 7065 7261 6e64 2028 6269 6e61 7279 2920 perand (binary) \n-0001f9c0: 666f 726d 732e 0a20 2020 2020 2020 2020 forms.. \n+0001f910: 2020 2020 2020 2020 2020 2020 2044 7261 Dra\n+0001f920: 7769 6e67 2061 2073 6f6c 6964 2e20 2020 wing a solid. \n+0001f930: 2020 286c 696e 6520 2033 3529 0a2a 2062 (line 35).* b\n+0001f940: 6173 656c 696e 6520 6672 6163 7469 6f6e aseline fraction\n+0001f950: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+0001f960: 2020 2020 2020 5069 6374 7572 6520 626f Picture bo\n+0001f970: 782e 2020 2020 2020 2020 2028 6c69 6e65 x. (line\n+0001f980: 2020 3130 290a 2a20 6269 6e61 7279 2066 10).* binary f\n+0001f990: 6f72 6d3a 2020 2020 2020 2020 2020 2020 orm: \n+0001f9a0: 2020 2020 2020 2020 2020 2020 2020 2054 T\n+0001f9b0: 776f 2d6f 7065 7261 6e64 2028 6269 6e61 wo-operand (bina\n+0001f9c0: 7279 2920 666f 726d 732e 0a20 2020 2020 ry) forms.. \n 0001f9d0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001f9e0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001f9f0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001fa00: 2020 2020 2028 6c69 6e65 2020 2036 290a (line 6).\n-0001fa10: 2a20 6269 6e61 7279 2073 7061 6365 2070 * binary space p\n-0001fa20: 6172 7469 7469 6f6e 3a20 2020 2020 2020 artition: \n-0001fa30: 2020 2020 2020 2020 2048 6964 6465 6e20 Hidden \n-0001fa40: 7375 7266 6163 6520 7265 6d6f 7661 6c2e surface removal.\n-0001fa50: 0a20 2020 2020 2020 2020 2020 2020 2020 . \n+0001fa00: 2020 2020 2020 2020 2028 6c69 6e65 2020 (line \n+0001fa10: 2036 290a 2a20 6269 6e61 7279 2073 7061 6).* binary spa\n+0001fa20: 6365 2070 6172 7469 7469 6f6e 3a20 2020 ce partition: \n+0001fa30: 2020 2020 2020 2020 2020 2020 2048 6964 Hid\n+0001fa40: 6465 6e20 7375 7266 6163 6520 7265 6d6f den surface remo\n+0001fa50: 7661 6c2e 0a20 2020 2020 2020 2020 2020 val.. \n 0001fa60: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001fa70: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001fa80: 2020 2020 2020 2020 2020 2020 2020 2028 (\n-0001fa90: 6c69 6e65 2020 3233 290a 2a20 6269 6e61 line 23).* bina\n-0001faa0: 7279 2073 7061 6365 2070 6172 7469 7469 ry space partiti\n-0001fab0: 6f6e 203c 313e 3a20 2020 2020 2020 2020 on <1>: \n-0001fac0: 2020 2053 7461 7469 7374 6963 732e 2020 Statistics. \n-0001fad0: 2020 2020 2020 2020 286c 696e 6520 2031 (line 1\n-0001fae0: 3329 0a2a 2062 696e 6172 7920 7370 6163 3).* binary spac\n-0001faf0: 6520 7061 7274 6974 696f 6e20 3c32 3e3a e partition <2>:\n-0001fb00: 2020 2020 2020 2020 2020 2020 436f 6d6d Comm\n-0001fb10: 616e 6420 6c69 6e65 2e20 2020 2020 2020 and line. \n-0001fb20: 2028 6c69 6e65 2020 3236 290a 2a20 626c (line 26).* bl\n-0001fb30: 6f63 6b3a 2020 2020 2020 2020 2020 2020 ock: \n+0001fa80: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001fa90: 2020 2028 6c69 6e65 2020 3233 290a 2a20 (line 23).* \n+0001faa0: 6269 6e61 7279 2073 7061 6365 2070 6172 binary space par\n+0001fab0: 7469 7469 6f6e 203c 313e 3a20 2020 2020 tition <1>: \n+0001fac0: 2020 2020 2020 2053 7461 7469 7374 6963 Statistic\n+0001fad0: 732e 2020 2020 2020 2020 2020 286c 696e s. (lin\n+0001fae0: 6520 2031 3329 0a2a 2062 696e 6172 7920 e 13).* binary \n+0001faf0: 7370 6163 6520 7061 7274 6974 696f 6e20 space partition \n+0001fb00: 3c32 3e3a 2020 2020 2020 2020 2020 2020 <2>: \n+0001fb10: 436f 6d6d 616e 6420 6c69 6e65 2e20 2020 Command line. \n+0001fb20: 2020 2020 2028 6c69 6e65 2020 3236 290a (line 26).\n+0001fb30: 2a20 626c 6f63 6b3a 2020 2020 2020 2020 * block: \n 0001fb40: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001fb50: 2020 2020 204f 626a 6563 7420 7472 616e Object tran\n-0001fb60: 7366 6f72 6d73 2e20 2020 286c 696e 6520 sforms. (line \n-0001fb70: 2032 3429 0a2a 2062 6c6f 636b 203c 313e 24).* block <1>\n-0001fb80: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n-0001fb90: 2020 2020 2020 2020 2020 2020 2020 5377 Sw\n-0001fba0: 6570 7420 626c 6f63 6b73 2e20 2020 2020 ept blocks. \n-0001fbb0: 2020 2028 6c69 6e65 2020 2037 290a 2a20 (line 7).* \n-0001fbc0: 626c 6f63 6b20 3c32 3e3a 2020 2020 2020 block <2>: \n+0001fb50: 2020 2020 2020 2020 204f 626a 6563 7420 Object \n+0001fb60: 7472 616e 7366 6f72 6d73 2e20 2020 286c transforms. (l\n+0001fb70: 696e 6520 2032 3429 0a2a 2062 6c6f 636b ine 24).* block\n+0001fb80: 203c 313e 3a20 2020 2020 2020 2020 2020 <1>: \n+0001fb90: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001fba0: 2020 5377 6570 7420 626c 6f63 6b73 2e20 Swept blocks. \n+0001fbb0: 2020 2020 2020 2028 6c69 6e65 2020 2037 (line 7\n+0001fbc0: 290a 2a20 626c 6f63 6b20 3c32 3e3a 2020 ).* block <2>: \n 0001fbd0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001fbe0: 2020 2020 2020 2042 6c6f 636b 732e 2020 Blocks. \n-0001fbf0: 2020 2020 2020 2020 2020 2020 286c 696e (lin\n-0001fc00: 6520 2020 3629 0a2a 2062 6c6f 636b 2073 e 6).* block s\n-0001fc10: 7765 6570 3a20 2020 2020 2020 2020 2020 weep: \n+0001fbe0: 2020 2020 2020 2020 2020 2042 6c6f 636b Block\n+0001fbf0: 732e 2020 2020 2020 2020 2020 2020 2020 s. \n+0001fc00: 286c 696e 6520 2020 3629 0a2a 2062 6c6f (line 6).* blo\n+0001fc10: 636b 2073 7765 6570 3a20 2020 2020 2020 ck sweep: \n 0001fc20: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001fc30: 5377 6570 7420 626c 6f63 6b73 2e20 2020 Swept blocks. \n-0001fc40: 2020 2020 2028 6c69 6e65 2020 2036 290a (line 6).\n-0001fc50: 2a20 626f 6479 2070 6f6c 7967 6f6e 3a20 * body polygon: \n-0001fc60: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001fc70: 2020 2020 2020 2020 2049 6e74 6572 6e61 Interna\n-0001fc80: 6c20 6f70 7469 6f6e 732e 2020 2020 286c l options. (l\n-0001fc90: 696e 6520 2031 3629 0a2a 2062 6f64 7920 ine 16).* body \n-0001fca0: 706f 6c79 676f 6e20 3c31 3e3a 2020 2020 polygon <1>: \n+0001fc30: 2020 2020 5377 6570 7420 626c 6f63 6b73 Swept blocks\n+0001fc40: 2e20 2020 2020 2020 2028 6c69 6e65 2020 . (line \n+0001fc50: 2036 290a 2a20 626f 6479 2070 6f6c 7967 6).* body polyg\n+0001fc60: 6f6e 3a20 2020 2020 2020 2020 2020 2020 on: \n+0001fc70: 2020 2020 2020 2020 2020 2020 2049 6e74 Int\n+0001fc80: 6572 6e61 6c20 6f70 7469 6f6e 732e 2020 ernal options. \n+0001fc90: 2020 286c 696e 6520 2031 3629 0a2a 2062 (line 16).* b\n+0001fca0: 6f64 7920 706f 6c79 676f 6e20 3c31 3e3a ody polygon <1>:\n 0001fcb0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001fcc0: 2020 5377 6570 7420 6c69 6e65 732e 2020 Swept lines. \n-0001fcd0: 2020 2020 2020 2028 6c69 6e65 2020 3130 (line 10\n-0001fce0: 290a 2a20 626f 6479 2070 6f6c 7967 6f6e ).* body polygon\n-0001fcf0: 203c 323e 3a20 2020 2020 2020 2020 2020 <2>: \n-0001fd00: 2020 2020 2020 2020 2020 2053 7765 7074 Swept\n-0001fd10: 2070 6f6c 7967 6f6e 732e 2020 2020 2020 polygons. \n-0001fd20: 286c 696e 6520 2020 3929 0a2a 2062 6f75 (line 9).* bou\n-0001fd30: 6e64 696e 6720 626f 783a 2020 2020 2020 nding box: \n+0001fcc0: 2020 2020 2020 5377 6570 7420 6c69 6e65 Swept line\n+0001fcd0: 732e 2020 2020 2020 2020 2028 6c69 6e65 s. (line\n+0001fce0: 2020 3130 290a 2a20 626f 6479 2070 6f6c 10).* body pol\n+0001fcf0: 7967 6f6e 203c 323e 3a20 2020 2020 2020 ygon <2>: \n+0001fd00: 2020 2020 2020 2020 2020 2020 2020 2053 S\n+0001fd10: 7765 7074 2070 6f6c 7967 6f6e 732e 2020 wept polygons. \n+0001fd20: 2020 2020 286c 696e 6520 2020 3929 0a2a (line 9).*\n+0001fd30: 2062 6f75 6e64 696e 6720 626f 783a 2020 bounding box: \n 0001fd40: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001fd50: 2020 2020 5069 6374 7572 6520 626f 782e Picture box.\n-0001fd60: 2020 2020 2020 2020 2028 6c69 6e65 2020 (line \n-0001fd70: 3135 290a 2a20 4253 502c 2062 696e 6172 15).* BSP, binar\n-0001fd80: 7920 7370 6163 6520 7061 7274 6974 696f y space partitio\n-0001fd90: 6e3a 2020 2020 2020 2020 2020 2048 6964 n: Hid\n-0001fda0: 6465 6e20 7375 7266 6163 6520 7265 6d6f den surface remo\n-0001fdb0: 7661 6c2e 0a20 2020 2020 2020 2020 2020 val.. \n+0001fd50: 2020 2020 2020 2020 5069 6374 7572 6520 Picture \n+0001fd60: 626f 782e 2020 2020 2020 2020 2028 6c69 box. (li\n+0001fd70: 6e65 2020 3135 290a 2a20 4253 502c 2062 ne 15).* BSP, b\n+0001fd80: 696e 6172 7920 7370 6163 6520 7061 7274 inary space part\n+0001fd90: 6974 696f 6e3a 2020 2020 2020 2020 2020 ition: \n+0001fda0: 2048 6964 6465 6e20 7375 7266 6163 6520 Hidden surface \n+0001fdb0: 7265 6d6f 7661 6c2e 0a20 2020 2020 2020 removal.. \n 0001fdc0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001fdd0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 0001fde0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001fdf0: 2020 2028 6c69 6e65 2020 3233 290a 2a20 (line 23).* \n-0001fe00: 4253 502c 2062 696e 6172 7920 7370 6163 BSP, binary spac\n-0001fe10: 6520 7061 7274 6974 696f 6e20 3c31 3e3a e partition <1>:\n-0001fe20: 2020 2020 2020 2043 6f6d 6d61 6e64 206c Command l\n-0001fe30: 696e 652e 2020 2020 2020 2020 286c 696e ine. (lin\n-0001fe40: 6520 2032 3629 0a2a 2063 616d 6572 613a e 26).* camera:\n-0001fe50: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001fdf0: 2020 2020 2020 2028 6c69 6e65 2020 3233 (line 23\n+0001fe00: 290a 2a20 4253 502c 2062 696e 6172 7920 ).* BSP, binary \n+0001fe10: 7370 6163 6520 7061 7274 6974 696f 6e20 space partition \n+0001fe20: 3c31 3e3a 2020 2020 2020 2043 6f6d 6d61 <1>: Comma\n+0001fe30: 6e64 206c 696e 652e 2020 2020 2020 2020 nd line. \n+0001fe40: 286c 696e 6520 2032 3629 0a2a 2063 616d (line 26).* cam\n+0001fe50: 6572 613a 2020 2020 2020 2020 2020 2020 era: \n 0001fe60: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001fe70: 4361 6d65 7261 2e20 2020 2020 2020 2020 Camera. \n-0001fe80: 2020 2020 2028 6c69 6e65 2020 2036 290a (line 6).\n-0001fe90: 2a20 6365 6e74 6572 206f 6620 726f 7461 * center of rota\n-0001fea0: 7469 6f6e 3a20 2020 2020 2020 2020 2020 tion: \n-0001feb0: 2020 2020 2020 2020 204f 626a 6563 7420 Object \n-0001fec0: 7472 616e 7366 6f72 6d73 2e20 2020 286c transforms. (l\n-0001fed0: 696e 6520 2020 3829 0a2a 2063 656e 7465 ine 8).* cente\n-0001fee0: 7220 6f66 2072 6f74 6174 696f 6e20 3c31 r of rotation <1\n-0001fef0: 3e3a 2020 2020 2020 2020 2020 2020 2020 >: \n-0001ff00: 2020 4e65 7374 6564 2073 7765 6570 732e Nested sweeps.\n-0001ff10: 2020 2020 2020 2028 6c69 6e65 2020 2036 (line 6\n-0001ff20: 290a 2a20 6365 6e74 726f 6964 3a20 2020 ).* centroid: \n-0001ff30: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001ff40: 2020 2020 2020 2020 2020 2041 2074 6563 A tec\n-0001ff50: 686e 6963 616c 2064 7261 7769 6e67 2e20 hnical drawing. \n-0001ff60: 286c 696e 6520 2036 3429 0a2a 2063 656e (line 64).* cen\n-0001ff70: 7472 6f69 6420 3c31 3e3a 2020 2020 2020 troid <1>: \n+0001fe70: 2020 2020 4361 6d65 7261 2e20 2020 2020 Camera. \n+0001fe80: 2020 2020 2020 2020 2028 6c69 6e65 2020 (line \n+0001fe90: 2036 290a 2a20 6365 6e74 6572 206f 6620 6).* center of \n+0001fea0: 726f 7461 7469 6f6e 3a20 2020 2020 2020 rotation: \n+0001feb0: 2020 2020 2020 2020 2020 2020 204f 626a Obj\n+0001fec0: 6563 7420 7472 616e 7366 6f72 6d73 2e20 ect transforms. \n+0001fed0: 2020 286c 696e 6520 2020 3829 0a2a 2063 (line 8).* c\n+0001fee0: 656e 7465 7220 6f66 2072 6f74 6174 696f enter of rotatio\n+0001fef0: 6e20 3c31 3e3a 2020 2020 2020 2020 2020 n <1>: \n+0001ff00: 2020 2020 2020 4e65 7374 6564 2073 7765 Nested swe\n+0001ff10: 6570 732e 2020 2020 2020 2028 6c69 6e65 eps. (line\n+0001ff20: 2020 2036 290a 2a20 6365 6e74 726f 6964 6).* centroid\n+0001ff30: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+0001ff40: 2020 2020 2020 2020 2020 2020 2020 2041 A\n+0001ff50: 2074 6563 686e 6963 616c 2064 7261 7769 technical drawi\n+0001ff60: 6e67 2e20 286c 696e 6520 2036 3429 0a2a ng. (line 64).*\n+0001ff70: 2063 656e 7472 6f69 6420 3c31 3e3a 2020 centroid <1>: \n 0001ff80: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001ff90: 2020 2020 4120 7465 6368 6e69 6361 6c20 A technical \n-0001ffa0: 6472 6177 696e 672e 2028 6c69 6e65 2031 drawing. (line 1\n-0001ffb0: 3032 290a 2a20 636c 6970 7069 6e67 3a20 02).* clipping: \n-0001ffc0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-0001ffd0: 2020 2020 2020 2020 2020 2020 2050 6963 Pic\n-0001ffe0: 7475 7265 2062 6f78 2e20 2020 2020 2020 ture box. \n-0001fff0: 2020 286c 696e 6520 2031 3729 0a2a 2063 (line 17).* c\n-00020000: 6c69 7070 696e 6720 3c31 3e3a 2020 2020 lipping <1>: \n+0001ff90: 2020 2020 2020 2020 4120 7465 6368 6e69 A techni\n+0001ffa0: 6361 6c20 6472 6177 696e 672e 2028 6c69 cal drawing. (li\n+0001ffb0: 6e65 2031 3032 290a 2a20 636c 6970 7069 ne 102).* clippi\n+0001ffc0: 6e67 3a20 2020 2020 2020 2020 2020 2020 ng: \n+0001ffd0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+0001ffe0: 2050 6963 7475 7265 2062 6f78 2e20 2020 Picture box. \n+0001fff0: 2020 2020 2020 286c 696e 6520 2031 3729 (line 17)\n+00020000: 0a2a 2063 6c69 7070 696e 6720 3c31 3e3a .* clipping <1>:\n 00020010: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00020020: 2020 2020 2020 436c 6970 7069 6e67 2e20 Clipping. \n-00020030: 2020 2020 2020 2020 2020 2028 6c69 6e65 (line\n-00020040: 2020 2036 290a 2a20 636c 6f73 7572 6520 6).* closure \n-00020050: 7461 672c 203c 3e3a 2020 2020 2020 2020 tag, <>: \n-00020060: 2020 2020 2020 2020 2020 2020 2020 2050 P\n-00020070: 6f69 6e74 2073 7765 6570 732e 2020 2020 oint sweeps. \n-00020080: 2020 2020 286c 696e 6520 2031 3829 0a2a (line 18).*\n-00020090: 2063 6c6f 7375 7265 2074 6167 2c20 3c3e closure tag, <>\n-000200a0: 203c 313e 3a20 2020 2020 2020 2020 2020 <1>: \n-000200b0: 2020 2020 2020 2020 506f 6c79 6c69 6e65 Polyline\n-000200c0: 2073 7765 6570 7320 7769 7468 2063 6c6f sweeps with clo\n-000200d0: 7375 7265 2e0a 2020 2020 2020 2020 2020 sure.. \n+00020020: 2020 2020 2020 2020 2020 436c 6970 7069 Clippi\n+00020030: 6e67 2e20 2020 2020 2020 2020 2020 2028 ng. (\n+00020040: 6c69 6e65 2020 2036 290a 2a20 636c 6f73 line 6).* clos\n+00020050: 7572 6520 7461 672c 203c 3e3a 2020 2020 ure tag, <>: \n+00020060: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00020070: 2020 2050 6f69 6e74 2073 7765 6570 732e Point sweeps.\n+00020080: 2020 2020 2020 2020 286c 696e 6520 2031 (line 1\n+00020090: 3829 0a2a 2063 6c6f 7375 7265 2074 6167 8).* closure tag\n+000200a0: 2c20 3c3e 203c 313e 3a20 2020 2020 2020 , <> <1>: \n+000200b0: 2020 2020 2020 2020 2020 2020 506f 6c79 Poly\n+000200c0: 6c69 6e65 2073 7765 6570 7320 7769 7468 line sweeps with\n+000200d0: 2063 6c6f 7375 7265 2e0a 2020 2020 2020 closure.. \n 000200e0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 000200f0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00020100: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00020110: 2020 2020 286c 696e 6520 2020 3629 0a2a (line 6).*\n-00020120: 2063 6c6f 7375 7265 2074 6167 2c20 3c3e closure tag, <>\n-00020130: 203c 323e 3a20 2020 2020 2020 2020 2020 <2>: \n-00020140: 2020 2020 2020 2020 5377 6565 7073 2e20 Sweeps. \n-00020150: 2020 2020 2020 2020 2020 2020 2028 6c69 (li\n-00020160: 6e65 2020 3139 290a 2a20 636c 6f73 7572 ne 19).* closur\n-00020170: 6520 7461 672c 203c 3e20 3c33 3e3a 2020 e tag, <> <3>: \n-00020180: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00020190: 2053 7765 7074 2070 6f69 6e74 732e 2020 Swept points. \n-000201a0: 2020 2020 2020 286c 696e 6520 2020 3629 (line 6)\n-000201b0: 0a2a 2063 6c6f 7375 7265 2074 6167 2c20 .* closure tag, \n-000201c0: 3c3e 203c 343e 3a20 2020 2020 2020 2020 <> <4>: \n-000201d0: 2020 2020 2020 2020 2020 5377 6570 7420 Swept \n-000201e0: 706f 696e 7473 2e20 2020 2020 2020 2028 points. 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(line \n+000201b0: 2020 3629 0a2a 2063 6c6f 7375 7265 2074 6).* closure t\n+000201c0: 6167 2c20 3c3e 203c 343e 3a20 2020 2020 ag, <> <4>: \n+000201d0: 2020 2020 2020 2020 2020 2020 2020 5377 Sw\n+000201e0: 6570 7420 706f 696e 7473 2e20 2020 2020 ept points. \n+000201f0: 2020 2028 6c69 6e65 2020 3133 290a 2a20 (line 13).* \n+00020200: 636c 6f73 7572 6520 7461 672c 203c 3e20 closure tag, <> \n+00020210: 3c35 3e3a 2020 2020 2020 2020 2020 2020 <5>: \n+00020220: 2020 2020 2020 2053 7765 7074 206c 696e Swept lin\n+00020230: 6573 2e20 2020 2020 2020 2020 286c 696e es. 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(li\n-00020430: 6e65 2020 2036 290a 2a20 636f 6d6d 656e ne 6).* commen\n-00020440: 7473 3a20 2020 2020 2020 2020 2020 2020 ts: \n+00020300: 2020 2020 2020 2020 2020 2020 2028 6c69 (li\n+00020310: 6e65 2020 3131 290a 2a20 636f 6d6d 616e ne 11).* comman\n+00020320: 6420 6c69 6e65 206f 7074 696f 6e3a 2020 d line option: \n+00020330: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00020340: 2048 656c 6c6f 2077 6f72 6c64 2e20 2020 Hello world. \n+00020350: 2020 2020 2020 286c 696e 6520 2033 3029 (line 30)\n+00020360: 0a2a 2063 6f6d 6d61 6e64 206c 696e 6520 .* command line \n+00020370: 6f70 7469 6f6e 733a 2020 2020 2020 2020 options: \n+00020380: 2020 2020 2020 2020 2020 436f 6d6d 616e Comman\n+00020390: 6420 6c69 6e65 2e20 2020 2020 2020 2028 d line. (\n+000203a0: 6c69 6e65 2020 3134 290a 2a20 636f 6d6d line 14).* comm\n+000203b0: 616e 6420 6c69 6e65 2c20 736b 6574 6368 and line, sketch\n+000203c0: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+000203d0: 2020 2048 656c 6c6f 2077 6f72 6c64 2e20 Hello world. \n+000203e0: 2020 2020 2020 2020 286c 696e 6520 2020 (line \n+000203f0: 3929 0a2a 2063 6f6d 6d61 6e64 206c 696e 9).* command lin\n+00020400: 652c 2073 6b65 7463 6820 3c31 3e3a 2020 e, sketch <1>: \n+00020410: 2020 2020 2020 2020 2020 2020 436f 6d6d Comm\n+00020420: 616e 6420 6c69 6e65 2e20 2020 2020 2020 and line. \n+00020430: 2028 6c69 6e65 2020 2036 290a 2a20 636f (line 6).* co\n+00020440: 6d6d 656e 7473 3a20 2020 2020 2020 2020 mments: \n 00020450: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00020460: 2044 7261 7769 6e67 2061 2073 6f6c 6964 Drawing a solid\n-00020470: 2e20 2020 2020 286c 696e 6520 2032 3829 . (line 28)\n-00020480: 0a2a 2063 6f6d 6d65 6e74 7320 3c31 3e3a .* comments <1>:\n-00020490: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000204a0: 2020 2020 2020 2020 2020 4c61 6e67 7561 Langua\n-000204b0: 6765 2062 6173 6963 732e 2020 2020 2028 ge basics. (\n-000204c0: 6c69 6e65 2020 3138 290a 2a20 636f 6e73 line 18).* cons\n-000204d0: 7472 7563 746f 723a 2020 2020 2020 2020 tructor: \n+00020460: 2020 2020 2044 7261 7769 6e67 2061 2073 Drawing a s\n+00020470: 6f6c 6964 2e20 2020 2020 286c 696e 6520 olid. (line \n+00020480: 2032 3829 0a2a 2063 6f6d 6d65 6e74 7320 28).* comments \n+00020490: 3c31 3e3a 2020 2020 2020 2020 2020 2020 <1>: \n+000204a0: 2020 2020 2020 2020 2020 2020 2020 4c61 La\n+000204b0: 6e67 7561 6765 2062 6173 6963 732e 2020 nguage basics. \n+000204c0: 2020 2028 6c69 6e65 2020 3138 290a 2a20 (line 18).* \n+000204d0: 636f 6e73 7472 7563 746f 723a 2020 2020 constructor: \n 000204e0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000204f0: 2020 2054 7261 6e73 666f 726d 206c 6974 Transform lit\n-00020500: 6572 616c 732e 2020 286c 696e 6520 2020 erals. 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(l\n-00020670: 696e 6520 2033 3529 0a2a 2063 756c 6c69 ine 35).* culli\n-00020680: 6e67 203c 323e 3a20 2020 2020 2020 2020 ng <2>: \n+00020610: 2020 2020 4865 6c6c 6f20 776f 726c 642e Hello world.\n+00020620: 2020 2020 2020 2020 2028 6c69 6e65 2020 (line \n+00020630: 3433 290a 2a20 6375 6c6c 696e 6720 3c31 43).* culling <1\n+00020640: 3e3a 2020 2020 2020 2020 2020 2020 2020 >: \n+00020650: 2020 2020 2020 2020 2020 2020 2044 7261 Dra\n+00020660: 7769 6e67 2061 2073 6f6c 6964 2e20 2020 wing a solid. \n+00020670: 2020 286c 696e 6520 2033 3529 0a2a 2063 (line 35).* c\n+00020680: 756c 6c69 6e67 203c 323e 3a20 2020 2020 ulling <2>: \n 00020690: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000206a0: 2020 4472 6177 696e 6720 6120 736f 6c69 Drawing a soli\n-000206b0: 642e 2020 2020 2028 6c69 6e65 2020 3431 d. (line 41\n-000206c0: 290a 2a20 6375 6c6c 696e 6720 3c33 3e3a ).* culling <3>:\n-000206d0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000206e0: 2020 2020 2020 2020 2020 2053 7765 7074 Swept\n-000206f0: 206c 696e 6573 2e20 2020 2020 2020 2020 lines. \n-00020700: 286c 696e 6520 2031 3629 0a2a 2064 6563 (line 16).* dec\n-00020710: 6c61 7261 7469 7665 206c 616e 6775 6167 larative languag\n-00020720: 653a 2020 2020 2020 2020 2020 2020 2020 e: \n-00020730: 2020 2020 4c61 6e67 7561 6765 2062 6173 Language bas\n-00020740: 6963 732e 2020 2020 2028 6c69 6e65 2020 ics. 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(line 22)\n+000207a0: 0a2a 2064 6566 696e 6974 696f 6e20 3c31 .* definition <1\n+000207b0: 3e3a 2020 2020 2020 2020 2020 2020 2020 >: \n+000207c0: 2020 2020 2020 2020 2020 4465 6669 6e69 Defini\n+000207d0: 7469 6f6e 732e 2020 2020 2020 2020 2028 tions. 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(lin\n-000208b0: 6520 2032 3729 0a2a 2064 6566 696e 6974 e 27).* definit\n-000208c0: 696f 6e2c 206f 7074 696f 6e73 3a20 2020 ion, options: \n-000208d0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000208e0: 506f 6c79 6c69 6e65 2073 7765 6570 7320 Polyline sweeps \n-000208f0: 7769 7468 2063 6c6f 7375 7265 2e0a 2020 with closure.. \n-00020900: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00020860: 2020 2020 2020 2028 6c69 6e65 2020 2036 (line 6\n+00020870: 290a 2a20 6465 6669 6e69 7469 6f6e 2c20 ).* definition, \n+00020880: 6472 6177 6162 6c65 3a20 2020 2020 2020 drawable: \n+00020890: 2020 2020 2020 2020 2020 204f 626a 6563 Objec\n+000208a0: 7420 7472 616e 7366 6f72 6d73 2e20 2020 t transforms. \n+000208b0: 286c 696e 6520 2032 3729 0a2a 2064 6566 (line 27).* def\n+000208c0: 696e 6974 696f 6e2c 206f 7074 696f 6e73 inition, options\n+000208d0: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+000208e0: 2020 2020 506f 6c79 6c69 6e65 2073 7765 Polyline swe\n+000208f0: 6570 7320 7769 7468 2063 6c6f 7375 7265 eps with closure\n+00020900: 2e0a 2020 2020 2020 2020 2020 2020 2020 .. \n 00020910: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00020920: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00020930: 2020 2020 2020 2020 2020 2020 286c 696e (lin\n-00020940: 6520 2032 3329 0a2a 2064 6566 696e 6974 e 23).* definit\n-00020950: 696f 6e2c 2070 6f69 6e74 3a20 2020 2020 ion, point: \n+00020930: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00020940: 286c 696e 6520 2032 3329 0a2a 2064 6566 (line 23).* def\n+00020950: 696e 6974 696f 6e2c 2070 6f69 6e74 3a20 inition, point: \n 00020960: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00020970: 4472 6177 696e 6720 6120 736f 6c69 642e Drawing a solid.\n-00020980: 2020 2020 2028 6c69 6e65 2020 3233 290a (line 23).\n-00020990: 2a20 6465 6669 6e69 7469 6f6e 2c20 7363 * definition, sc\n-000209a0: 616c 6172 3a20 2020 2020 2020 2020 2020 alar: \n-000209b0: 2020 2020 2020 2020 2053 7765 7074 206f Swept o\n-000209c0: 626a 6563 7473 2e20 2020 2020 2020 286c bjects. 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(line \n+00020990: 3233 290a 2a20 6465 6669 6e69 7469 6f6e 23).* definition\n+000209a0: 2c20 7363 616c 6172 3a20 2020 2020 2020 , scalar: \n+000209b0: 2020 2020 2020 2020 2020 2020 2053 7765 Swe\n+000209c0: 7074 206f 626a 6563 7473 2e20 2020 2020 pt objects. \n+000209d0: 2020 286c 696e 6520 2031 3329 0a2a 2064 (line 13).* d\n+000209e0: 6566 696e 6974 696f 6e2c 2073 696d 706c efinition, simpl\n+000209f0: 653a 2020 2020 2020 2020 2020 2020 2020 e: \n+00020a00: 2020 2020 2020 466f 726d 7320 6f66 2064 Forms of d\n+00020a10: 6566 696e 6974 696f 6e73 2e0a 2020 2020 efinitions.. \n 00020a20: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00020a30: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00020a40: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00020a50: 2020 2020 2020 286c 696e 6520 2020 3629 (line 6)\n-00020a60: 0a2a 2064 6566 696e 6974 696f 6e2c 2074 .* definition, t\n-00020a70: 6167 3a20 2020 2020 2020 2020 2020 2020 ag: \n-00020a80: 2020 2020 2020 2020 2020 4465 6669 6e69 Defini\n-00020a90: 7469 6f6e 732e 2020 2020 2020 2020 2028 tions. (\n-00020aa0: 6c69 6e65 2020 3138 290a 2a20 6465 6669 line 18).* defi\n-00020ab0: 6e69 7469 6f6e 2c20 7461 6720 3c31 3e3a nition, tag <1>:\n-00020ac0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00020ad0: 2020 2046 6f72 6d73 206f 6620 6465 6669 Forms of defi\n-00020ae0: 6e69 7469 6f6e 732e 0a20 2020 2020 2020 nitions.. \n+00020a50: 2020 2020 2020 2020 2020 286c 696e 6520 (line \n+00020a60: 2020 3629 0a2a 2064 6566 696e 6974 696f 6).* definitio\n+00020a70: 6e2c 2074 6167 3a20 2020 2020 2020 2020 n, tag: \n+00020a80: 2020 2020 2020 2020 2020 2020 2020 4465 De\n+00020a90: 6669 6e69 7469 6f6e 732e 2020 2020 2020 finitions. \n+00020aa0: 2020 2028 6c69 6e65 2020 3138 290a 2a20 (line 18).* \n+00020ab0: 6465 6669 6e69 7469 6f6e 2c20 7461 6720 definition, tag \n+00020ac0: 3c31 3e3a 2020 2020 2020 2020 2020 2020 <1>: \n+00020ad0: 2020 2020 2020 2046 6f72 6d73 206f 6620 Forms of \n+00020ae0: 6465 6669 6e69 7469 6f6e 732e 0a20 2020 definitions.. \n 00020af0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00020b00: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00020b10: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00020b20: 2020 2020 2020 2028 6c69 6e65 2020 2037 (line 7\n-00020b30: 290a 2a20 6465 6669 6e69 7469 6f6e 2c20 ).* definition, \n-00020b40: 7461 6720 3c32 3e3a 2020 2020 2020 2020 tag <2>: \n-00020b50: 2020 2020 2020 2020 2020 2043 6f6d 6d61 Comma\n-00020b60: 6e64 206c 696e 652e 2020 2020 2020 2020 nd line. \n-00020b70: 286c 696e 6520 2035 3329 0a2a 2064 6566 (line 53).* def\n-00020b80: 696e 6974 696f 6e2c 2074 7261 6e73 666f inition, transfo\n-00020b90: 726d 3a20 2020 2020 2020 2020 2020 2020 rm: \n-00020ba0: 2020 2020 4f76 6572 7669 6577 2e20 2020 Overview. \n-00020bb0: 2020 2020 2020 2020 2028 6c69 6e65 2020 (line \n-00020bc0: 3335 290a 2a20 6465 6669 6e69 7469 6f6e 35).* definition\n-00020bd0: 2c20 7665 6374 6f72 3a20 2020 2020 2020 , vector: \n-00020be0: 2020 2020 2020 2020 2020 2020 2050 6f6c Pol\n-00020bf0: 796c 696e 6520 7377 6565 7073 2e20 2020 yline sweeps. \n-00020c00: 2020 286c 696e 6520 2031 3929 0a2a 2064 (line 19).* d\n-00020c10: 6570 7468 2073 6f72 743a 2020 2020 2020 epth sort: \n+00020b20: 2020 2020 2020 2020 2020 2028 6c69 6e65 (line\n+00020b30: 2020 2037 290a 2a20 6465 6669 6e69 7469 7).* definiti\n+00020b40: 6f6e 2c20 7461 6720 3c32 3e3a 2020 2020 on, tag <2>: \n+00020b50: 2020 2020 2020 2020 2020 2020 2020 2043 C\n+00020b60: 6f6d 6d61 6e64 206c 696e 652e 2020 2020 ommand line. \n+00020b70: 2020 2020 286c 696e 6520 2035 3329 0a2a (line 53).*\n+00020b80: 2064 6566 696e 6974 696f 6e2c 2074 7261 definition, tra\n+00020b90: 6e73 666f 726d 3a20 2020 2020 2020 2020 nsform: \n+00020ba0: 2020 2020 2020 2020 4f76 6572 7669 6577 Overview\n+00020bb0: 2e20 2020 2020 2020 2020 2020 2028 6c69 . 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(\n+00020c50: 6c69 6e65 2020 3133 290a 2a20 6465 7074 line 13).* dept\n+00020c60: 6820 736f 7274 203c 313e 3a20 2020 2020 h sort <1>: \n+00020c70: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00020c80: 2020 2048 6964 6465 6e20 7375 7266 6163 Hidden surfac\n+00020c90: 6520 7265 6d6f 7661 6c2e 0a20 2020 2020 e removal.. \n 00020ca0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00020cb0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00020cc0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00020cd0: 2020 2020 2028 6c69 6e65 2020 2036 290a (line 6).\n-00020ce0: 2a20 6465 7074 6820 736f 7274 203c 323e * depth sort <2>\n-00020cf0: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n-00020d00: 2020 2020 2020 2020 2042 7567 7320 616e Bugs an\n-00020d10: 6420 616e 6f6d 616c 6965 732e 2020 286c d anomalies. (l\n-00020d20: 696e 6520 2020 3829 0a2a 2064 6972 6563 ine 8).* direc\n-00020d30: 7420 7472 616e 7366 6f72 6d3a 2020 2020 t transform: \n+00020cd0: 2020 2020 2020 2020 2028 6c69 6e65 2020 (line \n+00020ce0: 2036 290a 2a20 6465 7074 6820 736f 7274 6).* depth sort\n+00020cf0: 203c 323e 3a20 2020 2020 2020 2020 2020 <2>: \n+00020d00: 2020 2020 2020 2020 2020 2020 2042 7567 Bug\n+00020d10: 7320 616e 6420 616e 6f6d 616c 6965 732e s and anomalies.\n+00020d20: 2020 286c 696e 6520 2020 3829 0a2a 2064 (line 8).* d\n+00020d30: 6972 6563 7420 7472 616e 7366 6f72 6d3a irect transform:\n 00020d40: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00020d50: 2020 5472 616e 7366 6f72 6d20 6c69 7465 Transform lite\n-00020d60: 7261 6c73 2e20 2028 6c69 6e65 2020 3437 rals. 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(line\n+00020d70: 2020 3437 290a 2a20 6469 7265 6374 2074 47).* direct t\n+00020d80: 7261 6e73 666f 726d 203c 313e 3a20 2020 ransform <1>: \n+00020d90: 2020 2020 2020 2020 2020 2020 2020 2054 T\n+00020da0: 7261 6e73 666f 726d 206c 6974 6572 616c ransform literal\n+00020db0: 732e 2020 286c 696e 6520 2035 3329 0a2a s. (line 53).*\n+00020dc0: 2064 6f63 756d 656e 7420 7465 6d70 6c61 document templa\n+00020dd0: 7465 3a20 2020 2020 2020 2020 2020 2020 te: \n+00020de0: 2020 2020 2020 2020 4865 6c6c 6f20 776f Hello wo\n+00020df0: 726c 642e 2020 2020 2020 2020 2028 6c69 rld. 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(\n+00020e90: 6c69 6e65 2020 3237 290a 2a20 6472 6177 line 27).* draw\n+00020ea0: 6162 6c65 203c 313e 3a20 2020 2020 2020 able <1>: \n+00020eb0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00020ec0: 2020 2044 7261 7761 626c 6573 2e20 2020 Drawables. \n+00020ed0: 2020 2020 2020 2020 286c 696e 6520 2020 (line \n+00020ee0: 3629 0a2a 2064 7261 7761 626c 6520 6465 6).* drawable de\n+00020ef0: 6669 6e69 7469 6f6e 3a20 2020 2020 2020 finition: \n+00020f00: 2020 2020 2020 2020 2020 2020 4f62 6a65 Obje\n+00020f10: 6374 2074 7261 6e73 666f 726d 732e 2020 ct transforms. \n+00020f20: 2028 6c69 6e65 2020 3237 290a 2a20 6472 (line 27).* dr\n+00020f30: 6177 6162 6c65 2072 6566 6572 656e 6365 awable reference\n+00020f40: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+00020f50: 2020 2020 2046 6f72 6d73 206f 6620 7265 Forms of re\n+00020f60: 6665 7265 6e63 6573 2e20 286c 696e 6520 ferences. 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(lin\n+00021000: 6520 2031 3529 0a2a 2066 6163 6573 3a20 e 15).* faces: \n 00021010: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00021020: 2020 2020 2020 2020 2020 2020 4472 6177 Draw\n-00021030: 696e 6720 6120 736f 6c69 642e 2020 2020 ing a solid. \n-00021040: 2028 6c69 6e65 2020 2037 290a 2a20 6661 (line 7).* fa\n-00021050: 6365 7320 3c31 3e3a 2020 2020 2020 2020 ces <1>: \n+00021020: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00021030: 4472 6177 696e 6720 6120 736f 6c69 642e Drawing a solid.\n+00021040: 2020 2020 2028 6c69 6e65 2020 2037 290a (line 7).\n+00021050: 2a20 6661 6365 7320 3c31 3e3a 2020 2020 * faces <1>: \n 00021060: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00021070: 2020 2020 2050 6f6c 796c 696e 6520 7377 Polyline sw\n-00021080: 6565 7073 2e20 2020 2020 286c 696e 6520 eeps. 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(lin\n-00021120: 6520 2020 3629 0a2a 2066 7261 6d65 2062 e 6).* frame b\n-00021130: 6f78 3a20 2020 2020 2020 2020 2020 2020 ox: \n+00021100: 2020 2020 2020 2020 2020 204c 616e 6775 Langu\n+00021110: 6167 6520 6261 7369 6373 2e20 2020 2020 age basics. \n+00021120: 286c 696e 6520 2020 3629 0a2a 2066 7261 (line 6).* fra\n+00021130: 6d65 2062 6f78 3a20 2020 2020 2020 2020 me box: \n 00021140: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00021150: 4672 616d 652e 2020 2020 2020 2020 2020 Frame. \n-00021160: 2020 2020 2028 6c69 6e65 2020 2036 290a (line 6).\n-00021170: 2a20 676c 6f62 616c 206f 7074 696f 6e73 * global options\n-00021180: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n-00021190: 2020 2020 2020 2020 2044 7261 7769 6e67 Drawing\n-000211a0: 206f 7074 696f 6e73 2e20 2020 2020 286c options. (l\n-000211b0: 696e 6520 2031 3829 0a2a 2067 6c6f 6261 ine 18).* globa\n-000211c0: 6c20 6f70 7469 6f6e 7320 3c31 3e3a 2020 l options <1>: \n-000211d0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000211e0: 2020 476c 6f62 616c 206f 7074 696f 6e73 Global options\n-000211f0: 2e20 2020 2020 2028 6c69 6e65 2020 2036 . (line 6\n-00021200: 290a 2a20 6865 6c69 783a 2020 2020 2020 ).* helix: \n+00021150: 2020 2020 4672 616d 652e 2020 2020 2020 Frame. \n+00021160: 2020 2020 2020 2020 2028 6c69 6e65 2020 (line \n+00021170: 2036 290a 2a20 676c 6f62 616c 206f 7074 6).* global opt\n+00021180: 696f 6e73 3a20 2020 2020 2020 2020 2020 ions: \n+00021190: 2020 2020 2020 2020 2020 2020 2044 7261 Dra\n+000211a0: 7769 6e67 206f 7074 696f 6e73 2e20 2020 wing options. \n+000211b0: 2020 286c 696e 6520 2031 3829 0a2a 2067 (line 18).* g\n+000211c0: 6c6f 6261 6c20 6f70 7469 6f6e 7320 3c31 lobal options <1\n+000211d0: 3e3a 2020 2020 2020 2020 2020 2020 2020 >: \n+000211e0: 2020 2020 2020 476c 6f62 616c 206f 7074 Global opt\n+000211f0: 696f 6e73 2e20 2020 2020 2028 6c69 6e65 ions. (line\n+00021200: 2020 2036 290a 2a20 6865 6c69 783a 2020 6).* helix: \n 00021210: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00021220: 2020 2020 2020 2020 2020 2044 7261 7769 Drawi\n-00021230: 6e67 2061 2073 6f6c 6964 2e20 2020 2020 ng a solid. \n-00021240: 286c 696e 6520 2034 3729 0a2a 2068 656c (line 47).* hel\n-00021250: 6978 203c 313e 3a20 2020 2020 2020 2020 ix <1>: \n+00021220: 2020 2020 2020 2020 2020 2020 2020 2044 D\n+00021230: 7261 7769 6e67 2061 2073 6f6c 6964 2e20 rawing a solid. \n+00021240: 2020 2020 286c 696e 6520 2034 3729 0a2a (line 47).*\n+00021250: 2068 656c 6978 203c 313e 3a20 2020 2020 helix <1>: \n 00021260: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00021270: 2020 2020 506f 6c79 6c69 6e65 2073 7765 Polyline swe\n-00021280: 6570 732e 2020 2020 2028 6c69 6e65 2020 eps. (line \n-00021290: 3135 290a 2a20 6865 6c6c 6f20 776f 726c 15).* hello worl\n-000212a0: 643a 2020 2020 2020 2020 2020 2020 2020 d: \n-000212b0: 2020 2020 2020 2020 2020 2020 2048 656c Hel\n-000212c0: 6c6f 2077 6f72 6c64 2e20 2020 2020 2020 lo world. \n-000212d0: 2020 286c 696e 6520 2020 3729 0a2a 2068 (line 7).* h\n-000212e0: 6964 6465 6e20 7375 7266 6163 6520 616c idden surface al\n-000212f0: 676f 7269 7468 6d3a 2020 2020 2020 2020 gorithm: \n-00021300: 2020 2020 2020 4865 6c6c 6f20 776f 726c Hello worl\n-00021310: 642e 2020 2020 2020 2020 2028 6c69 6e65 d. (line\n-00021320: 2020 3139 290a 2a20 6869 6464 656e 2073 19).* hidden s\n-00021330: 7572 6661 6365 2061 6c67 6f72 6974 686d urface algorithm\n-00021340: 203c 313e 3a20 2020 2020 2020 2020 2053 <1>: S\n-00021350: 7065 6369 616c 206f 626a 6563 7473 2e20 pecial objects. \n-00021360: 2020 2020 286c 696e 6520 2034 3829 0a2a (line 48).*\n-00021370: 2068 6964 6465 6e20 7375 7266 6163 6520 hidden surface \n-00021380: 616c 676f 7269 7468 6d20 3c32 3e3a 2020 algorithm <2>: \n-00021390: 2020 2020 2020 2020 5472 616e 7366 6f72 Transfor\n-000213a0: 6d20 6c69 7465 7261 6c73 2e20 2028 6c69 m literals. (li\n-000213b0: 6e65 2020 3234 290a 2a20 6869 6464 656e ne 24).* hidden\n-000213c0: 2073 7572 6661 6365 2061 6c67 6f72 6974 surface algorit\n-000213d0: 686d 203c 333e 3a20 2020 2020 2020 2020 hm <3>: \n-000213e0: 2041 2074 6563 686e 6963 616c 2064 7261 A technical dra\n-000213f0: 7769 6e67 2e20 286c 696e 6520 2032 3529 wing. (line 25)\n-00021400: 0a2a 2068 6964 6465 6e20 7375 7266 6163 .* hidden surfac\n-00021410: 6520 616c 676f 7269 7468 6d20 3c34 3e3a e algorithm <4>:\n-00021420: 2020 2020 2020 2020 2020 4869 6464 656e Hidden\n-00021430: 2073 7572 6661 6365 2072 656d 6f76 616c surface removal\n-00021440: 2e0a 2020 2020 2020 2020 2020 2020 2020 .. \n+00021270: 2020 2020 2020 2020 506f 6c79 6c69 6e65 Polyline\n+00021280: 2073 7765 6570 732e 2020 2020 2028 6c69 sweeps. (li\n+00021290: 6e65 2020 3135 290a 2a20 6865 6c6c 6f20 ne 15).* hello \n+000212a0: 776f 726c 643a 2020 2020 2020 2020 2020 world: \n+000212b0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+000212c0: 2048 656c 6c6f 2077 6f72 6c64 2e20 2020 Hello world. \n+000212d0: 2020 2020 2020 286c 696e 6520 2020 3729 (line 7)\n+000212e0: 0a2a 2068 6964 6465 6e20 7375 7266 6163 .* hidden surfac\n+000212f0: 6520 616c 676f 7269 7468 6d3a 2020 2020 e algorithm: \n+00021300: 2020 2020 2020 2020 2020 4865 6c6c 6f20 Hello \n+00021310: 776f 726c 642e 2020 2020 2020 2020 2028 world. (\n+00021320: 6c69 6e65 2020 3139 290a 2a20 6869 6464 line 19).* hidd\n+00021330: 656e 2073 7572 6661 6365 2061 6c67 6f72 en surface algor\n+00021340: 6974 686d 203c 313e 3a20 2020 2020 2020 ithm <1>: \n+00021350: 2020 2053 7065 6369 616c 206f 626a 6563 Special objec\n+00021360: 7473 2e20 2020 2020 286c 696e 6520 2034 ts. (line 4\n+00021370: 3829 0a2a 2068 6964 6465 6e20 7375 7266 8).* hidden surf\n+00021380: 6163 6520 616c 676f 7269 7468 6d20 3c32 ace algorithm <2\n+00021390: 3e3a 2020 2020 2020 2020 2020 5472 616e >: Tran\n+000213a0: 7366 6f72 6d20 6c69 7465 7261 6c73 2e20 sform literals. \n+000213b0: 2028 6c69 6e65 2020 3234 290a 2a20 6869 (line 24).* hi\n+000213c0: 6464 656e 2073 7572 6661 6365 2061 6c67 dden surface alg\n+000213d0: 6f72 6974 686d 203c 333e 3a20 2020 2020 orithm <3>: \n+000213e0: 2020 2020 2041 2074 6563 686e 6963 616c A technical\n+000213f0: 2064 7261 7769 6e67 2e20 286c 696e 6520 drawing. (line \n+00021400: 2032 3529 0a2a 2068 6964 6465 6e20 7375 25).* hidden su\n+00021410: 7266 6163 6520 616c 676f 7269 7468 6d20 rface algorithm \n+00021420: 3c34 3e3a 2020 2020 2020 2020 2020 4869 <4>: Hi\n+00021430: 6464 656e 2073 7572 6661 6365 2072 656d dden surface rem\n+00021440: 6f76 616c 2e0a 2020 2020 2020 2020 2020 oval.. \n 00021450: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00021460: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00021470: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00021480: 286c 696e 6520 2020 3629 0a2a 2068 6965 (line 6).* hie\n-00021490: 7261 7263 6869 6361 6c20 6d6f 6465 6c3a rarchical model:\n-000214a0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000214b0: 2020 2020 4120 6869 6572 6172 6368 6963 A hierarchic\n-000214c0: 616c 206d 6f64 656c 2e0a 2020 2020 2020 al model.. \n+00021480: 2020 2020 286c 696e 6520 2020 3629 0a2a (line 6).*\n+00021490: 2068 6965 7261 7263 6869 6361 6c20 6d6f hierarchical mo\n+000214a0: 6465 6c3a 2020 2020 2020 2020 2020 2020 del: \n+000214b0: 2020 2020 2020 2020 4120 6869 6572 6172 A hierar\n+000214c0: 6368 6963 616c 206d 6f64 656c 2e0a 2020 chical model.. \n 000214d0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 000214e0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 000214f0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00021500: 2020 2020 2020 2020 286c 696e 6520 2020 (line \n-00021510: 3629 0a2a 2069 6465 6e74 6966 6965 7273 6).* identifiers\n-00021520: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n-00021530: 2020 2020 2020 2020 2020 2020 4964 656e Iden\n-00021540: 7469 6669 6572 732e 2020 2020 2020 2020 tifiers. \n-00021550: 2028 6c69 6e65 2020 2037 290a 2a20 696e (line 7).* in\n-00021560: 636c 7564 6520 6669 6c65 3a20 2020 2020 clude file: \n+00021500: 2020 2020 2020 2020 2020 2020 286c 696e (lin\n+00021510: 6520 2020 3629 0a2a 2069 6465 6e74 6966 e 6).* identif\n+00021520: 6965 7273 3a20 2020 2020 2020 2020 2020 iers: \n+00021530: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00021540: 4964 656e 7469 6669 6572 732e 2020 2020 Identifiers. \n+00021550: 2020 2020 2028 6c69 6e65 2020 2037 290a (line 7).\n+00021560: 2a20 696e 636c 7564 6520 6669 6c65 3a20 * include file: \n 00021570: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00021580: 2020 2020 204c 616e 6775 6167 6520 6261 Language ba\n-00021590: 7369 6373 2e20 2020 2020 286c 696e 6520 sics. (line \n-000215a0: 2031 3929 0a2a 2069 6e70 7574 2066 696c 19).* input fil\n-000215b0: 653a 2020 2020 2020 2020 2020 2020 2020 e: \n-000215c0: 2020 2020 2020 2020 2020 2020 2020 4c61 La\n-000215d0: 6e67 7561 6765 2062 6173 6963 732e 2020 nguage basics. \n-000215e0: 2020 2028 6c69 6e65 2020 2036 290a 2a20 (line 6).* \n-000215f0: 696e 7465 726e 616c 206f 7074 696f 6e3a internal option:\n-00021600: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00021610: 2020 2020 2020 2053 7065 6369 616c 206f Special o\n-00021620: 626a 6563 7473 2e20 2020 2020 286c 696e bjects. (lin\n-00021630: 6520 2034 3729 0a2a 2069 6e74 6572 6e61 e 47).* interna\n-00021640: 6c20 6f70 7469 6f6e 733a 2020 2020 2020 l options: \n+00021580: 2020 2020 2020 2020 204c 616e 6775 6167 Languag\n+00021590: 6520 6261 7369 6373 2e20 2020 2020 286c e basics. (l\n+000215a0: 696e 6520 2031 3929 0a2a 2069 6e70 7574 ine 19).* input\n+000215b0: 2066 696c 653a 2020 2020 2020 2020 2020 file: \n+000215c0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+000215d0: 2020 4c61 6e67 7561 6765 2062 6173 6963 Language basic\n+000215e0: 732e 2020 2020 2028 6c69 6e65 2020 2036 s. (line 6\n+000215f0: 290a 2a20 696e 7465 726e 616c 206f 7074 ).* internal opt\n+00021600: 696f 6e3a 2020 2020 2020 2020 2020 2020 ion: \n+00021610: 2020 2020 2020 2020 2020 2053 7065 6369 Speci\n+00021620: 616c 206f 626a 6563 7473 2e20 2020 2020 al objects. \n+00021630: 286c 696e 6520 2034 3729 0a2a 2069 6e74 (line 47).* int\n+00021640: 6572 6e61 6c20 6f70 7469 6f6e 733a 2020 ernal options: \n 00021650: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00021660: 496e 7465 726e 616c 206f 7074 696f 6e73 Internal options\n-00021670: 2e20 2020 2028 6c69 6e65 2020 2036 290a . (line 6).\n-00021680: 2a20 696e 7465 726e 616c 206f 7074 696f * internal optio\n-00021690: 6e73 203c 313e 3a20 2020 2020 2020 2020 ns <1>: \n-000216a0: 2020 2020 2020 2020 2047 6c6f 6261 6c20 Global \n-000216b0: 6f70 7469 6f6e 732e 2020 2020 2020 286c options. (l\n-000216c0: 696e 6520 2031 3229 0a2a 206b 6579 776f ine 12).* keywo\n-000216d0: 7264 733a 2020 2020 2020 2020 2020 2020 rds: \n+00021660: 2020 2020 496e 7465 726e 616c 206f 7074 Internal opt\n+00021670: 696f 6e73 2e20 2020 2028 6c69 6e65 2020 ions. (line \n+00021680: 2036 290a 2a20 696e 7465 726e 616c 206f 6).* internal o\n+00021690: 7074 696f 6e73 203c 313e 3a20 2020 2020 ptions <1>: \n+000216a0: 2020 2020 2020 2020 2020 2020 2047 6c6f Glo\n+000216b0: 6261 6c20 6f70 7469 6f6e 732e 2020 2020 bal options. \n+000216c0: 2020 286c 696e 6520 2031 3229 0a2a 206b (line 12).* k\n+000216d0: 6579 776f 7264 733a 2020 2020 2020 2020 eywords: \n 000216e0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000216f0: 2020 4b65 7920 616e 6420 7265 7365 7276 Key and reserv\n-00021700: 6564 2077 6f72 6473 2e0a 2020 2020 2020 ed words.. \n+000216f0: 2020 2020 2020 4b65 7920 616e 6420 7265 Key and re\n+00021700: 7365 7276 6564 2077 6f72 6473 2e0a 2020 served words.. \n 00021710: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00021720: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00021730: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00021740: 2020 2020 2020 2020 286c 696e 6520 2020 (line \n-00021750: 3629 0a2a 206c 6162 656c 733a 2020 2020 6).* labels: \n+00021740: 2020 2020 2020 2020 2020 2020 286c 696e (lin\n+00021750: 6520 2020 3629 0a2a 206c 6162 656c 733a e 6).* labels:\n 00021760: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00021770: 2020 2020 2020 2020 2020 2020 5370 6563 Spec\n-00021780: 6961 6c20 6f62 6a65 6374 732e 2020 2020 ial objects. \n-00021790: 2028 6c69 6e65 2020 2036 290a 2a20 6c61 (line 6).* la\n-000217a0: 6e67 7561 6765 2c20 6465 636c 6172 6174 nguage, declarat\n-000217b0: 6976 653a 2020 2020 2020 2020 2020 2020 ive: \n-000217c0: 2020 2020 204c 616e 6775 6167 6520 6261 Language ba\n-000217d0: 7369 6373 2e20 2020 2020 286c 696e 6520 sics. (line \n-000217e0: 2020 3829 0a2a 206c 616e 6775 6167 652c 8).* language,\n-000217f0: 206f 7574 7075 743a 2020 2020 2020 2020 output: \n-00021800: 2020 2020 2020 2020 2020 2020 2020 4c61 La\n-00021810: 6e67 7561 6765 2e20 2020 2020 2020 2020 nguage. \n-00021820: 2020 2028 6c69 6e65 2020 2036 290a 2a20 (line 6).* \n-00021830: 6c61 6e67 7561 6765 2c20 7363 656e 6520 language, scene \n-00021840: 6465 7363 7269 7074 696f 6e3a 2020 2020 description: \n-00021850: 2020 2020 2020 204c 616e 6775 6167 6520 Language \n-00021860: 6261 7369 6373 2e20 2020 2020 286c 696e basics. (lin\n-00021870: 6520 2020 3829 0a2a 206c 6578 6963 616c e 8).* lexical\n-00021880: 2073 636f 7065 3a20 2020 2020 2020 2020 scope: \n+00021770: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00021780: 5370 6563 6961 6c20 6f62 6a65 6374 732e Special objects.\n+00021790: 2020 2020 2028 6c69 6e65 2020 2036 290a (line 6).\n+000217a0: 2a20 6c61 6e67 7561 6765 2c20 6465 636c * language, decl\n+000217b0: 6172 6174 6976 653a 2020 2020 2020 2020 arative: \n+000217c0: 2020 2020 2020 2020 204c 616e 6775 6167 Languag\n+000217d0: 6520 6261 7369 6373 2e20 2020 2020 286c e basics. (l\n+000217e0: 696e 6520 2020 3829 0a2a 206c 616e 6775 ine 8).* langu\n+000217f0: 6167 652c 206f 7574 7075 743a 2020 2020 age, output: \n+00021800: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00021810: 2020 4c61 6e67 7561 6765 2e20 2020 2020 Language. \n+00021820: 2020 2020 2020 2028 6c69 6e65 2020 2036 (line 6\n+00021830: 290a 2a20 6c61 6e67 7561 6765 2c20 7363 ).* language, sc\n+00021840: 656e 6520 6465 7363 7269 7074 696f 6e3a ene description:\n+00021850: 2020 2020 2020 2020 2020 204c 616e 6775 Langu\n+00021860: 6167 6520 6261 7369 6373 2e20 2020 2020 age basics. \n+00021870: 286c 696e 6520 2020 3829 0a2a 206c 6578 (line 8).* lex\n+00021880: 6963 616c 2073 636f 7065 3a20 2020 2020 ical scope: \n 00021890: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000218a0: 4f62 6a65 6374 2074 7261 6e73 666f 726d Object transform\n-000218b0: 732e 2020 2028 6c69 6e65 2020 3236 290a s. (line 26).\n-000218c0: 2a20 6c65 7869 6361 6c20 7363 6f70 6520 * lexical scope \n-000218d0: 3c31 3e3a 2020 2020 2020 2020 2020 2020 <1>: \n-000218e0: 2020 2020 2020 2020 2042 6c6f 636b 732e Blocks.\n-000218f0: 2020 2020 2020 2020 2020 2020 2020 286c (l\n-00021900: 696e 6520 2031 3329 0a2a 206c 696e 6520 ine 13).* line \n-00021910: 7377 6565 703a 2020 2020 2020 2020 2020 sweep: \n+000218a0: 2020 2020 4f62 6a65 6374 2074 7261 6e73 Object trans\n+000218b0: 666f 726d 732e 2020 2028 6c69 6e65 2020 forms. (line \n+000218c0: 3236 290a 2a20 6c65 7869 6361 6c20 7363 26).* lexical sc\n+000218d0: 6f70 6520 3c31 3e3a 2020 2020 2020 2020 ope <1>: \n+000218e0: 2020 2020 2020 2020 2020 2020 2042 6c6f Blo\n+000218f0: 636b 732e 2020 2020 2020 2020 2020 2020 cks. \n+00021900: 2020 286c 696e 6520 2031 3329 0a2a 206c (line 13).* l\n+00021910: 696e 6520 7377 6565 703a 2020 2020 2020 ine sweep: \n 00021920: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00021930: 2020 506f 6c79 6c69 6e65 2073 7765 6570 Polyline sweep\n-00021940: 732e 2020 2020 2028 6c69 6e65 2020 2036 s. (line 6\n-00021950: 290a 2a20 6c69 6e65 2073 7765 6570 203c ).* line sweep <\n-00021960: 313e 3a20 2020 2020 2020 2020 2020 2020 1>: \n-00021970: 2020 2020 2020 2020 2020 2053 7765 7074 Swept\n-00021980: 206c 696e 6573 2e20 2020 2020 2020 2020 lines. \n-00021990: 286c 696e 6520 2020 3629 0a2a 206c 696e (line 6).* lin\n-000219a0: 6520 7377 6565 7020 3c32 3e3a 2020 2020 e sweep <2>: \n+00021930: 2020 2020 2020 506f 6c79 6c69 6e65 2073 Polyline s\n+00021940: 7765 6570 732e 2020 2020 2028 6c69 6e65 weeps. (line\n+00021950: 2020 2036 290a 2a20 6c69 6e65 2073 7765 6).* line swe\n+00021960: 6570 203c 313e 3a20 2020 2020 2020 2020 ep <1>: \n+00021970: 2020 2020 2020 2020 2020 2020 2020 2053 S\n+00021980: 7765 7074 206c 696e 6573 2e20 2020 2020 wept lines. \n+00021990: 2020 2020 286c 696e 6520 2020 3629 0a2a (line 6).*\n+000219a0: 206c 696e 6520 7377 6565 7020 3c32 3e3a line sweep <2>:\n 000219b0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000219c0: 2020 2020 4120 7465 6368 6e69 6361 6c20 A technical \n-000219d0: 6472 6177 696e 672e 2028 6c69 6e65 2020 drawing. (line \n-000219e0: 3230 290a 2a20 6c69 7465 7261 6c2c 2070 20).* literal, p\n-000219f0: 6f69 6e74 3a20 2020 2020 2020 2020 2020 oint: \n-00021a00: 2020 2020 2020 2020 2020 2020 2050 6f69 Poi\n-00021a10: 6e74 2061 6e64 2076 6563 746f 7220 6c69 nt and vector li\n-00021a20: 7465 7261 6c73 2e0a 2020 2020 2020 2020 terals.. \n+000219c0: 2020 2020 2020 2020 4120 7465 6368 6e69 A techni\n+000219d0: 6361 6c20 6472 6177 696e 672e 2028 6c69 cal drawing. (li\n+000219e0: 6e65 2020 3230 290a 2a20 6c69 7465 7261 ne 20).* litera\n+000219f0: 6c2c 2070 6f69 6e74 3a20 2020 2020 2020 l, point: \n+00021a00: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00021a10: 2050 6f69 6e74 2061 6e64 2076 6563 746f Point and vecto\n+00021a20: 7220 6c69 7465 7261 6c73 2e0a 2020 2020 r literals.. \n 00021a30: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00021a40: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00021a50: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00021a60: 2020 2020 2020 286c 696e 6520 2020 3629 (line 6)\n-00021a70: 0a2a 206c 6974 6572 616c 2c20 7363 616c .* literal, scal\n-00021a80: 6172 3a20 2020 2020 2020 2020 2020 2020 ar: \n-00021a90: 2020 2020 2020 2020 2020 5363 616c 6172 Scalar\n-00021aa0: 206c 6974 6572 616c 732e 2020 2020 2028 literals. (\n-00021ab0: 6c69 6e65 2020 2036 290a 2a20 6c69 7465 line 6).* lite\n-00021ac0: 7261 6c2c 2074 7261 6e73 666f 726d 3a20 ral, transform: \n-00021ad0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00021ae0: 2020 2054 7261 6e73 666f 726d 206c 6974 Transform lit\n-00021af0: 6572 616c 732e 2020 286c 696e 6520 2020 erals. (line \n-00021b00: 3629 0a2a 206c 6974 6572 616c 2c20 7665 6).* literal, ve\n-00021b10: 6374 6f72 3a20 2020 2020 2020 2020 2020 ctor: \n-00021b20: 2020 2020 2020 2020 2020 2020 506f 696e Poin\n-00021b30: 7420 616e 6420 7665 6374 6f72 206c 6974 t and vector lit\n-00021b40: 6572 616c 732e 0a20 2020 2020 2020 2020 erals.. \n+00021a60: 2020 2020 2020 2020 2020 286c 696e 6520 (line \n+00021a70: 2020 3629 0a2a 206c 6974 6572 616c 2c20 6).* literal, \n+00021a80: 7363 616c 6172 3a20 2020 2020 2020 2020 scalar: \n+00021a90: 2020 2020 2020 2020 2020 2020 2020 5363 Sc\n+00021aa0: 616c 6172 206c 6974 6572 616c 732e 2020 alar literals. \n+00021ab0: 2020 2028 6c69 6e65 2020 2036 290a 2a20 (line 6).* \n+00021ac0: 6c69 7465 7261 6c2c 2074 7261 6e73 666f literal, transfo\n+00021ad0: 726d 3a20 2020 2020 2020 2020 2020 2020 rm: \n+00021ae0: 2020 2020 2020 2054 7261 6e73 666f 726d Transform\n+00021af0: 206c 6974 6572 616c 732e 2020 286c 696e literals. (lin\n+00021b00: 6520 2020 3629 0a2a 206c 6974 6572 616c e 6).* literal\n+00021b10: 2c20 7665 6374 6f72 3a20 2020 2020 2020 , vector: \n+00021b20: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00021b30: 506f 696e 7420 616e 6420 7665 6374 6f72 Point and vector\n+00021b40: 206c 6974 6572 616c 732e 0a20 2020 2020 literals.. \n 00021b50: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00021b60: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00021b70: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00021b80: 2020 2020 2028 6c69 6e65 2020 2036 290a (line 6).\n-00021b90: 2a20 6e65 7374 696e 672c 2073 7765 7074 * nesting, swept\n-00021ba0: 206f 626a 6563 743a 2020 2020 2020 2020 object: \n-00021bb0: 2020 2020 2020 2020 204e 6573 7465 6420 Nested \n-00021bc0: 7377 6565 7073 2e20 2020 2020 2020 286c sweeps. (l\n-00021bd0: 696e 6520 2032 3229 0a2a 206f 6e65 2d6f ine 22).* one-o\n-00021be0: 7065 7261 6e64 2066 6f72 6d3a 2020 2020 perand form: \n+00021b80: 2020 2020 2020 2020 2028 6c69 6e65 2020 (line \n+00021b90: 2036 290a 2a20 6e65 7374 696e 672c 2073 6).* nesting, s\n+00021ba0: 7765 7074 206f 626a 6563 743a 2020 2020 wept object: \n+00021bb0: 2020 2020 2020 2020 2020 2020 204e 6573 Nes\n+00021bc0: 7465 6420 7377 6565 7073 2e20 2020 2020 ted sweeps. \n+00021bd0: 2020 286c 696e 6520 2032 3229 0a2a 206f (line 22).* o\n+00021be0: 6e65 2d6f 7065 7261 6e64 2066 6f72 6d3a ne-operand form:\n 00021bf0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00021c00: 2020 556e 6172 7920 666f 726d 732e 2020 Unary forms. \n-00021c10: 2020 2020 2020 2028 6c69 6e65 2020 2036 (line 6\n-00021c20: 290a 2a20 6f70 6572 6174 6f72 2061 7373 ).* operator ass\n-00021c30: 6f63 6961 7469 7669 7479 3a20 2020 2020 ociativity: \n-00021c40: 2020 2020 2020 2020 2020 2054 776f 2d6f Two-o\n-00021c50: 7065 7261 6e64 2028 6269 6e61 7279 2920 perand (binary) \n-00021c60: 666f 726d 732e 0a20 2020 2020 2020 2020 forms.. \n+00021c00: 2020 2020 2020 556e 6172 7920 666f 726d Unary form\n+00021c10: 732e 2020 2020 2020 2020 2028 6c69 6e65 s. (line\n+00021c20: 2020 2036 290a 2a20 6f70 6572 6174 6f72 6).* operator\n+00021c30: 2061 7373 6f63 6961 7469 7669 7479 3a20 associativity: \n+00021c40: 2020 2020 2020 2020 2020 2020 2020 2054 T\n+00021c50: 776f 2d6f 7065 7261 6e64 2028 6269 6e61 wo-operand (bina\n+00021c60: 7279 2920 666f 726d 732e 0a20 2020 2020 ry) forms.. \n 00021c70: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00021c80: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00021c90: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00021ca0: 2020 2020 2028 6c69 6e65 2020 3538 290a (line 58).\n-00021cb0: 2a20 6f70 6572 6174 6f72 2070 7265 6365 * operator prece\n-00021cc0: 6465 6e63 653a 2020 2020 2020 2020 2020 dence: \n-00021cd0: 2020 2020 2020 2020 2054 776f 2d6f 7065 Two-ope\n-00021ce0: 7261 6e64 2028 6269 6e61 7279 2920 666f rand (binary) fo\n-00021cf0: 726d 732e 0a20 2020 2020 2020 2020 2020 rms.. \n+00021ca0: 2020 2020 2020 2020 2028 6c69 6e65 2020 (line \n+00021cb0: 3538 290a 2a20 6f70 6572 6174 6f72 2070 58).* operator p\n+00021cc0: 7265 6365 6465 6e63 653a 2020 2020 2020 recedence: \n+00021cd0: 2020 2020 2020 2020 2020 2020 2054 776f Two\n+00021ce0: 2d6f 7065 7261 6e64 2028 6269 6e61 7279 -operand (binary\n+00021cf0: 2920 666f 726d 732e 0a20 2020 2020 2020 ) forms.. \n 00021d00: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00021d10: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00021d20: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00021d30: 2020 2028 6c69 6e65 2020 3435 290a 2a20 (line 45).* \n-00021d40: 6f70 7469 6f6e 3a20 2020 2020 2020 2020 option: \n+00021d30: 2020 2020 2020 2028 6c69 6e65 2020 3435 (line 45\n+00021d40: 290a 2a20 6f70 7469 6f6e 3a20 2020 2020 ).* option: \n 00021d50: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00021d60: 2020 2020 2020 2044 7261 7769 6e67 206f Drawing o\n-00021d70: 7074 696f 6e73 2e20 2020 2020 286c 696e ptions. (lin\n-00021d80: 6520 2020 3629 0a2a 206f 7074 696f 6e2c e 6).* option,\n-00021d90: 2063 6f6d 6d61 6e64 206c 696e 653a 2020 command line: \n-00021da0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00021db0: 4865 6c6c 6f20 776f 726c 642e 2020 2020 Hello world. \n-00021dc0: 2020 2020 2028 6c69 6e65 2020 3330 290a (line 30).\n-00021dd0: 2a20 6f70 7469 6f6e 2c20 696e 7465 726e * option, intern\n-00021de0: 616c 3a20 2020 2020 2020 2020 2020 2020 al: \n-00021df0: 2020 2020 2020 2020 2053 7065 6369 616c Special\n-00021e00: 206f 626a 6563 7473 2e20 2020 2020 286c objects. (l\n-00021e10: 696e 6520 2034 3729 0a2a 206f 7074 696f ine 47).* optio\n-00021e20: 6e73 3a20 2020 2020 2020 2020 2020 2020 ns: \n+00021d60: 2020 2020 2020 2020 2020 2044 7261 7769 Drawi\n+00021d70: 6e67 206f 7074 696f 6e73 2e20 2020 2020 ng options. \n+00021d80: 286c 696e 6520 2020 3629 0a2a 206f 7074 (line 6).* opt\n+00021d90: 696f 6e2c 2063 6f6d 6d61 6e64 206c 696e ion, command lin\n+00021da0: 653a 2020 2020 2020 2020 2020 2020 2020 e: \n+00021db0: 2020 2020 4865 6c6c 6f20 776f 726c 642e Hello world.\n+00021dc0: 2020 2020 2020 2020 2028 6c69 6e65 2020 (line \n+00021dd0: 3330 290a 2a20 6f70 7469 6f6e 2c20 696e 30).* option, in\n+00021de0: 7465 726e 616c 3a20 2020 2020 2020 2020 ternal: \n+00021df0: 2020 2020 2020 2020 2020 2020 2053 7065 Spe\n+00021e00: 6369 616c 206f 626a 6563 7473 2e20 2020 cial objects. \n+00021e10: 2020 286c 696e 6520 2034 3729 0a2a 206f (line 47).* o\n+00021e20: 7074 696f 6e73 3a20 2020 2020 2020 2020 ptions: \n 00021e30: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00021e40: 2020 4f70 7469 6f6e 732e 2020 2020 2020 Options. \n-00021e50: 2020 2020 2020 2028 6c69 6e65 2020 2036 (line 6\n-00021e60: 290a 2a20 6f70 7469 6f6e 7320 6465 6669 ).* options defi\n-00021e70: 6e69 7469 6f6e 3a20 2020 2020 2020 2020 nition: \n-00021e80: 2020 2020 2020 2020 2020 2050 6f6c 796c Polyl\n-00021e90: 696e 6520 7377 6565 7073 2077 6974 6820 ine sweeps with \n-00021ea0: 636c 6f73 7572 652e 0a20 2020 2020 2020 closure.. \n+00021e40: 2020 2020 2020 4f70 7469 6f6e 732e 2020 Options. \n+00021e50: 2020 2020 2020 2020 2020 2028 6c69 6e65 (line\n+00021e60: 2020 2036 290a 2a20 6f70 7469 6f6e 7320 6).* options \n+00021e70: 6465 6669 6e69 7469 6f6e 3a20 2020 2020 definition: \n+00021e80: 2020 2020 2020 2020 2020 2020 2020 2050 P\n+00021e90: 6f6c 796c 696e 6520 7377 6565 7073 2077 olyline sweeps w\n+00021ea0: 6974 6820 636c 6f73 7572 652e 0a20 2020 ith closure.. \n 00021eb0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00021ec0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00021ed0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00021ee0: 2020 2020 2020 2028 6c69 6e65 2020 3233 (line 23\n-00021ef0: 290a 2a20 6f70 7469 6f6e 7320 6d75 6c74 ).* options mult\n-00021f00: 6970 6c65 2072 6566 6572 656e 6365 3a20 iple reference: \n-00021f10: 2020 2020 2020 2020 2020 2046 6f72 6d73 Forms\n-00021f20: 206f 6620 7265 6665 7265 6e63 6573 2e20 of references. \n-00021f30: 286c 696e 6520 2031 3829 0a2a 206f 7074 (line 18).* opt\n-00021f40: 696f 6e73 2072 6566 6572 656e 6365 3a20 ions reference: \n-00021f50: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00021f60: 2020 2020 466f 726d 7320 6f66 2072 6566 Forms of ref\n-00021f70: 6572 656e 6365 732e 2028 6c69 6e65 2020 erences. (line \n-00021f80: 3138 290a 2a20 6f70 7469 6f6e 732c 2063 18).* options, c\n-00021f90: 6f6d 6d61 6e64 206c 696e 653a 2020 2020 ommand line: \n-00021fa0: 2020 2020 2020 2020 2020 2020 2043 6f6d Com\n-00021fb0: 6d61 6e64 206c 696e 652e 2020 2020 2020 mand line. \n-00021fc0: 2020 286c 696e 6520 2031 3429 0a2a 206f (line 14).* o\n-00021fd0: 7074 696f 6e73 2c20 676c 6f62 616c 3a20 ptions, global: \n-00021fe0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00021ff0: 2020 2020 2020 4472 6177 696e 6720 6f70 Drawing op\n-00022000: 7469 6f6e 732e 2020 2020 2028 6c69 6e65 tions. (line\n-00022010: 2020 3138 290a 2a20 6f70 7469 6f6e 732c 18).* options,\n-00022020: 2067 6c6f 6261 6c20 3c31 3e3a 2020 2020 global <1>: \n-00022030: 2020 2020 2020 2020 2020 2020 2020 2047 G\n-00022040: 6c6f 6261 6c20 6f70 7469 6f6e 732e 2020 lobal options. \n-00022050: 2020 2020 286c 696e 6520 2020 3629 0a2a (line 6).*\n-00022060: 206f 7074 696f 6e73 2c20 696e 7465 726e options, intern\n-00022070: 616c 3a20 2020 2020 2020 2020 2020 2020 al: \n-00022080: 2020 2020 2020 2020 496e 7465 726e 616c Internal\n-00022090: 206f 7074 696f 6e73 2e20 2020 2028 6c69 options. (li\n-000220a0: 6e65 2020 2036 290a 2a20 6f70 7469 6f6e ne 6).* option\n-000220b0: 732c 2069 6e74 6572 6e61 6c20 3c31 3e3a s, internal <1>:\n-000220c0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000220d0: 2047 6c6f 6261 6c20 6f70 7469 6f6e 732e Global options.\n-000220e0: 2020 2020 2020 286c 696e 6520 2031 3229 (line 12)\n-000220f0: 0a2a 206f 7074 696f 6e73 2c20 7377 6565 .* options, swee\n-00022100: 703a 2020 2020 2020 2020 2020 2020 2020 p: \n-00022110: 2020 2020 2020 2020 2020 506f 6c79 6c69 Polyli\n-00022120: 6e65 2073 7765 6570 732e 2020 2020 2028 ne sweeps. (\n-00022130: 6c69 6e65 2020 3136 290a 2a20 6f70 7469 line 16).* opti\n-00022140: 6f6e 732c 2073 7765 6570 203c 313e 3a20 ons, sweep <1>: \n-00022150: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00022160: 2020 2050 6f6c 7967 6f6e 2073 7765 6570 Polygon sweep\n-00022170: 732e 2020 2020 2020 286c 696e 6520 2020 s. (line \n-00022180: 3929 0a2a 206f 7074 696f 6e73 2c20 7377 9).* options, sw\n-00022190: 6565 7020 3c32 3e3a 2020 2020 2020 2020 eep <2>: \n-000221a0: 2020 2020 2020 2020 2020 2020 5377 6570 Swep\n-000221b0: 7420 706f 696e 7473 2e20 2020 2020 2020 t points. \n-000221c0: 2028 6c69 6e65 2020 3231 290a 2a20 6f70 (line 21).* op\n-000221d0: 7469 6f6e 732c 2073 7765 6570 203c 333e tions, sweep <3>\n-000221e0: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n-000221f0: 2020 2020 2053 7765 7074 206c 696e 6573 Swept lines\n-00022200: 2e20 2020 2020 2020 2020 286c 696e 6520 . (line \n-00022210: 2032 3529 0a2a 206f 7074 696f 6e73 2c20 25).* options, \n-00022220: 7377 6565 7020 3c34 3e3a 2020 2020 2020 sweep <4>: \n-00022230: 2020 2020 2020 2020 2020 2020 2020 5377 Sw\n-00022240: 6570 7420 6c69 6e65 732e 2020 2020 2020 ept lines. \n-00022250: 2020 2028 6c69 6e65 2020 3438 290a 2a20 (line 48).* \n-00022260: 6f70 7469 6f6e 732c 2073 7765 7074 206f options, swept o\n-00022270: 626a 6563 743a 2020 2020 2020 2020 2020 bject: \n-00022280: 2020 2020 2020 2050 6f6c 7967 6f6e 2073 Polygon s\n-00022290: 7765 6570 732e 2020 2020 2020 286c 696e weeps. (lin\n-000222a0: 6520 2020 3829 0a2a 206f 7074 696f 6e73 e 8).* options\n-000222b0: 2c20 7377 6570 7420 6f62 6a65 6374 203c , swept object <\n-000222c0: 313e 3a20 2020 2020 2020 2020 2020 2020 1>: \n-000222d0: 506f 6c79 6c69 6e65 2073 7765 6570 7320 Polyline sweeps \n-000222e0: 7769 7468 2063 6c6f 7375 7265 2e0a 2020 with closure.. \n-000222f0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00021ee0: 2020 2020 2020 2020 2020 2028 6c69 6e65 (line\n+00021ef0: 2020 3233 290a 2a20 6f70 7469 6f6e 7320 23).* options \n+00021f00: 6d75 6c74 6970 6c65 2072 6566 6572 656e multiple referen\n+00021f10: 6365 3a20 2020 2020 2020 2020 2020 2046 ce: F\n+00021f20: 6f72 6d73 206f 6620 7265 6665 7265 6e63 orms of referenc\n+00021f30: 6573 2e20 286c 696e 6520 2031 3829 0a2a es. (line 18).*\n+00021f40: 206f 7074 696f 6e73 2072 6566 6572 656e options referen\n+00021f50: 6365 3a20 2020 2020 2020 2020 2020 2020 ce: \n+00021f60: 2020 2020 2020 2020 466f 726d 7320 6f66 Forms of\n+00021f70: 2072 6566 6572 656e 6365 732e 2028 6c69 references. (li\n+00021f80: 6e65 2020 3138 290a 2a20 6f70 7469 6f6e ne 18).* option\n+00021f90: 732c 2063 6f6d 6d61 6e64 206c 696e 653a s, command line:\n+00021fa0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00021fb0: 2043 6f6d 6d61 6e64 206c 696e 652e 2020 Command line. \n+00021fc0: 2020 2020 2020 286c 696e 6520 2031 3429 (line 14)\n+00021fd0: 0a2a 206f 7074 696f 6e73 2c20 676c 6f62 .* options, glob\n+00021fe0: 616c 3a20 2020 2020 2020 2020 2020 2020 al: \n+00021ff0: 2020 2020 2020 2020 2020 4472 6177 696e Drawin\n+00022000: 6720 6f70 7469 6f6e 732e 2020 2020 2028 g options. (\n+00022010: 6c69 6e65 2020 3138 290a 2a20 6f70 7469 line 18).* opti\n+00022020: 6f6e 732c 2067 6c6f 6261 6c20 3c31 3e3a ons, global <1>:\n+00022030: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00022040: 2020 2047 6c6f 6261 6c20 6f70 7469 6f6e Global option\n+00022050: 732e 2020 2020 2020 286c 696e 6520 2020 s. (line \n+00022060: 3629 0a2a 206f 7074 696f 6e73 2c20 696e 6).* options, in\n+00022070: 7465 726e 616c 3a20 2020 2020 2020 2020 ternal: \n+00022080: 2020 2020 2020 2020 2020 2020 496e 7465 Inte\n+00022090: 726e 616c 206f 7074 696f 6e73 2e20 2020 rnal options. \n+000220a0: 2028 6c69 6e65 2020 2036 290a 2a20 6f70 (line 6).* op\n+000220b0: 7469 6f6e 732c 2069 6e74 6572 6e61 6c20 tions, internal \n+000220c0: 3c31 3e3a 2020 2020 2020 2020 2020 2020 <1>: \n+000220d0: 2020 2020 2047 6c6f 6261 6c20 6f70 7469 Global opti\n+000220e0: 6f6e 732e 2020 2020 2020 286c 696e 6520 ons. (line \n+000220f0: 2031 3229 0a2a 206f 7074 696f 6e73 2c20 12).* options, \n+00022100: 7377 6565 703a 2020 2020 2020 2020 2020 sweep: \n+00022110: 2020 2020 2020 2020 2020 2020 2020 506f Po\n+00022120: 6c79 6c69 6e65 2073 7765 6570 732e 2020 lyline sweeps. \n+00022130: 2020 2028 6c69 6e65 2020 3136 290a 2a20 (line 16).* \n+00022140: 6f70 7469 6f6e 732c 2073 7765 6570 203c options, sweep <\n+00022150: 313e 3a20 2020 2020 2020 2020 2020 2020 1>: \n+00022160: 2020 2020 2020 2050 6f6c 7967 6f6e 2073 Polygon s\n+00022170: 7765 6570 732e 2020 2020 2020 286c 696e weeps. (lin\n+00022180: 6520 2020 3929 0a2a 206f 7074 696f 6e73 e 9).* options\n+00022190: 2c20 7377 6565 7020 3c32 3e3a 2020 2020 , sweep <2>: \n+000221a0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+000221b0: 5377 6570 7420 706f 696e 7473 2e20 2020 Swept points. \n+000221c0: 2020 2020 2028 6c69 6e65 2020 3231 290a (line 21).\n+000221d0: 2a20 6f70 7469 6f6e 732c 2073 7765 6570 * options, sweep\n+000221e0: 203c 333e 3a20 2020 2020 2020 2020 2020 <3>: \n+000221f0: 2020 2020 2020 2020 2053 7765 7074 206c Swept l\n+00022200: 696e 6573 2e20 2020 2020 2020 2020 286c ines. (l\n+00022210: 696e 6520 2032 3529 0a2a 206f 7074 696f ine 25).* optio\n+00022220: 6e73 2c20 7377 6565 7020 3c34 3e3a 2020 ns, sweep <4>: \n+00022230: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00022240: 2020 5377 6570 7420 6c69 6e65 732e 2020 Swept lines. \n+00022250: 2020 2020 2020 2028 6c69 6e65 2020 3438 (line 48\n+00022260: 290a 2a20 6f70 7469 6f6e 732c 2073 7765 ).* options, swe\n+00022270: 7074 206f 626a 6563 743a 2020 2020 2020 pt object: \n+00022280: 2020 2020 2020 2020 2020 2050 6f6c 7967 Polyg\n+00022290: 6f6e 2073 7765 6570 732e 2020 2020 2020 on sweeps. \n+000222a0: 286c 696e 6520 2020 3829 0a2a 206f 7074 (line 8).* opt\n+000222b0: 696f 6e73 2c20 7377 6570 7420 6f62 6a65 ions, swept obje\n+000222c0: 6374 203c 313e 3a20 2020 2020 2020 2020 ct <1>: \n+000222d0: 2020 2020 506f 6c79 6c69 6e65 2073 7765 Polyline swe\n+000222e0: 6570 7320 7769 7468 2063 6c6f 7375 7265 eps with closure\n+000222f0: 2e0a 2020 2020 2020 2020 2020 2020 2020 .. \n 00022300: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00022310: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00022320: 2020 2020 2020 2020 2020 2020 286c 696e (lin\n-00022330: 6520 2032 3229 0a2a 206f 7074 696f 6e73 e 22).* options\n-00022340: 2c20 7377 6570 7420 6f62 6a65 6374 203c , swept object <\n-00022350: 323e 3a20 2020 2020 2020 2020 2020 2020 2>: \n-00022360: 5377 6570 7420 6c69 6e65 732e 2020 2020 Swept lines. \n-00022370: 2020 2020 2028 6c69 6e65 2020 3236 290a (line 26).\n-00022380: 2a20 6f70 7469 6f6e 732c 2073 7765 7074 * options, swept\n-00022390: 206f 626a 6563 7420 3c33 3e3a 2020 2020 object <3>: \n-000223a0: 2020 2020 2020 2020 2053 7765 7074 206c Swept l\n-000223b0: 696e 6573 2e20 2020 2020 2020 2020 286c ines. (l\n-000223c0: 696e 6520 2034 3829 0a2a 206f 7264 6572 ine 48).* order\n-000223d0: 2c20 706f 6c79 676f 6e20 7665 7274 6578 , polygon vertex\n-000223e0: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n-000223f0: 2020 4865 6c6c 6f20 776f 726c 642e 2020 Hello world. \n-00022400: 2020 2020 2020 2028 6c69 6e65 2020 3339 (line 39\n-00022410: 290a 2a20 6f72 6465 722c 2070 6f6c 7967 ).* order, polyg\n-00022420: 6f6e 2076 6572 7465 7820 3c31 3e3a 2020 on vertex <1>: \n-00022430: 2020 2020 2020 2020 2020 2053 7765 7074 Swept\n-00022440: 206c 696e 6573 2e20 2020 2020 2020 2020 lines. \n-00022450: 286c 696e 6520 2031 3429 0a2a 206f 7264 (line 14).* ord\n-00022460: 6572 2c20 706f 6c79 676f 6e20 7665 7274 er, polygon vert\n-00022470: 6578 203c 323e 3a20 2020 2020 2020 2020 ex <2>: \n-00022480: 2020 2020 5377 6570 7420 6c69 6e65 732e Swept lines.\n-00022490: 2020 2020 2020 2020 2028 6c69 6e65 2020 (line \n-000224a0: 3430 290a 2a20 6f72 6465 722c 2070 6f6c 40).* order, pol\n-000224b0: 7967 6f6e 2076 6572 7465 7820 3c33 3e3a ygon vertex <3>:\n-000224c0: 2020 2020 2020 2020 2020 2020 2053 7765 Swe\n-000224d0: 7074 2070 6f6c 7967 6f6e 732e 2020 2020 pt polygons. \n-000224e0: 2020 286c 696e 6520 2031 3929 0a2a 206f (line 19).* o\n-000224f0: 7264 6572 2c20 706f 6c79 676f 6e20 7665 rder, polygon ve\n-00022500: 7274 6578 203c 343e 3a20 2020 2020 2020 rtex <4>: \n-00022510: 2020 2020 2020 4f76 6572 7669 6577 2e20 Overview. \n-00022520: 2020 2020 2020 2020 2020 2028 6c69 6e65 (line\n-00022530: 2020 3337 290a 2a20 6f72 6465 722c 2070 37).* order, p\n-00022540: 6f6c 7967 6f6e 2076 6572 7465 7820 3c35 olygon vertex <5\n-00022550: 3e3a 2020 2020 2020 2020 2020 2020 204f >: O\n-00022560: 7665 7276 6965 772e 2020 2020 2020 2020 verview. \n-00022570: 2020 2020 286c 696e 6520 2035 3529 0a2a (line 55).*\n-00022580: 206f 7574 7075 7420 6c61 6e67 7561 6765 output language\n-00022590: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n-000225a0: 2020 2020 2020 2020 4c61 6e67 7561 6765 Language\n-000225b0: 2e20 2020 2020 2020 2020 2020 2028 6c69 . (li\n-000225c0: 6e65 2020 2036 290a 2a20 6f75 7473 6964 ne 6).* outsid\n-000225d0: 6520 6f66 2061 2070 6f6c 7967 6f6e 3a20 e of a polygon: \n-000225e0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000225f0: 2048 656c 6c6f 2077 6f72 6c64 2e20 2020 Hello world. \n-00022600: 2020 2020 2020 286c 696e 6520 2033 3729 (line 37)\n-00022610: 0a2a 2070 6172 616c 6c65 6c20 7072 6f6a .* parallel proj\n-00022620: 6563 7469 6f6e 3a20 2020 2020 2020 2020 ection: \n-00022630: 2020 2020 2020 2020 2020 4865 6c6c 6f20 Hello \n-00022640: 776f 726c 642e 2020 2020 2020 2020 2028 world. (\n-00022650: 6c69 6e65 2020 3236 290a 2a20 7061 7261 line 26).* para\n-00022660: 6c6c 656c 2070 726f 6a65 6374 696f 6e20 llel projection \n-00022670: 3c31 3e3a 2020 2020 2020 2020 2020 2020 <1>: \n-00022680: 2020 2054 7261 6e73 666f 726d 206c 6974 Transform lit\n-00022690: 6572 616c 732e 2020 286c 696e 6520 2031 erals. (line 1\n-000226a0: 3829 0a2a 2070 6172 616c 6c65 6c20 7072 8).* parallel pr\n-000226b0: 6f6a 6563 7469 6f6e 203c 323e 3a20 2020 ojection <2>: \n-000226c0: 2020 2020 2020 2020 2020 2020 5069 6374 Pict\n-000226d0: 7572 6520 626f 782e 2020 2020 2020 2020 ure box. \n-000226e0: 2028 6c69 6e65 2020 3135 290a 2a20 7061 (line 15).* pa\n-000226f0: 7265 6e74 6865 7365 733a 2020 2020 2020 rentheses: \n+00022320: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00022330: 286c 696e 6520 2032 3229 0a2a 206f 7074 (line 22).* opt\n+00022340: 696f 6e73 2c20 7377 6570 7420 6f62 6a65 ions, swept obje\n+00022350: 6374 203c 323e 3a20 2020 2020 2020 2020 ct <2>: \n+00022360: 2020 2020 5377 6570 7420 6c69 6e65 732e Swept lines.\n+00022370: 2020 2020 2020 2020 2028 6c69 6e65 2020 (line \n+00022380: 3236 290a 2a20 6f70 7469 6f6e 732c 2073 26).* options, s\n+00022390: 7765 7074 206f 626a 6563 7420 3c33 3e3a wept object <3>:\n+000223a0: 2020 2020 2020 2020 2020 2020 2053 7765 Swe\n+000223b0: 7074 206c 696e 6573 2e20 2020 2020 2020 pt lines. \n+000223c0: 2020 286c 696e 6520 2034 3829 0a2a 206f (line 48).* o\n+000223d0: 7264 6572 2c20 706f 6c79 676f 6e20 7665 rder, polygon ve\n+000223e0: 7274 6578 3a20 2020 2020 2020 2020 2020 rtex: \n+000223f0: 2020 2020 2020 4865 6c6c 6f20 776f 726c Hello worl\n+00022400: 642e 2020 2020 2020 2020 2028 6c69 6e65 d. (line\n+00022410: 2020 3339 290a 2a20 6f72 6465 722c 2070 39).* order, p\n+00022420: 6f6c 7967 6f6e 2076 6572 7465 7820 3c31 olygon vertex <1\n+00022430: 3e3a 2020 2020 2020 2020 2020 2020 2053 >: S\n+00022440: 7765 7074 206c 696e 6573 2e20 2020 2020 wept lines. \n+00022450: 2020 2020 286c 696e 6520 2031 3429 0a2a (line 14).*\n+00022460: 206f 7264 6572 2c20 706f 6c79 676f 6e20 order, polygon \n+00022470: 7665 7274 6578 203c 323e 3a20 2020 2020 vertex <2>: \n+00022480: 2020 2020 2020 2020 5377 6570 7420 6c69 Swept li\n+00022490: 6e65 732e 2020 2020 2020 2020 2028 6c69 nes. (li\n+000224a0: 6e65 2020 3430 290a 2a20 6f72 6465 722c ne 40).* order,\n+000224b0: 2070 6f6c 7967 6f6e 2076 6572 7465 7820 polygon vertex \n+000224c0: 3c33 3e3a 2020 2020 2020 2020 2020 2020 <3>: \n+000224d0: 2053 7765 7074 2070 6f6c 7967 6f6e 732e Swept polygons.\n+000224e0: 2020 2020 2020 286c 696e 6520 2031 3929 (line 19)\n+000224f0: 0a2a 206f 7264 6572 2c20 706f 6c79 676f .* order, polygo\n+00022500: 6e20 7665 7274 6578 203c 343e 3a20 2020 n vertex <4>: \n+00022510: 2020 2020 2020 2020 2020 4f76 6572 7669 Overvi\n+00022520: 6577 2e20 2020 2020 2020 2020 2020 2028 ew. (\n+00022530: 6c69 6e65 2020 3337 290a 2a20 6f72 6465 line 37).* orde\n+00022540: 722c 2070 6f6c 7967 6f6e 2076 6572 7465 r, polygon verte\n+00022550: 7820 3c35 3e3a 2020 2020 2020 2020 2020 x <5>: \n+00022560: 2020 204f 7665 7276 6965 772e 2020 2020 Overview. \n+00022570: 2020 2020 2020 2020 286c 696e 6520 2035 (line 5\n+00022580: 3529 0a2a 206f 7574 7075 7420 6c61 6e67 5).* output lang\n+00022590: 7561 6765 3a20 2020 2020 2020 2020 2020 uage: \n+000225a0: 2020 2020 2020 2020 2020 2020 4c61 6e67 Lang\n+000225b0: 7561 6765 2e20 2020 2020 2020 2020 2020 uage. \n+000225c0: 2028 6c69 6e65 2020 2036 290a 2a20 6f75 (line 6).* ou\n+000225d0: 7473 6964 6520 6f66 2061 2070 6f6c 7967 tside of a polyg\n+000225e0: 6f6e 3a20 2020 2020 2020 2020 2020 2020 on: \n+000225f0: 2020 2020 2048 656c 6c6f 2077 6f72 6c64 Hello world\n+00022600: 2e20 2020 2020 2020 2020 286c 696e 6520 . (line \n+00022610: 2033 3729 0a2a 2070 6172 616c 6c65 6c20 37).* parallel \n+00022620: 7072 6f6a 6563 7469 6f6e 3a20 2020 2020 projection: \n+00022630: 2020 2020 2020 2020 2020 2020 2020 4865 He\n+00022640: 6c6c 6f20 776f 726c 642e 2020 2020 2020 llo world. \n+00022650: 2020 2028 6c69 6e65 2020 3236 290a 2a20 (line 26).* \n+00022660: 7061 7261 6c6c 656c 2070 726f 6a65 6374 parallel project\n+00022670: 696f 6e20 3c31 3e3a 2020 2020 2020 2020 ion <1>: \n+00022680: 2020 2020 2020 2054 7261 6e73 666f 726d Transform\n+00022690: 206c 6974 6572 616c 732e 2020 286c 696e literals. (lin\n+000226a0: 6520 2031 3829 0a2a 2070 6172 616c 6c65 e 18).* paralle\n+000226b0: 6c20 7072 6f6a 6563 7469 6f6e 203c 323e l projection <2>\n+000226c0: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+000226d0: 5069 6374 7572 6520 626f 782e 2020 2020 Picture box. \n+000226e0: 2020 2020 2028 6c69 6e65 2020 3135 290a (line 15).\n+000226f0: 2a20 7061 7265 6e74 6865 7365 733a 2020 * parentheses: \n 00022700: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00022710: 2020 2020 2054 776f 2d6f 7065 7261 6e64 Two-operand\n-00022720: 2028 6269 6e61 7279 2920 666f 726d 732e (binary) forms.\n-00022730: 0a20 2020 2020 2020 2020 2020 2020 2020 . \n+00022710: 2020 2020 2020 2020 2054 776f 2d6f 7065 Two-ope\n+00022720: 7261 6e64 2028 6269 6e61 7279 2920 666f rand (binary) fo\n+00022730: 726d 732e 0a20 2020 2020 2020 2020 2020 rms.. \n 00022740: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00022750: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00022760: 2020 2020 2020 2020 2020 2020 2020 2028 (\n-00022770: 6c69 6e65 2020 3538 290a 2a20 7065 7273 line 58).* pers\n-00022780: 7065 6374 6976 6520 7072 6f6a 6563 7469 pective projecti\n-00022790: 6f6e 3a20 2020 2020 2020 2020 2020 2020 on: \n-000227a0: 2020 2054 7261 6e73 666f 726d 206c 6974 Transform lit\n-000227b0: 6572 616c 732e 2020 286c 696e 6520 2031 erals. (line 1\n-000227c0: 3929 0a2a 2070 6572 7370 6563 7469 7665 9).* perspective\n-000227d0: 2070 726f 6a65 6374 696f 6e20 3c31 3e3a projection <1>:\n-000227e0: 2020 2020 2020 2020 2020 2020 5472 616e Tran\n-000227f0: 7366 6f72 6d20 6c69 7465 7261 6c73 2e20 sform literals. \n-00022800: 2028 6c69 6e65 2020 3231 290a 2a20 7065 (line 21).* pe\n-00022810: 7273 7065 6374 6976 6520 7072 6f6a 6563 rspective projec\n-00022820: 7469 6f6e 203c 323e 3a20 2020 2020 2020 tion <2>: \n-00022830: 2020 2020 204f 7665 7276 6965 772e 2020 Overview. \n-00022840: 2020 2020 2020 2020 2020 286c 696e 6520 (line \n-00022850: 2020 3729 0a2a 2070 6963 7475 7265 2062 7).* picture b\n-00022860: 6f78 3a20 2020 2020 2020 2020 2020 2020 ox: \n-00022870: 2020 2020 2020 2020 2020 2020 2020 5069 Pi\n-00022880: 6374 7572 6520 626f 782e 2020 2020 2020 cture box. \n-00022890: 2020 2028 6c69 6e65 2020 2036 290a 2a20 (line 6).* \n-000228a0: 706c 616e 6172 6974 7920 6f66 2070 6f6c planarity of pol\n-000228b0: 7967 6f6e 733a 2020 2020 2020 2020 2020 ygons: \n-000228c0: 2020 2020 2020 2050 6f6c 7967 6f6e 732e Polygons.\n-000228d0: 2020 2020 2020 2020 2020 2020 286c 696e (lin\n-000228e0: 6520 2031 3029 0a2a 2070 6c61 6e61 7269 e 10).* planari\n-000228f0: 7479 206f 6620 706f 6c79 676f 6e73 203c ty of polygons <\n-00022900: 313e 3a20 2020 2020 2020 2020 2020 2020 1>: \n-00022910: 5377 6565 7020 6661 6365 2073 706c 6974 Sweep face split\n-00022920: 7469 6e67 2e0a 2020 2020 2020 2020 2020 ting.. \n+00022760: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00022770: 2020 2028 6c69 6e65 2020 3538 290a 2a20 (line 58).* \n+00022780: 7065 7273 7065 6374 6976 6520 7072 6f6a perspective proj\n+00022790: 6563 7469 6f6e 3a20 2020 2020 2020 2020 ection: \n+000227a0: 2020 2020 2020 2054 7261 6e73 666f 726d Transform\n+000227b0: 206c 6974 6572 616c 732e 2020 286c 696e literals. (lin\n+000227c0: 6520 2031 3929 0a2a 2070 6572 7370 6563 e 19).* perspec\n+000227d0: 7469 7665 2070 726f 6a65 6374 696f 6e20 tive projection \n+000227e0: 3c31 3e3a 2020 2020 2020 2020 2020 2020 <1>: \n+000227f0: 5472 616e 7366 6f72 6d20 6c69 7465 7261 Transform litera\n+00022800: 6c73 2e20 2028 6c69 6e65 2020 3231 290a ls. (line 21).\n+00022810: 2a20 7065 7273 7065 6374 6976 6520 7072 * perspective pr\n+00022820: 6f6a 6563 7469 6f6e 203c 323e 3a20 2020 ojection <2>: \n+00022830: 2020 2020 2020 2020 204f 7665 7276 6965 Overvie\n+00022840: 772e 2020 2020 2020 2020 2020 2020 286c w. (l\n+00022850: 696e 6520 2020 3729 0a2a 2070 6963 7475 ine 7).* pictu\n+00022860: 7265 2062 6f78 3a20 2020 2020 2020 2020 re box: \n+00022870: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00022880: 2020 5069 6374 7572 6520 626f 782e 2020 Picture box. \n+00022890: 2020 2020 2020 2028 6c69 6e65 2020 2036 (line 6\n+000228a0: 290a 2a20 706c 616e 6172 6974 7920 6f66 ).* planarity of\n+000228b0: 2070 6f6c 7967 6f6e 733a 2020 2020 2020 polygons: \n+000228c0: 2020 2020 2020 2020 2020 2050 6f6c 7967 Polyg\n+000228d0: 6f6e 732e 2020 2020 2020 2020 2020 2020 ons. \n+000228e0: 286c 696e 6520 2031 3029 0a2a 2070 6c61 (line 10).* pla\n+000228f0: 6e61 7269 7479 206f 6620 706f 6c79 676f narity of polygo\n+00022900: 6e73 203c 313e 3a20 2020 2020 2020 2020 ns <1>: \n+00022910: 2020 2020 5377 6565 7020 6661 6365 2073 Sweep face s\n+00022920: 706c 6974 7469 6e67 2e0a 2020 2020 2020 plitting.. \n 00022930: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00022940: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00022950: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00022960: 2020 2020 286c 696e 6520 2020 3729 0a2a (line 7).*\n-00022970: 2070 6f69 6e74 2064 6566 696e 6974 696f point definitio\n-00022980: 6e3a 2020 2020 2020 2020 2020 2020 2020 n: \n-00022990: 2020 2020 2020 2020 4472 6177 696e 6720 Drawing \n-000229a0: 6120 736f 6c69 642e 2020 2020 2028 6c69 a solid. (li\n-000229b0: 6e65 2020 3233 290a 2a20 706f 696e 7420 ne 23).* point \n-000229c0: 6c69 7374 3a20 2020 2020 2020 2020 2020 list: \n+00022960: 2020 2020 2020 2020 286c 696e 6520 2020 (line \n+00022970: 3729 0a2a 2070 6f69 6e74 2064 6566 696e 7).* point defin\n+00022980: 6974 696f 6e3a 2020 2020 2020 2020 2020 ition: \n+00022990: 2020 2020 2020 2020 2020 2020 4472 6177 Draw\n+000229a0: 696e 6720 6120 736f 6c69 642e 2020 2020 ing a solid. \n+000229b0: 2028 6c69 6e65 2020 3233 290a 2a20 706f (line 23).* po\n+000229c0: 696e 7420 6c69 7374 3a20 2020 2020 2020 int list: \n 000229d0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000229e0: 2053 7765 7074 2070 6f69 6e74 732e 2020 Swept points. \n-000229f0: 2020 2020 2020 286c 696e 6520 2020 3929 (line 9)\n-00022a00: 0a2a 2070 6f69 6e74 206c 6974 6572 616c .* point literal\n-00022a10: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n-00022a20: 2020 2020 2020 2020 2020 506f 696e 7420 Point \n-00022a30: 616e 6420 7665 6374 6f72 206c 6974 6572 and vector liter\n-00022a40: 616c 732e 0a20 2020 2020 2020 2020 2020 als.. \n+000229e0: 2020 2020 2053 7765 7074 2070 6f69 6e74 Swept point\n+000229f0: 732e 2020 2020 2020 2020 286c 696e 6520 s. (line \n+00022a00: 2020 3929 0a2a 2070 6f69 6e74 206c 6974 9).* point lit\n+00022a10: 6572 616c 3a20 2020 2020 2020 2020 2020 eral: \n+00022a20: 2020 2020 2020 2020 2020 2020 2020 506f Po\n+00022a30: 696e 7420 616e 6420 7665 6374 6f72 206c int and vector l\n+00022a40: 6974 6572 616c 732e 0a20 2020 2020 2020 iterals.. \n 00022a50: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00022a60: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00022a70: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00022a80: 2020 2028 6c69 6e65 2020 2036 290a 2a20 (line 6).* \n-00022a90: 706f 696e 7420 7265 6665 7265 6e63 653a point reference:\n-00022aa0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00022ab0: 2020 2020 2020 2046 6f72 6d73 206f 6620 Forms of \n-00022ac0: 7265 6665 7265 6e63 6573 2e20 286c 696e references. (lin\n-00022ad0: 6520 2031 3429 0a2a 2070 6f69 6e74 2073 e 14).* point s\n-00022ae0: 7765 6570 3a20 2020 2020 2020 2020 2020 weep: \n+00022a80: 2020 2020 2020 2028 6c69 6e65 2020 2036 (line 6\n+00022a90: 290a 2a20 706f 696e 7420 7265 6665 7265 ).* point refere\n+00022aa0: 6e63 653a 2020 2020 2020 2020 2020 2020 nce: \n+00022ab0: 2020 2020 2020 2020 2020 2046 6f72 6d73 Forms\n+00022ac0: 206f 6620 7265 6665 7265 6e63 6573 2e20 of references. \n+00022ad0: 286c 696e 6520 2031 3429 0a2a 2070 6f69 (line 14).* poi\n+00022ae0: 6e74 2073 7765 6570 3a20 2020 2020 2020 nt sweep: \n 00022af0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00022b00: 5377 6570 7420 6f62 6a65 6374 732e 2020 Swept objects. \n-00022b10: 2020 2020 2028 6c69 6e65 2020 3131 290a (line 11).\n-00022b20: 2a20 706f 696e 7420 7377 6565 7020 3c31 * point sweep <1\n-00022b30: 3e3a 2020 2020 2020 2020 2020 2020 2020 >: \n-00022b40: 2020 2020 2020 2020 2053 7765 7074 2070 Swept p\n-00022b50: 6f69 6e74 732e 2020 2020 2020 2020 286c oints. (l\n-00022b60: 696e 6520 2020 3629 0a2a 2070 6f6c 7967 ine 6).* polyg\n-00022b70: 6f6e 2073 7765 6570 3a20 2020 2020 2020 on sweep: \n+00022b00: 2020 2020 5377 6570 7420 6f62 6a65 6374 Swept object\n+00022b10: 732e 2020 2020 2020 2028 6c69 6e65 2020 s. (line \n+00022b20: 3131 290a 2a20 706f 696e 7420 7377 6565 11).* point swee\n+00022b30: 7020 3c31 3e3a 2020 2020 2020 2020 2020 p <1>: \n+00022b40: 2020 2020 2020 2020 2020 2020 2053 7765 Swe\n+00022b50: 7074 2070 6f69 6e74 732e 2020 2020 2020 pt points. \n+00022b60: 2020 286c 696e 6520 2020 3629 0a2a 2070 (line 6).* p\n+00022b70: 6f6c 7967 6f6e 2073 7765 6570 3a20 2020 olygon sweep: \n 00022b80: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00022b90: 2020 506f 6c79 676f 6e20 7377 6565 7073 Polygon sweeps\n-00022ba0: 2e20 2020 2020 2028 6c69 6e65 2020 2036 . (line 6\n-00022bb0: 290a 2a20 706f 6c79 676f 6e20 7377 6565 ).* polygon swee\n-00022bc0: 7020 3c31 3e3a 2020 2020 2020 2020 2020 p <1>: \n-00022bd0: 2020 2020 2020 2020 2020 2053 7765 7074 Swept\n-00022be0: 2070 6f6c 7967 6f6e 732e 2020 2020 2020 polygons. \n-00022bf0: 286c 696e 6520 2020 3629 0a2a 2070 6f6c (line 6).* pol\n-00022c00: 7967 6f6e 2073 7765 6570 203c 323e 3a20 ygon sweep <2>: \n-00022c10: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00022c20: 2020 2020 4120 6869 6572 6172 6368 6963 A hierarchic\n-00022c30: 616c 206d 6f64 656c 2e0a 2020 2020 2020 al model.. \n+00022b90: 2020 2020 2020 506f 6c79 676f 6e20 7377 Polygon sw\n+00022ba0: 6565 7073 2e20 2020 2020 2028 6c69 6e65 eeps. (line\n+00022bb0: 2020 2036 290a 2a20 706f 6c79 676f 6e20 6).* polygon \n+00022bc0: 7377 6565 7020 3c31 3e3a 2020 2020 2020 sweep <1>: \n+00022bd0: 2020 2020 2020 2020 2020 2020 2020 2053 S\n+00022be0: 7765 7074 2070 6f6c 7967 6f6e 732e 2020 wept polygons. \n+00022bf0: 2020 2020 286c 696e 6520 2020 3629 0a2a (line 6).*\n+00022c00: 2070 6f6c 7967 6f6e 2073 7765 6570 203c polygon sweep <\n+00022c10: 323e 3a20 2020 2020 2020 2020 2020 2020 2>: \n+00022c20: 2020 2020 2020 2020 4120 6869 6572 6172 A hierar\n+00022c30: 6368 6963 616c 206d 6f64 656c 2e0a 2020 chical model.. \n 00022c40: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00022c50: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00022c60: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00022c70: 2020 2020 2020 2020 286c 696e 6520 3132 (line 12\n-00022c80: 3229 0a2a 2070 6f6c 7967 6f6e 2076 6572 2).* polygon ver\n-00022c90: 7465 7820 6f72 6465 723a 2020 2020 2020 tex order: \n-00022ca0: 2020 2020 2020 2020 2020 2020 4865 6c6c Hell\n-00022cb0: 6f20 776f 726c 642e 2020 2020 2020 2020 o world. \n-00022cc0: 2028 6c69 6e65 2020 3339 290a 2a20 706f (line 39).* po\n-00022cd0: 6c79 676f 6e20 7665 7274 6578 206f 7264 lygon vertex ord\n-00022ce0: 6572 203c 313e 3a20 2020 2020 2020 2020 er <1>: \n-00022cf0: 2020 2020 2053 7765 7074 206c 696e 6573 Swept lines\n-00022d00: 2e20 2020 2020 2020 2020 286c 696e 6520 . (line \n-00022d10: 2031 3429 0a2a 2070 6f6c 7967 6f6e 2076 14).* polygon v\n-00022d20: 6572 7465 7820 6f72 6465 7220 3c32 3e3a ertex order <2>:\n-00022d30: 2020 2020 2020 2020 2020 2020 2020 5377 Sw\n-00022d40: 6570 7420 6c69 6e65 732e 2020 2020 2020 ept lines. \n-00022d50: 2020 2028 6c69 6e65 2020 3430 290a 2a20 (line 40).* \n-00022d60: 706f 6c79 676f 6e20 7665 7274 6578 206f polygon vertex o\n-00022d70: 7264 6572 203c 333e 3a20 2020 2020 2020 rder <3>: \n-00022d80: 2020 2020 2020 2053 7765 7074 2070 6f6c Swept pol\n-00022d90: 7967 6f6e 732e 2020 2020 2020 286c 696e ygons. 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(li\n+00023300: 6e65 2020 3231 290a 2a20 7072 6f6a 6563 ne 21).* projec\n+00023310: 7469 6f6e 2c20 7065 7273 7065 6374 6976 tion, perspectiv\n+00023320: 6520 3c32 3e3a 2020 2020 2020 2020 2020 e <2>: \n+00023330: 204f 7665 7276 6965 772e 2020 2020 2020 Overview. \n+00023340: 2020 2020 2020 286c 696e 6520 2020 3729 (line 7)\n+00023350: 0a2a 2070 7365 7564 6f2d 6f70 7469 6f6e .* pseudo-option\n+00023360: 733a 2020 2020 2020 2020 2020 2020 2020 s: \n+00023370: 2020 2020 2020 2020 2020 5469 6b5a 2f50 TikZ/P\n+00023380: 4746 2075 7365 722d 6465 6669 6e65 6420 GF user-defined \n+00023390: 7374 796c 6573 2e0a 2020 2020 2020 2020 styles.. \n 000233a0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 000233b0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 000233c0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000233d0: 2020 286c 696e 6520 2032 3529 0a2a 2070 (line 25).* p\n-000233e0: 7366 7261 6d65 626f 783a 2020 2020 2020 sframebox: \n+000233d0: 2020 2020 2020 286c 696e 6520 2032 3529 (line 25)\n+000233e0: 0a2a 2070 7366 7261 6d65 626f 783a 2020 .* psframebox: \n 000233f0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00023400: 2020 2020 2020 4672 616d 652e 2020 2020 Frame. \n-00023410: 2020 2020 2020 2020 2020 2028 6c69 6e65 (line\n-00023420: 2020 2038 290a 2a20 7073 7069 6374 7572 8).* pspictur\n-00023430: 653a 2020 2020 2020 2020 2020 2020 2020 e: \n-00023440: 2020 2020 2020 2020 2020 2020 2020 2050 P\n-00023450: 6963 7475 7265 2062 6f78 2e20 2020 2020 icture box. \n-00023460: 2020 2020 286c 696e 6520 2031 3029 0a2a (line 10).*\n-00023470: 2050 5354 7269 636b 733a 2020 2020 2020 PSTricks: \n+00023400: 2020 2020 2020 2020 2020 4672 616d 652e Frame.\n+00023410: 2020 2020 2020 2020 2020 2020 2020 2028 (\n+00023420: 6c69 6e65 2020 2038 290a 2a20 7073 7069 line 8).* pspi\n+00023430: 6374 7572 653a 2020 2020 2020 2020 2020 cture: \n+00023440: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00023450: 2020 2050 6963 7475 7265 2062 6f78 2e20 Picture box. \n+00023460: 2020 2020 2020 2020 286c 696e 6520 2031 (line 1\n+00023470: 3029 0a2a 2050 5354 7269 636b 733a 2020 0).* PSTricks: \n 00023480: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00023490: 2020 2020 2020 2020 4162 6f75 7420 736b About sk\n-000234a0: 6574 6368 2e20 2020 2020 2020 2028 6c69 etch. 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(lin\n-000236b0: 6520 2031 3829 0a2a 2072 6566 6572 656e e 18).* referen\n-000236c0: 6365 2c20 6f70 7469 6f6e 733a 2020 2020 ce, options: \n+000234e0: 2020 2020 2053 7065 6369 616c 206f 626a Special obj\n+000234f0: 6563 7473 2e20 2020 2020 286c 696e 6520 ects. (line \n+00023500: 2032 3029 0a2a 2071 756f 7469 6e67 2c20 20).* quoting, \n+00023510: 7370 6563 6961 6c20 3c31 3e3a 2020 2020 special <1>: \n+00023520: 2020 2020 2020 2020 2020 2020 2020 5370 Sp\n+00023530: 6563 6961 6c73 2e20 2020 2020 2020 2020 ecials. \n+00023540: 2020 2028 6c69 6e65 2020 2038 290a 2a20 (line 8).* \n+00023550: 7265 6665 7265 6e63 652c 2064 7261 7761 reference, drawa\n+00023560: 626c 653a 2020 2020 2020 2020 2020 2020 ble: \n+00023570: 2020 2020 2020 204f 626a 6563 7420 7472 Object tr\n+00023580: 616e 7366 6f72 6d73 2e20 2020 286c 696e ansforms. (lin\n+00023590: 6520 2033 3329 0a2a 2072 6566 6572 656e e 33).* referen\n+000235a0: 6365 2c20 6472 6177 6162 6c65 203c 313e ce, drawable <1>\n+000235b0: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+000235c0: 466f 726d 7320 6f66 2072 6566 6572 656e Forms of referen\n+000235d0: 6365 732e 2028 6c69 6e65 2020 3137 290a ces. (line 17).\n+000235e0: 2a20 7265 6665 7265 6e63 652c 2064 7261 * reference, dra\n+000235f0: 7761 626c 6520 3c32 3e3a 2020 2020 2020 wable <2>: \n+00023600: 2020 2020 2020 2020 204f 7665 7276 6965 Overvie\n+00023610: 772e 2020 2020 2020 2020 2020 2020 286c w. (l\n+00023620: 696e 6520 2034 3729 0a2a 2072 6566 6572 ine 47).* refer\n+00023630: 656e 6365 2c20 6472 6177 6162 6c65 203c ence, drawable <\n+00023640: 333e 3a20 2020 2020 2020 2020 2020 2020 3>: \n+00023650: 2020 4f76 6572 7669 6577 2e20 2020 2020 Overview. \n+00023660: 2020 2020 2020 2028 6c69 6e65 2020 3537 (line 57\n+00023670: 290a 2a20 7265 6665 7265 6e63 652c 206d ).* reference, m\n+00023680: 756c 7469 706c 6520 6f70 7469 6f6e 733a ultiple options:\n+00023690: 2020 2020 2020 2020 2020 2046 6f72 6d73 Forms\n+000236a0: 206f 6620 7265 6665 7265 6e63 6573 2e20 of references. \n+000236b0: 286c 696e 6520 2031 3829 0a2a 2072 6566 (line 18).* ref\n+000236c0: 6572 656e 6365 2c20 6f70 7469 6f6e 733a erence, options:\n 000236d0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000236e0: 506f 6c79 6c69 6e65 2073 7765 6570 7320 Polyline sweeps \n-000236f0: 7769 7468 2063 6c6f 7375 7265 2e0a 2020 with closure.. \n-00023700: 2020 2020 2020 2020 2020 2020 2020 2020 \n+000236e0: 2020 2020 506f 6c79 6c69 6e65 2073 7765 Polyline swe\n+000236f0: 6570 7320 7769 7468 2063 6c6f 7375 7265 eps with closure\n+00023700: 2e0a 2020 2020 2020 2020 2020 2020 2020 .. \n 00023710: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00023720: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00023730: 2020 2020 2020 2020 2020 2020 286c 696e (lin\n-00023740: 6520 2032 3429 0a2a 2072 6566 6572 656e e 24).* referen\n-00023750: 6365 2c20 6f70 7469 6f6e 7320 3c31 3e3a ce, options <1>:\n-00023760: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00023770: 466f 726d 7320 6f66 2072 6566 6572 656e Forms of referen\n-00023780: 6365 732e 2028 6c69 6e65 2020 3138 290a ces. (line 18).\n-00023790: 2a20 7265 6665 7265 6e63 652c 2070 6f69 * reference, poi\n-000237a0: 6e74 3a20 2020 2020 2020 2020 2020 2020 nt: \n-000237b0: 2020 2020 2020 2020 2044 7261 7769 6e67 Drawing\n-000237c0: 2061 2073 6f6c 6964 2e20 2020 2020 286c a solid. (l\n-000237d0: 696e 6520 2032 3329 0a2a 2072 6566 6572 ine 23).* refer\n-000237e0: 656e 6365 2c20 706f 696e 7420 3c31 3e3a ence, point <1>:\n-000237f0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00023800: 2020 466f 726d 7320 6f66 2072 6566 6572 Forms of refer\n-00023810: 656e 6365 732e 2028 6c69 6e65 2020 3134 ences. (line 14\n-00023820: 290a 2a20 7265 6665 7265 6e63 652c 2073 ).* reference, s\n-00023830: 6361 6c61 723a 2020 2020 2020 2020 2020 calar: \n-00023840: 2020 2020 2020 2020 2020 2053 7765 7074 Swept\n-00023850: 206f 626a 6563 7473 2e20 2020 2020 2020 objects. \n-00023860: 286c 696e 6520 2031 3329 0a2a 2072 6566 (line 13).* ref\n-00023870: 6572 656e 6365 2c20 7363 616c 6172 203c erence, scalar <\n-00023880: 313e 3a20 2020 2020 2020 2020 2020 2020 1>: \n-00023890: 2020 2020 466f 726d 7320 6f66 2072 6566 Forms of ref\n-000238a0: 6572 656e 6365 732e 2028 6c69 6e65 2020 erences. (line \n-000238b0: 3133 290a 2a20 7265 6665 7265 6e63 652c 13).* reference,\n-000238c0: 2074 6167 3a20 2020 2020 2020 2020 2020 tag: \n-000238d0: 2020 2020 2020 2020 2020 2020 2046 6f72 For\n-000238e0: 6d73 206f 6620 6465 6669 6e69 7469 6f6e ms of definition\n-000238f0: 732e 0a20 2020 2020 2020 2020 2020 2020 s.. \n+00023730: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00023740: 286c 696e 6520 2032 3429 0a2a 2072 6566 (line 24).* ref\n+00023750: 6572 656e 6365 2c20 6f70 7469 6f6e 7320 erence, options \n+00023760: 3c31 3e3a 2020 2020 2020 2020 2020 2020 <1>: \n+00023770: 2020 2020 466f 726d 7320 6f66 2072 6566 Forms of ref\n+00023780: 6572 656e 6365 732e 2028 6c69 6e65 2020 erences. (line \n+00023790: 3138 290a 2a20 7265 6665 7265 6e63 652c 18).* reference,\n+000237a0: 2070 6f69 6e74 3a20 2020 2020 2020 2020 point: \n+000237b0: 2020 2020 2020 2020 2020 2020 2044 7261 Dra\n+000237c0: 7769 6e67 2061 2073 6f6c 6964 2e20 2020 wing a solid. \n+000237d0: 2020 286c 696e 6520 2032 3329 0a2a 2072 (line 23).* r\n+000237e0: 6566 6572 656e 6365 2c20 706f 696e 7420 eference, point \n+000237f0: 3c31 3e3a 2020 2020 2020 2020 2020 2020 <1>: \n+00023800: 2020 2020 2020 466f 726d 7320 6f66 2072 Forms of r\n+00023810: 6566 6572 656e 6365 732e 2028 6c69 6e65 eferences. (line\n+00023820: 2020 3134 290a 2a20 7265 6665 7265 6e63 14).* referenc\n+00023830: 652c 2073 6361 6c61 723a 2020 2020 2020 e, scalar: \n+00023840: 2020 2020 2020 2020 2020 2020 2020 2053 S\n+00023850: 7765 7074 206f 626a 6563 7473 2e20 2020 wept objects. \n+00023860: 2020 2020 286c 696e 6520 2031 3329 0a2a (line 13).*\n+00023870: 2072 6566 6572 656e 6365 2c20 7363 616c reference, scal\n+00023880: 6172 203c 313e 3a20 2020 2020 2020 2020 ar <1>: \n+00023890: 2020 2020 2020 2020 466f 726d 7320 6f66 Forms of\n+000238a0: 2072 6566 6572 656e 6365 732e 2028 6c69 references. (li\n+000238b0: 6e65 2020 3133 290a 2a20 7265 6665 7265 ne 13).* refere\n+000238c0: 6e63 652c 2074 6167 3a20 2020 2020 2020 nce, tag: \n+000238d0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+000238e0: 2046 6f72 6d73 206f 6620 6465 6669 6e69 Forms of defini\n+000238f0: 7469 6f6e 732e 0a20 2020 2020 2020 2020 tions.. \n 00023900: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00023910: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00023920: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00023930: 2028 6c69 6e65 2020 3230 290a 2a20 7265 (line 20).* re\n-00023940: 6665 7265 6e63 652c 2074 6167 203c 313e ference, tag <1>\n-00023950: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n-00023960: 2020 2020 2046 6f72 6d73 206f 6620 7265 Forms of re\n-00023970: 6665 7265 6e63 6573 2e20 286c 696e 6520 ferences. (line \n-00023980: 2031 3929 0a2a 2072 6566 6572 656e 6365 19).* reference\n-00023990: 2c20 7472 616e 7366 6f72 6d3a 2020 2020 , transform: \n-000239a0: 2020 2020 2020 2020 2020 2020 2020 466f Fo\n-000239b0: 726d 7320 6f66 2072 6566 6572 656e 6365 rms of reference\n-000239c0: 732e 2028 6c69 6e65 2020 3136 290a 2a20 s. (line 16).* \n-000239d0: 7265 6665 7265 6e63 652c 2074 7261 6e73 reference, trans\n-000239e0: 666f 726d 203c 313e 3a20 2020 2020 2020 form <1>: \n-000239f0: 2020 2020 2020 204f 7665 7276 6965 772e Overview.\n-00023a00: 2020 2020 2020 2020 2020 2020 286c 696e (lin\n-00023a10: 6520 2033 3529 0a2a 2072 6566 6572 656e e 35).* referen\n-00023a20: 6365 2c20 7665 6374 6f72 3a20 2020 2020 ce, vector: \n+00023930: 2020 2020 2028 6c69 6e65 2020 3230 290a (line 20).\n+00023940: 2a20 7265 6665 7265 6e63 652c 2074 6167 * reference, tag\n+00023950: 203c 313e 3a20 2020 2020 2020 2020 2020 <1>: \n+00023960: 2020 2020 2020 2020 2046 6f72 6d73 206f Forms o\n+00023970: 6620 7265 6665 7265 6e63 6573 2e20 286c f references. (l\n+00023980: 696e 6520 2031 3929 0a2a 2072 6566 6572 ine 19).* refer\n+00023990: 656e 6365 2c20 7472 616e 7366 6f72 6d3a ence, transform:\n+000239a0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+000239b0: 2020 466f 726d 7320 6f66 2072 6566 6572 Forms of refer\n+000239c0: 656e 6365 732e 2028 6c69 6e65 2020 3136 ences. (line 16\n+000239d0: 290a 2a20 7265 6665 7265 6e63 652c 2074 ).* reference, t\n+000239e0: 7261 6e73 666f 726d 203c 313e 3a20 2020 ransform <1>: \n+000239f0: 2020 2020 2020 2020 2020 204f 7665 7276 Overv\n+00023a00: 6965 772e 2020 2020 2020 2020 2020 2020 iew. \n+00023a10: 286c 696e 6520 2033 3529 0a2a 2072 6566 (line 35).* ref\n+00023a20: 6572 656e 6365 2c20 7665 6374 6f72 3a20 erence, vector: \n 00023a30: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00023a40: 506f 6c79 6c69 6e65 2073 7765 6570 732e Polyline sweeps.\n-00023a50: 2020 2020 2028 6c69 6e65 2020 3230 290a (line 20).\n-00023a60: 2a20 7265 6665 7265 6e63 652c 2076 6563 * reference, vec\n-00023a70: 746f 7220 3c31 3e3a 2020 2020 2020 2020 tor <1>: \n-00023a80: 2020 2020 2020 2020 2046 6f72 6d73 206f Forms o\n-00023a90: 6620 7265 6665 7265 6e63 6573 2e20 286c f references. (l\n-00023aa0: 696e 6520 2031 3529 0a2a 2072 6570 6561 ine 15).* repea\n-00023ab0: 7465 6420 6f62 6a65 6374 3a20 2020 2020 ted object: \n+00023a40: 2020 2020 506f 6c79 6c69 6e65 2073 7765 Polyline swe\n+00023a50: 6570 732e 2020 2020 2028 6c69 6e65 2020 eps. (line \n+00023a60: 3230 290a 2a20 7265 6665 7265 6e63 652c 20).* reference,\n+00023a70: 2076 6563 746f 7220 3c31 3e3a 2020 2020 vector <1>: \n+00023a80: 2020 2020 2020 2020 2020 2020 2046 6f72 For\n+00023a90: 6d73 206f 6620 7265 6665 7265 6e63 6573 ms of references\n+00023aa0: 2e20 286c 696e 6520 2031 3529 0a2a 2072 . (line 15).* r\n+00023ab0: 6570 6561 7465 6420 6f62 6a65 6374 3a20 epeated object: \n 00023ac0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00023ad0: 2020 5265 7065 6174 732e 2020 2020 2020 Repeats. \n-00023ae0: 2020 2020 2020 2028 6c69 6e65 2020 2036 (line 6\n-00023af0: 290a 2a20 7265 7365 7276 6564 2077 6f72 ).* reserved wor\n-00023b00: 6473 3a20 2020 2020 2020 2020 2020 2020 ds: \n-00023b10: 2020 2020 2020 2020 2020 204b 6579 2061 Key a\n-00023b20: 6e64 2072 6573 6572 7665 6420 776f 7264 nd reserved word\n-00023b30: 732e 0a20 2020 2020 2020 2020 2020 2020 s.. \n+00023ad0: 2020 2020 2020 5265 7065 6174 732e 2020 Repeats. \n+00023ae0: 2020 2020 2020 2020 2020 2028 6c69 6e65 (line\n+00023af0: 2020 2036 290a 2a20 7265 7365 7276 6564 6).* reserved\n+00023b00: 2077 6f72 6473 3a20 2020 2020 2020 2020 words: \n+00023b10: 2020 2020 2020 2020 2020 2020 2020 204b K\n+00023b20: 6579 2061 6e64 2072 6573 6572 7665 6420 ey and reserved \n+00023b30: 776f 7264 732e 0a20 2020 2020 2020 2020 words.. \n 00023b40: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00023b50: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00023b60: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00023b70: 2028 6c69 6e65 2020 3131 290a 2a20 7269 (line 11).* ri\n-00023b80: 6768 7420 6861 6e64 2072 756c 653a 2020 ght hand rule: \n-00023b90: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00023ba0: 2020 2020 204f 626a 6563 7420 7472 616e Object tran\n-00023bb0: 7366 6f72 6d73 2e20 2020 286c 696e 6520 sforms. (line \n-00023bc0: 2034 3129 0a2a 2072 6967 6874 2d68 616e 41).* right-han\n-00023bd0: 6420 636f 6f72 6469 6e61 7465 2073 7973 d coordinate sys\n-00023be0: 7465 6d3a 2020 2020 2020 2020 2020 496e tem: In\n-00023bf0: 7472 6f64 7563 7469 6f6e 2062 7920 6578 troduction by ex\n-00023c00: 616d 706c 652e 0a20 2020 2020 2020 2020 ample.. \n+00023b70: 2020 2020 2028 6c69 6e65 2020 3131 290a (line 11).\n+00023b80: 2a20 7269 6768 7420 6861 6e64 2072 756c * right hand rul\n+00023b90: 653a 2020 2020 2020 2020 2020 2020 2020 e: \n+00023ba0: 2020 2020 2020 2020 204f 626a 6563 7420 Object \n+00023bb0: 7472 616e 7366 6f72 6d73 2e20 2020 286c transforms. (l\n+00023bc0: 696e 6520 2034 3129 0a2a 2072 6967 6874 ine 41).* right\n+00023bd0: 2d68 616e 6420 636f 6f72 6469 6e61 7465 -hand coordinate\n+00023be0: 2073 7973 7465 6d3a 2020 2020 2020 2020 system: \n+00023bf0: 2020 496e 7472 6f64 7563 7469 6f6e 2062 Introduction b\n+00023c00: 7920 6578 616d 706c 652e 0a20 2020 2020 y example.. \n 00023c10: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00023c20: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00023c30: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00023c40: 2020 2020 2028 6c69 6e65 2020 3131 290a (line 11).\n-00023c50: 2a20 726f 7461 7469 6f6e 3a20 2020 2020 * rotation: \n+00023c40: 2020 2020 2020 2020 2028 6c69 6e65 2020 (line \n+00023c50: 3131 290a 2a20 726f 7461 7469 6f6e 3a20 11).* rotation: \n 00023c60: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00023c70: 2020 2020 2020 2020 204f 626a 6563 7420 Object \n-00023c80: 7472 616e 7366 6f72 6d73 2e20 2020 286c transforms. (l\n-00023c90: 696e 6520 2033 3929 0a2a 2072 6f74 6174 ine 39).* rotat\n-00023ca0: 696f 6e20 7472 616e 7366 6f72 6d3a 2020 ion transform: \n-00023cb0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00023cc0: 2020 5472 616e 7366 6f72 6d20 6c69 7465 Transform lite\n-00023cd0: 7261 6c73 2e20 2028 6c69 6e65 2020 3130 rals. (line 10\n-00023ce0: 290a 2a20 726f 7461 7469 6f6e 2074 7261 ).* rotation tra\n-00023cf0: 6e73 666f 726d 203c 313e 3a20 2020 2020 nsform <1>: \n-00023d00: 2020 2020 2020 2020 2020 2041 2068 6965 A hie\n-00023d10: 7261 7263 6869 6361 6c20 6d6f 6465 6c2e rarchical model.\n-00023d20: 0a20 2020 2020 2020 2020 2020 2020 2020 . \n+00023c70: 2020 2020 2020 2020 2020 2020 204f 626a Obj\n+00023c80: 6563 7420 7472 616e 7366 6f72 6d73 2e20 ect transforms. \n+00023c90: 2020 286c 696e 6520 2033 3929 0a2a 2072 (line 39).* r\n+00023ca0: 6f74 6174 696f 6e20 7472 616e 7366 6f72 otation transfor\n+00023cb0: 6d3a 2020 2020 2020 2020 2020 2020 2020 m: \n+00023cc0: 2020 2020 2020 5472 616e 7366 6f72 6d20 Transform \n+00023cd0: 6c69 7465 7261 6c73 2e20 2028 6c69 6e65 literals. (line\n+00023ce0: 2020 3130 290a 2a20 726f 7461 7469 6f6e 10).* rotation\n+00023cf0: 2074 7261 6e73 666f 726d 203c 313e 3a20 transform <1>: \n+00023d00: 2020 2020 2020 2020 2020 2020 2020 2041 A\n+00023d10: 2068 6965 7261 7263 6869 6361 6c20 6d6f hierarchical mo\n+00023d20: 6465 6c2e 0a20 2020 2020 2020 2020 2020 del.. \n 00023d30: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00023d40: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00023d50: 2020 2020 2020 2020 2020 2020 2020 2028 (\n-00023d60: 6c69 6e65 2020 3438 290a 2a20 726f 7461 line 48).* rota\n-00023d70: 7469 6f6e 2074 7261 6e73 666f 726d 203c tion transform <\n-00023d80: 323e 3a20 2020 2020 2020 2020 2020 2020 2>: \n-00023d90: 2020 2041 2068 6965 7261 7263 6869 6361 A hierarchica\n-00023da0: 6c20 6d6f 6465 6c2e 0a20 2020 2020 2020 l model.. \n+00023d50: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00023d60: 2020 2028 6c69 6e65 2020 3438 290a 2a20 (line 48).* \n+00023d70: 726f 7461 7469 6f6e 2074 7261 6e73 666f rotation transfo\n+00023d80: 726d 203c 323e 3a20 2020 2020 2020 2020 rm <2>: \n+00023d90: 2020 2020 2020 2041 2068 6965 7261 7263 A hierarc\n+00023da0: 6869 6361 6c20 6d6f 6465 6c2e 0a20 2020 hical model.. \n 00023db0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00023dc0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00023dd0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00023de0: 2020 2020 2020 2028 6c69 6e65 2020 3538 (line 58\n-00023df0: 290a 2a20 726f 7461 7469 6f6e 2074 7261 ).* rotation tra\n-00023e00: 6e73 666f 726d 203c 333e 3a20 2020 2020 nsform <3>: \n-00023e10: 2020 2020 2020 2020 2020 2041 2068 6965 A hie\n-00023e20: 7261 7263 6869 6361 6c20 6d6f 6465 6c2e rarchical model.\n-00023e30: 0a20 2020 2020 2020 2020 2020 2020 2020 . \n+00023de0: 2020 2020 2020 2020 2020 2028 6c69 6e65 (line\n+00023df0: 2020 3538 290a 2a20 726f 7461 7469 6f6e 58).* rotation\n+00023e00: 2074 7261 6e73 666f 726d 203c 333e 3a20 transform <3>: \n+00023e10: 2020 2020 2020 2020 2020 2020 2020 2041 A\n+00023e20: 2068 6965 7261 7263 6869 6361 6c20 6d6f hierarchical mo\n+00023e30: 6465 6c2e 0a20 2020 2020 2020 2020 2020 del.. \n 00023e40: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00023e50: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00023e60: 2020 2020 2020 2020 2020 2020 2020 2028 (\n-00023e70: 6c69 6e65 2020 3738 290a 2a20 726f 7461 line 78).* rota\n-00023e80: 7469 6f6e 2074 7261 6e73 666f 726d 203c tion transform <\n-00023e90: 343e 3a20 2020 2020 2020 2020 2020 2020 4>: \n-00023ea0: 2020 2041 2068 6965 7261 7263 6869 6361 A hierarchica\n-00023eb0: 6c20 6d6f 6465 6c2e 0a20 2020 2020 2020 l model.. \n+00023e60: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00023e70: 2020 2028 6c69 6e65 2020 3738 290a 2a20 (line 78).* \n+00023e80: 726f 7461 7469 6f6e 2074 7261 6e73 666f rotation transfo\n+00023e90: 726d 203c 343e 3a20 2020 2020 2020 2020 rm <4>: \n+00023ea0: 2020 2020 2020 2041 2068 6965 7261 7263 A hierarc\n+00023eb0: 6869 6361 6c20 6d6f 6465 6c2e 0a20 2020 hical model.. \n 00023ec0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00023ed0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00023ee0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00023ef0: 2020 2020 2020 2028 6c69 6e65 2020 3933 (line 93\n-00023f00: 290a 2a20 726f 7461 7469 6f6e 2c20 6365 ).* rotation, ce\n-00023f10: 6e74 6572 206f 663a 2020 2020 2020 2020 nter of: \n-00023f20: 2020 2020 2020 2020 2020 204f 626a 6563 Objec\n-00023f30: 7420 7472 616e 7366 6f72 6d73 2e20 2020 t transforms. \n-00023f40: 286c 696e 6520 2020 3829 0a2a 2072 6f74 (line 8).* rot\n-00023f50: 6174 696f 6e2c 2063 656e 7465 7220 6f66 ation, center of\n-00023f60: 203c 313e 3a20 2020 2020 2020 2020 2020 <1>: \n-00023f70: 2020 2020 4e65 7374 6564 2073 7765 6570 Nested sweep\n-00023f80: 732e 2020 2020 2020 2028 6c69 6e65 2020 s. (line \n-00023f90: 2036 290a 2a20 7275 6e6e 696e 6720 736b 6).* running sk\n-00023fa0: 6574 6368 3a20 2020 2020 2020 2020 2020 etch: \n-00023fb0: 2020 2020 2020 2020 2020 2020 2048 656c Hel\n-00023fc0: 6c6f 2077 6f72 6c64 2e20 2020 2020 2020 lo world. \n-00023fd0: 2020 286c 696e 6520 2020 3929 0a2a 2073 (line 9).* s\n-00023fe0: 6361 6c61 7220 6465 6669 6e69 7469 6f6e calar definition\n-00023ff0: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n-00024000: 2020 2020 2020 5377 6570 7420 6f62 6a65 Swept obje\n-00024010: 6374 732e 2020 2020 2020 2028 6c69 6e65 cts. (line\n-00024020: 2020 3133 290a 2a20 7363 616c 6172 206c 13).* scalar l\n-00024030: 6974 6572 616c 3a20 2020 2020 2020 2020 iteral: \n-00024040: 2020 2020 2020 2020 2020 2020 2020 2053 S\n-00024050: 6361 6c61 7220 6c69 7465 7261 6c73 2e20 calar literals. \n-00024060: 2020 2020 286c 696e 6520 2020 3629 0a2a (line 6).*\n-00024070: 2073 6361 6c61 7220 7265 6665 7265 6e63 scalar referenc\n-00024080: 653a 2020 2020 2020 2020 2020 2020 2020 e: \n-00024090: 2020 2020 2020 2020 466f 726d 7320 6f66 Forms of\n-000240a0: 2072 6566 6572 656e 6365 732e 2028 6c69 references. (li\n-000240b0: 6e65 2020 3133 290a 2a20 7363 616c 6520 ne 13).* scale \n-000240c0: 7472 616e 7366 6f72 6d3a 2020 2020 2020 transform: \n+00023ef0: 2020 2020 2020 2020 2020 2028 6c69 6e65 (line\n+00023f00: 2020 3933 290a 2a20 726f 7461 7469 6f6e 93).* rotation\n+00023f10: 2c20 6365 6e74 6572 206f 663a 2020 2020 , center of: \n+00023f20: 2020 2020 2020 2020 2020 2020 2020 204f O\n+00023f30: 626a 6563 7420 7472 616e 7366 6f72 6d73 bject transforms\n+00023f40: 2e20 2020 286c 696e 6520 2020 3829 0a2a . (line 8).*\n+00023f50: 2072 6f74 6174 696f 6e2c 2063 656e 7465 rotation, cente\n+00023f60: 7220 6f66 203c 313e 3a20 2020 2020 2020 r of <1>: \n+00023f70: 2020 2020 2020 2020 4e65 7374 6564 2073 Nested s\n+00023f80: 7765 6570 732e 2020 2020 2020 2028 6c69 weeps. (li\n+00023f90: 6e65 2020 2036 290a 2a20 7275 6e6e 696e ne 6).* runnin\n+00023fa0: 6720 736b 6574 6368 3a20 2020 2020 2020 g sketch: \n+00023fb0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00023fc0: 2048 656c 6c6f 2077 6f72 6c64 2e20 2020 Hello world. \n+00023fd0: 2020 2020 2020 286c 696e 6520 2020 3929 (line 9)\n+00023fe0: 0a2a 2073 6361 6c61 7220 6465 6669 6e69 .* scalar defini\n+00023ff0: 7469 6f6e 3a20 2020 2020 2020 2020 2020 tion: \n+00024000: 2020 2020 2020 2020 2020 5377 6570 7420 Swept \n+00024010: 6f62 6a65 6374 732e 2020 2020 2020 2028 objects. (\n+00024020: 6c69 6e65 2020 3133 290a 2a20 7363 616c line 13).* scal\n+00024030: 6172 206c 6974 6572 616c 3a20 2020 2020 ar literal: \n+00024040: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00024050: 2020 2053 6361 6c61 7220 6c69 7465 7261 Scalar litera\n+00024060: 6c73 2e20 2020 2020 286c 696e 6520 2020 ls. (line \n+00024070: 3629 0a2a 2073 6361 6c61 7220 7265 6665 6).* scalar refe\n+00024080: 7265 6e63 653a 2020 2020 2020 2020 2020 rence: \n+00024090: 2020 2020 2020 2020 2020 2020 466f 726d Form\n+000240a0: 7320 6f66 2072 6566 6572 656e 6365 732e s of references.\n+000240b0: 2028 6c69 6e65 2020 3133 290a 2a20 7363 (line 13).* sc\n+000240c0: 616c 6520 7472 616e 7366 6f72 6d3a 2020 ale transform: \n 000240d0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000240e0: 2054 7261 6e73 666f 726d 206c 6974 6572 Transform liter\n-000240f0: 616c 732e 2020 286c 696e 6520 2031 3629 als. (line 16)\n-00024100: 0a2a 2073 6361 6c65 2074 7261 6e73 666f .* scale transfo\n-00024110: 726d 203c 313e 3a20 2020 2020 2020 2020 rm <1>: \n-00024120: 2020 2020 2020 2020 2020 5472 616e 7366 Transf\n-00024130: 6f72 6d20 6c69 7465 7261 6c73 2e20 2028 orm literals. (\n-00024140: 6c69 6e65 2020 3137 290a 2a20 7363 656e line 17).* scen\n-00024150: 6520 6465 7363 7269 7074 696f 6e20 6c61 e description la\n-00024160: 6e67 7561 6765 3a20 2020 2020 2020 2020 nguage: \n-00024170: 2020 204c 616e 6775 6167 6520 6261 7369 Language basi\n-00024180: 6373 2e20 2020 2020 286c 696e 6520 2020 cs. (line \n-00024190: 3829 0a2a 2073 6365 6e65 2064 6573 6372 8).* scene descr\n-000241a0: 6970 7469 6f6e 206c 616e 6775 6167 6520 iption language \n-000241b0: 3c31 3e3a 2020 2020 2020 2020 4275 696c <1>: Buil\n-000241c0: 6469 6e67 2061 2064 7261 7769 6e67 2e20 ding a drawing. \n-000241d0: 2028 6c69 6e65 2020 2037 290a 2a20 7363 (line 7).* sc\n-000241e0: 6f70 652c 2069 6465 6e74 6966 6965 723a ope, identifier:\n-000241f0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00024200: 2020 2020 204f 626a 6563 7420 7472 616e Object tran\n-00024210: 7366 6f72 6d73 2e20 2020 286c 696e 6520 sforms. (line \n-00024220: 2032 3629 0a2a 2073 636f 7065 2c20 6964 26).* scope, id\n-00024230: 656e 7469 6669 6572 203c 313e 3a20 2020 entifier <1>: \n-00024240: 2020 2020 2020 2020 2020 2020 2020 426c Bl\n-00024250: 6f63 6b73 2e20 2020 2020 2020 2020 2020 ocks. \n-00024260: 2020 2028 6c69 6e65 2020 3133 290a 2a20 (line 13).* \n-00024270: 7369 6d70 6c65 2064 6566 696e 6974 696f simple definitio\n-00024280: 6e3a 2020 2020 2020 2020 2020 2020 2020 n: \n-00024290: 2020 2020 2020 2046 6f72 6d73 206f 6620 Forms of \n-000242a0: 6465 6669 6e69 7469 6f6e 732e 0a20 2020 definitions.. \n-000242b0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+000240e0: 2020 2020 2054 7261 6e73 666f 726d 206c Transform l\n+000240f0: 6974 6572 616c 732e 2020 286c 696e 6520 iterals. (line \n+00024100: 2031 3629 0a2a 2073 6361 6c65 2074 7261 16).* scale tra\n+00024110: 6e73 666f 726d 203c 313e 3a20 2020 2020 nsform <1>: \n+00024120: 2020 2020 2020 2020 2020 2020 2020 5472 Tr\n+00024130: 616e 7366 6f72 6d20 6c69 7465 7261 6c73 ansform literals\n+00024140: 2e20 2028 6c69 6e65 2020 3137 290a 2a20 . (line 17).* \n+00024150: 7363 656e 6520 6465 7363 7269 7074 696f scene descriptio\n+00024160: 6e20 6c61 6e67 7561 6765 3a20 2020 2020 n language: \n+00024170: 2020 2020 2020 204c 616e 6775 6167 6520 Language \n+00024180: 6261 7369 6373 2e20 2020 2020 286c 696e basics. (lin\n+00024190: 6520 2020 3829 0a2a 2073 6365 6e65 2064 e 8).* scene d\n+000241a0: 6573 6372 6970 7469 6f6e 206c 616e 6775 escription langu\n+000241b0: 6167 6520 3c31 3e3a 2020 2020 2020 2020 age <1>: \n+000241c0: 4275 696c 6469 6e67 2061 2064 7261 7769 Building a drawi\n+000241d0: 6e67 2e20 2028 6c69 6e65 2020 2037 290a ng. (line 7).\n+000241e0: 2a20 7363 6f70 652c 2069 6465 6e74 6966 * scope, identif\n+000241f0: 6965 723a 2020 2020 2020 2020 2020 2020 ier: \n+00024200: 2020 2020 2020 2020 204f 626a 6563 7420 Object \n+00024210: 7472 616e 7366 6f72 6d73 2e20 2020 286c transforms. (l\n+00024220: 696e 6520 2032 3629 0a2a 2073 636f 7065 ine 26).* scope\n+00024230: 2c20 6964 656e 7469 6669 6572 203c 313e , identifier <1>\n+00024240: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+00024250: 2020 426c 6f63 6b73 2e20 2020 2020 2020 Blocks. \n+00024260: 2020 2020 2020 2028 6c69 6e65 2020 3133 (line 13\n+00024270: 290a 2a20 7369 6d70 6c65 2064 6566 696e ).* simple defin\n+00024280: 6974 696f 6e3a 2020 2020 2020 2020 2020 ition: \n+00024290: 2020 2020 2020 2020 2020 2046 6f72 6d73 Forms\n+000242a0: 206f 6620 6465 6669 6e69 7469 6f6e 732e of definitions.\n+000242b0: 0a20 2020 2020 2020 2020 2020 2020 2020 . \n 000242c0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 000242d0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000242e0: 2020 2020 2020 2020 2020 2028 6c69 6e65 (line\n-000242f0: 2020 2036 290a 2a20 736f 6c69 643a 2020 6).* solid: \n-00024300: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00024310: 2020 2020 2020 2020 2020 2020 2020 2044 D\n-00024320: 7261 7769 6e67 2061 2073 6f6c 6964 2e20 rawing a solid. \n-00024330: 2020 2020 286c 696e 6520 2020 3629 0a2a (line 6).*\n-00024340: 2073 7065 6369 616c 2061 7267 756d 656e special argumen\n-00024350: 7420 7375 6273 7469 7475 7469 6f6e 3a20 t substitution: \n-00024360: 2020 2020 2020 2020 5370 6563 6961 6c20 Special \n-00024370: 6f62 6a65 6374 732e 2020 2020 2028 6c69 objects. (li\n-00024380: 6e65 2020 3238 290a 2a20 7370 6563 6961 ne 28).* specia\n-00024390: 6c20 6172 6775 6d65 6e74 2073 7562 7374 l argument subst\n-000243a0: 6974 7574 696f 6e20 3c31 3e3a 2020 2020 itution <1>: \n-000243b0: 2053 7065 6369 616c 732e 2020 2020 2020 Specials. \n-000243c0: 2020 2020 2020 286c 696e 6520 2031 3029 (line 10)\n-000243d0: 0a2a 2073 7065 6369 616c 206f 626a 6563 .* special objec\n-000243e0: 743a 2020 2020 2020 2020 2020 2020 2020 t: \n-000243f0: 2020 2020 2020 2020 2020 5370 6563 6961 Specia\n-00024400: 6c20 6f62 6a65 6374 732e 2020 2020 2028 l objects. (\n-00024410: 6c69 6e65 2020 2036 290a 2a20 7370 6563 line 6).* spec\n-00024420: 6961 6c20 6f62 6a65 6374 733a 2020 2020 ial objects: \n+000242e0: 2020 2020 2020 2020 2020 2020 2020 2028 (\n+000242f0: 6c69 6e65 2020 2036 290a 2a20 736f 6c69 line 6).* soli\n+00024300: 643a 2020 2020 2020 2020 2020 2020 2020 d: \n+00024310: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00024320: 2020 2044 7261 7769 6e67 2061 2073 6f6c Drawing a sol\n+00024330: 6964 2e20 2020 2020 286c 696e 6520 2020 id. (line \n+00024340: 3629 0a2a 2073 7065 6369 616c 2061 7267 6).* special arg\n+00024350: 756d 656e 7420 7375 6273 7469 7475 7469 ument substituti\n+00024360: 6f6e 3a20 2020 2020 2020 2020 5370 6563 on: Spec\n+00024370: 6961 6c20 6f62 6a65 6374 732e 2020 2020 ial objects. \n+00024380: 2028 6c69 6e65 2020 3238 290a 2a20 7370 (line 28).* sp\n+00024390: 6563 6961 6c20 6172 6775 6d65 6e74 2073 ecial argument s\n+000243a0: 7562 7374 6974 7574 696f 6e20 3c31 3e3a ubstitution <1>:\n+000243b0: 2020 2020 2053 7065 6369 616c 732e 2020 Specials. \n+000243c0: 2020 2020 2020 2020 2020 286c 696e 6520 (line \n+000243d0: 2031 3029 0a2a 2073 7065 6369 616c 206f 10).* special o\n+000243e0: 626a 6563 743a 2020 2020 2020 2020 2020 bject: \n+000243f0: 2020 2020 2020 2020 2020 2020 2020 5370 Sp\n+00024400: 6563 6961 6c20 6f62 6a65 6374 732e 2020 ecial objects. \n+00024410: 2020 2028 6c69 6e65 2020 2036 290a 2a20 (line 6).* \n+00024420: 7370 6563 6961 6c20 6f62 6a65 6374 733a special objects:\n 00024430: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00024440: 2020 2041 2074 6563 686e 6963 616c 2064 A technical d\n-00024450: 7261 7769 6e67 2e20 286c 696e 6520 2033 rawing. (line 3\n-00024460: 3829 0a2a 2073 706c 6974 7469 6e67 2c20 8).* splitting, \n-00024470: 6c69 6e65 2061 6e64 2073 7572 6661 6365 line and surface\n-00024480: 3a20 2020 2020 2020 2020 2020 4865 6c6c : Hell\n-00024490: 6f20 776f 726c 642e 2020 2020 2020 2020 o world. \n-000244a0: 2028 6c69 6e65 2020 3139 290a 2a20 7370 (line 19).* sp\n-000244b0: 6c69 7474 696e 672c 206c 696e 6520 616e litting, line an\n-000244c0: 6420 7375 7266 6163 6520 3c31 3e3a 2020 d surface <1>: \n-000244d0: 2020 2020 2048 6964 6465 6e20 7375 7266 Hidden surf\n-000244e0: 6163 6520 7265 6d6f 7661 6c2e 0a20 2020 ace removal.. \n-000244f0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00024440: 2020 2020 2020 2041 2074 6563 686e 6963 A technic\n+00024450: 616c 2064 7261 7769 6e67 2e20 286c 696e al drawing. (lin\n+00024460: 6520 2033 3829 0a2a 2073 706c 6974 7469 e 38).* splitti\n+00024470: 6e67 2c20 6c69 6e65 2061 6e64 2073 7572 ng, line and sur\n+00024480: 6661 6365 3a20 2020 2020 2020 2020 2020 face: \n+00024490: 4865 6c6c 6f20 776f 726c 642e 2020 2020 Hello world. \n+000244a0: 2020 2020 2028 6c69 6e65 2020 3139 290a (line 19).\n+000244b0: 2a20 7370 6c69 7474 696e 672c 206c 696e * splitting, lin\n+000244c0: 6520 616e 6420 7375 7266 6163 6520 3c31 e and surface <1\n+000244d0: 3e3a 2020 2020 2020 2048 6964 6465 6e20 >: Hidden \n+000244e0: 7375 7266 6163 6520 7265 6d6f 7661 6c2e surface removal.\n+000244f0: 0a20 2020 2020 2020 2020 2020 2020 2020 . \n 00024500: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00024510: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00024520: 2020 2020 2020 2020 2020 2028 6c69 6e65 (line\n-00024530: 2020 3231 290a 2a20 7375 6273 7469 7475 21).* substitu\n-00024540: 7469 6f6e 2c20 7370 6563 6961 6c3a 2020 tion, special: \n-00024550: 2020 2020 2020 2020 2020 2020 2020 2053 S\n-00024560: 7065 6369 616c 206f 626a 6563 7473 2e20 pecial objects. \n-00024570: 2020 2020 286c 696e 6520 2032 3829 0a2a (line 28).*\n-00024580: 2073 7572 6661 6365 3a20 2020 2020 2020 surface: \n+00024520: 2020 2020 2020 2020 2020 2020 2020 2028 (\n+00024530: 6c69 6e65 2020 3231 290a 2a20 7375 6273 line 21).* subs\n+00024540: 7469 7475 7469 6f6e 2c20 7370 6563 6961 titution, specia\n+00024550: 6c3a 2020 2020 2020 2020 2020 2020 2020 l: \n+00024560: 2020 2053 7065 6369 616c 206f 626a 6563 Special objec\n+00024570: 7473 2e20 2020 2020 286c 696e 6520 2032 ts. (line 2\n+00024580: 3829 0a2a 2073 7572 6661 6365 3a20 2020 8).* surface: \n 00024590: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000245a0: 2020 2020 2020 2020 506f 6c79 6c69 6e65 Polyline\n-000245b0: 2073 7765 6570 732e 2020 2020 2028 6c69 sweeps. (li\n-000245c0: 6e65 2020 2036 290a 2a20 7377 6565 703a ne 6).* sweep:\n-000245d0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+000245a0: 2020 2020 2020 2020 2020 2020 506f 6c79 Poly\n+000245b0: 6c69 6e65 2073 7765 6570 732e 2020 2020 line sweeps. \n+000245c0: 2028 6c69 6e65 2020 2036 290a 2a20 7377 (line 6).* sw\n+000245d0: 6565 703a 2020 2020 2020 2020 2020 2020 eep: \n 000245e0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000245f0: 2053 7765 7074 206f 626a 6563 7473 2e20 Swept objects. \n-00024600: 2020 2020 2020 286c 696e 6520 2020 3629 (line 6)\n-00024610: 0a2a 2073 7765 7074 2062 6f63 6b3a 2020 .* swept bock: \n-00024620: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00024630: 2020 2020 2020 2020 2020 5377 6570 7420 Swept \n-00024640: 626c 6f63 6b73 2e20 2020 2020 2020 2028 blocks. (\n-00024650: 6c69 6e65 2020 2036 290a 2a20 7377 6570 line 6).* swep\n-00024660: 7420 6c69 6e65 3a20 2020 2020 2020 2020 t line: \n+000245f0: 2020 2020 2053 7765 7074 206f 626a 6563 Swept objec\n+00024600: 7473 2e20 2020 2020 2020 286c 696e 6520 ts. (line \n+00024610: 2020 3629 0a2a 2073 7765 7074 2062 6f63 6).* swept boc\n+00024620: 6b3a 2020 2020 2020 2020 2020 2020 2020 k: \n+00024630: 2020 2020 2020 2020 2020 2020 2020 5377 Sw\n+00024640: 6570 7420 626c 6f63 6b73 2e20 2020 2020 ept blocks. \n+00024650: 2020 2028 6c69 6e65 2020 2036 290a 2a20 (line 6).* \n+00024660: 7377 6570 7420 6c69 6e65 3a20 2020 2020 swept line: \n 00024670: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00024680: 2020 2050 6f6c 796c 696e 6520 7377 6565 Polyline swee\n-00024690: 7073 2e20 2020 2020 286c 696e 6520 2020 ps. (line \n-000246a0: 3629 0a2a 2073 7765 7074 206c 696e 6520 6).* swept line \n-000246b0: 3c31 3e3a 2020 2020 2020 2020 2020 2020 <1>: \n-000246c0: 2020 2020 2020 2020 2020 2020 5377 6570 Swep\n-000246d0: 7420 6c69 6e65 732e 2020 2020 2020 2020 t lines. \n-000246e0: 2028 6c69 6e65 2020 2036 290a 2a20 7377 (line 6).* sw\n-000246f0: 6570 7420 6c69 6e65 203c 323e 3a20 2020 ept line <2>: \n-00024700: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00024710: 2020 2020 2041 2074 6563 686e 6963 616c A technical\n-00024720: 2064 7261 7769 6e67 2e20 286c 696e 6520 drawing. (line \n-00024730: 2032 3029 0a2a 2073 7765 7074 206f 626a 20).* swept obj\n-00024740: 6563 743a 2020 2020 2020 2020 2020 2020 ect: \n-00024750: 2020 2020 2020 2020 2020 2020 2020 5377 Sw\n-00024760: 6570 7420 6f62 6a65 6374 732e 2020 2020 ept objects. \n-00024770: 2020 2028 6c69 6e65 2020 2036 290a 2a20 (line 6).* \n-00024780: 7377 6570 7420 6f62 6a65 6374 203c 313e swept object <1>\n-00024790: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n-000247a0: 2020 2020 2020 2053 7765 6570 732e 2020 Sweeps. \n-000247b0: 2020 2020 2020 2020 2020 2020 286c 696e (lin\n-000247c0: 6520 2020 3929 0a2a 2073 7765 7074 2070 e 9).* swept p\n-000247d0: 6f69 6e74 3a20 2020 2020 2020 2020 2020 oint: \n+00024680: 2020 2020 2020 2050 6f6c 796c 696e 6520 Polyline \n+00024690: 7377 6565 7073 2e20 2020 2020 286c 696e sweeps. (lin\n+000246a0: 6520 2020 3629 0a2a 2073 7765 7074 206c e 6).* swept l\n+000246b0: 696e 6520 3c31 3e3a 2020 2020 2020 2020 ine <1>: \n+000246c0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+000246d0: 5377 6570 7420 6c69 6e65 732e 2020 2020 Swept lines. \n+000246e0: 2020 2020 2028 6c69 6e65 2020 2036 290a (line 6).\n+000246f0: 2a20 7377 6570 7420 6c69 6e65 203c 323e * swept line <2>\n+00024700: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+00024710: 2020 2020 2020 2020 2041 2074 6563 686e A techn\n+00024720: 6963 616c 2064 7261 7769 6e67 2e20 286c ical drawing. 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(l\n-00024850: 696e 6520 2020 3629 0a2a 2073 7765 7074 ine 6).* swept\n-00024860: 2070 6f6c 7967 6f6e 3a20 2020 2020 2020 polygon: \n+000247f0: 2020 2020 5377 6570 7420 6f62 6a65 6374 Swept object\n+00024800: 732e 2020 2020 2020 2028 6c69 6e65 2020 s. (line \n+00024810: 3131 290a 2a20 7377 6570 7420 706f 696e 11).* swept poin\n+00024820: 7420 3c31 3e3a 2020 2020 2020 2020 2020 t <1>: \n+00024830: 2020 2020 2020 2020 2020 2020 2053 7765 Swe\n+00024840: 7074 2070 6f69 6e74 732e 2020 2020 2020 pt points. \n+00024850: 2020 286c 696e 6520 2020 3629 0a2a 2073 (line 6).* s\n+00024860: 7765 7074 2070 6f6c 7967 6f6e 3a20 2020 wept polygon: \n 00024870: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00024880: 2020 506f 6c79 676f 6e20 7377 6565 7073 Polygon sweeps\n-00024890: 2e20 2020 2020 2028 6c69 6e65 2020 2036 . (line 6\n-000248a0: 290a 2a20 7377 6570 7420 706f 6c79 676f ).* swept polygo\n-000248b0: 6e20 3c31 3e3a 2020 2020 2020 2020 2020 n <1>: \n-000248c0: 2020 2020 2020 2020 2020 2041 2068 6965 A hie\n-000248d0: 7261 7263 6869 6361 6c20 6d6f 6465 6c2e rarchical model.\n-000248e0: 0a20 2020 2020 2020 2020 2020 2020 2020 . \n+00024880: 2020 2020 2020 506f 6c79 676f 6e20 7377 Polygon sw\n+00024890: 6565 7073 2e20 2020 2020 2028 6c69 6e65 eeps. (line\n+000248a0: 2020 2036 290a 2a20 7377 6570 7420 706f 6).* swept po\n+000248b0: 6c79 676f 6e20 3c31 3e3a 2020 2020 2020 lygon <1>: \n+000248c0: 2020 2020 2020 2020 2020 2020 2020 2041 A\n+000248d0: 2068 6965 7261 7263 6869 6361 6c20 6d6f hierarchical mo\n+000248e0: 6465 6c2e 0a20 2020 2020 2020 2020 2020 del.. \n 000248f0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00024900: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00024910: 2020 2020 2020 2020 2020 2020 2020 2028 (\n-00024920: 6c69 6e65 2031 3232 290a 2a20 7461 6720 line 122).* tag \n-00024930: 6465 6669 6e69 7469 6f6e 3a20 2020 2020 definition: \n+00024910: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00024920: 2020 2028 6c69 6e65 2031 3232 290a 2a20 (line 122).* \n+00024930: 7461 6720 6465 6669 6e69 7469 6f6e 3a20 tag definition: \n 00024940: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00024950: 2020 2044 6566 696e 6974 696f 6e73 2e20 Definitions. \n-00024960: 2020 2020 2020 2020 286c 696e 6520 2031 (line 1\n-00024970: 3829 0a2a 2074 6167 2064 6566 696e 6974 8).* tag definit\n-00024980: 696f 6e20 3c31 3e3a 2020 2020 2020 2020 ion <1>: \n-00024990: 2020 2020 2020 2020 2020 2020 466f 726d Form\n-000249a0: 7320 6f66 2064 6566 696e 6974 696f 6e73 s of definitions\n-000249b0: 2e0a 2020 2020 2020 2020 2020 2020 2020 .. \n+00024950: 2020 2020 2020 2044 6566 696e 6974 696f Definitio\n+00024960: 6e73 2e20 2020 2020 2020 2020 286c 696e ns. (lin\n+00024970: 6520 2031 3829 0a2a 2074 6167 2064 6566 e 18).* tag def\n+00024980: 696e 6974 696f 6e20 3c31 3e3a 2020 2020 inition <1>: \n+00024990: 2020 2020 2020 2020 2020 2020 2020 2020 \n+000249a0: 466f 726d 7320 6f66 2064 6566 696e 6974 Forms of definit\n+000249b0: 696f 6e73 2e0a 2020 2020 2020 2020 2020 ions.. \n 000249c0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 000249d0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 000249e0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000249f0: 286c 696e 6520 2020 3729 0a2a 2074 6167 (line 7).* tag\n-00024a00: 2064 6566 696e 6974 696f 6e20 3c32 3e3a definition <2>:\n-00024a10: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00024a20: 2020 2020 436f 6d6d 616e 6420 6c69 6e65 Command line\n-00024a30: 2e20 2020 2020 2020 2028 6c69 6e65 2020 . (line \n-00024a40: 3533 290a 2a20 7461 6720 7265 6665 7265 53).* tag refere\n-00024a50: 6e63 653a 2020 2020 2020 2020 2020 2020 nce: \n-00024a60: 2020 2020 2020 2020 2020 2020 2046 6f72 For\n-00024a70: 6d73 206f 6620 6465 6669 6e69 7469 6f6e ms of definition\n-00024a80: 732e 0a20 2020 2020 2020 2020 2020 2020 s.. \n+000249f0: 2020 2020 286c 696e 6520 2020 3729 0a2a (line 7).*\n+00024a00: 2074 6167 2064 6566 696e 6974 696f 6e20 tag definition \n+00024a10: 3c32 3e3a 2020 2020 2020 2020 2020 2020 <2>: \n+00024a20: 2020 2020 2020 2020 436f 6d6d 616e 6420 Command \n+00024a30: 6c69 6e65 2e20 2020 2020 2020 2028 6c69 line. (li\n+00024a40: 6e65 2020 3533 290a 2a20 7461 6720 7265 ne 53).* tag re\n+00024a50: 6665 7265 6e63 653a 2020 2020 2020 2020 ference: \n+00024a60: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00024a70: 2046 6f72 6d73 206f 6620 6465 6669 6e69 Forms of defini\n+00024a80: 7469 6f6e 732e 0a20 2020 2020 2020 2020 tions.. \n 00024a90: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00024aa0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00024ab0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00024ac0: 2028 6c69 6e65 2020 3230 290a 2a20 7461 (line 20).* ta\n-00024ad0: 6720 7265 6665 7265 6e63 6520 3c31 3e3a g reference <1>:\n-00024ae0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00024af0: 2020 2020 2046 6f72 6d73 206f 6620 7265 Forms of re\n-00024b00: 6665 7265 6e63 6573 2e20 286c 696e 6520 ferences. (line \n-00024b10: 2031 3929 0a2a 2074 656d 706c 6174 652c 19).* template,\n-00024b20: 2064 6f63 756d 656e 743a 2020 2020 2020 document: \n-00024b30: 2020 2020 2020 2020 2020 2020 2020 4865 He\n-00024b40: 6c6c 6f20 776f 726c 642e 2020 2020 2020 llo world. \n-00024b50: 2020 2028 6c69 6e65 2020 3332 290a 2a20 (line 32).* \n-00024b60: 7465 6d70 6c61 7465 2c20 646f 6375 6d65 template, docume\n-00024b70: 6e74 203c 313e 3a20 2020 2020 2020 2020 nt <1>: \n-00024b80: 2020 2020 2020 2043 6f6d 6d61 6e64 206c Command l\n-00024b90: 696e 652e 2020 2020 2020 2020 286c 696e ine. (lin\n-00024ba0: 6520 2033 3629 0a2a 2074 6574 7261 6865 e 36).* tetrahe\n-00024bb0: 6472 6f6e 3a20 2020 2020 2020 2020 2020 dron: \n+00024ac0: 2020 2020 2028 6c69 6e65 2020 3230 290a (line 20).\n+00024ad0: 2a20 7461 6720 7265 6665 7265 6e63 6520 * tag reference \n+00024ae0: 3c31 3e3a 2020 2020 2020 2020 2020 2020 <1>: \n+00024af0: 2020 2020 2020 2020 2046 6f72 6d73 206f Forms o\n+00024b00: 6620 7265 6665 7265 6e63 6573 2e20 286c f references. (l\n+00024b10: 696e 6520 2031 3929 0a2a 2074 656d 706c ine 19).* templ\n+00024b20: 6174 652c 2064 6f63 756d 656e 743a 2020 ate, document: \n+00024b30: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00024b40: 2020 4865 6c6c 6f20 776f 726c 642e 2020 Hello world. \n+00024b50: 2020 2020 2020 2028 6c69 6e65 2020 3332 (line 32\n+00024b60: 290a 2a20 7465 6d70 6c61 7465 2c20 646f ).* template, do\n+00024b70: 6375 6d65 6e74 203c 313e 3a20 2020 2020 cument <1>: \n+00024b80: 2020 2020 2020 2020 2020 2043 6f6d 6d61 Comma\n+00024b90: 6e64 206c 696e 652e 2020 2020 2020 2020 nd line. \n+00024ba0: 286c 696e 6520 2033 3629 0a2a 2074 6574 (line 36).* tet\n+00024bb0: 7261 6865 6472 6f6e 3a20 2020 2020 2020 rahedron: \n 00024bc0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00024bd0: 4472 6177 696e 6720 6120 736f 6c69 642e Drawing a solid.\n-00024be0: 2020 2020 2028 6c69 6e65 2020 2039 290a (line 9).\n-00024bf0: 2a20 7469 636b 206f 7065 7261 746f 7220 * tick operator \n-00024c00: 2827 293a 2020 2020 2020 2020 2020 2020 ('): \n-00024c10: 2020 2020 2020 2020 2041 6666 696e 6520 Affine \n-00024c20: 6172 6974 686d 6574 6963 2e20 2020 286c arithmetic. (l\n-00024c30: 696e 6520 2033 3729 0a2a 2074 6963 6b20 ine 37).* tick \n-00024c40: 6f70 6572 6174 6f72 2028 2729 203c 313e operator (') <1>\n-00024c50: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n-00024c60: 2020 5477 6f2d 6f70 6572 616e 6420 2862 Two-operand (b\n-00024c70: 696e 6172 7929 2066 6f72 6d73 2e0a 2020 inary) forms.. \n-00024c80: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00024bd0: 2020 2020 4472 6177 696e 6720 6120 736f Drawing a so\n+00024be0: 6c69 642e 2020 2020 2028 6c69 6e65 2020 lid. (line \n+00024bf0: 2039 290a 2a20 7469 636b 206f 7065 7261 9).* tick opera\n+00024c00: 746f 7220 2827 293a 2020 2020 2020 2020 tor ('): \n+00024c10: 2020 2020 2020 2020 2020 2020 2041 6666 Aff\n+00024c20: 696e 6520 6172 6974 686d 6574 6963 2e20 ine arithmetic. \n+00024c30: 2020 286c 696e 6520 2033 3729 0a2a 2074 (line 37).* t\n+00024c40: 6963 6b20 6f70 6572 6174 6f72 2028 2729 ick operator (')\n+00024c50: 203c 313e 3a20 2020 2020 2020 2020 2020 <1>: \n+00024c60: 2020 2020 2020 5477 6f2d 6f70 6572 616e Two-operan\n+00024c70: 6420 2862 696e 6172 7929 2066 6f72 6d73 d (binary) forms\n+00024c80: 2e0a 2020 2020 2020 2020 2020 2020 2020 .. \n 00024c90: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00024ca0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00024cb0: 2020 2020 2020 2020 2020 2020 286c 696e (lin\n-00024cc0: 6520 2034 3129 0a2a 2054 696b 5a2f 5047 e 41).* TikZ/PG\n-00024cd0: 463a 2020 2020 2020 2020 2020 2020 2020 F: \n+00024cb0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00024cc0: 286c 696e 6520 2034 3129 0a2a 2054 696b (line 41).* Tik\n+00024cd0: 5a2f 5047 463a 2020 2020 2020 2020 2020 Z/PGF: \n 00024ce0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00024cf0: 4162 6f75 7420 736b 6574 6368 2e20 2020 About sketch. \n-00024d00: 2020 2020 2028 6c69 6e65 2020 3232 290a (line 22).\n-00024d10: 2a20 746f 726f 6964 3a20 2020 2020 2020 * toroid: \n+00024cf0: 2020 2020 4162 6f75 7420 736b 6574 6368 About sketch\n+00024d00: 2e20 2020 2020 2020 2028 6c69 6e65 2020 . (line \n+00024d10: 3232 290a 2a20 746f 726f 6964 3a20 2020 22).* toroid: \n 00024d20: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00024d30: 2020 2020 2020 2020 204e 6573 7465 6420 Nested \n-00024d40: 7377 6565 7073 2e20 2020 2020 2020 286c sweeps. (l\n-00024d50: 696e 6520 2020 3829 0a2a 2074 7261 6e73 ine 8).* trans\n-00024d60: 666f 726d 3a20 2020 2020 2020 2020 2020 form: \n+00024d30: 2020 2020 2020 2020 2020 2020 204e 6573 Nes\n+00024d40: 7465 6420 7377 6565 7073 2e20 2020 2020 ted sweeps. \n+00024d50: 2020 286c 696e 6520 2020 3829 0a2a 2074 (line 8).* t\n+00024d60: 7261 6e73 666f 726d 3a20 2020 2020 2020 ransform: \n 00024d70: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00024d80: 2020 4f62 6a65 6374 2074 7261 6e73 666f Object transfo\n-00024d90: 726d 732e 2020 2028 6c69 6e65 2020 2036 rms. (line 6\n-00024da0: 290a 2a20 7472 616e 7366 6f72 6d20 3c31 ).* transform <1\n-00024db0: 3e3a 2020 2020 2020 2020 2020 2020 2020 >: \n-00024dc0: 2020 2020 2020 2020 2020 2054 776f 2d6f Two-o\n-00024dd0: 7065 7261 6e64 2028 6269 6e61 7279 2920 perand (binary) \n-00024de0: 666f 726d 732e 0a20 2020 2020 2020 2020 forms.. \n+00024d80: 2020 2020 2020 4f62 6a65 6374 2074 7261 Object tra\n+00024d90: 6e73 666f 726d 732e 2020 2028 6c69 6e65 nsforms. (line\n+00024da0: 2020 2036 290a 2a20 7472 616e 7366 6f72 6).* transfor\n+00024db0: 6d20 3c31 3e3a 2020 2020 2020 2020 2020 m <1>: \n+00024dc0: 2020 2020 2020 2020 2020 2020 2020 2054 T\n+00024dd0: 776f 2d6f 7065 7261 6e64 2028 6269 6e61 wo-operand (bina\n+00024de0: 7279 2920 666f 726d 732e 0a20 2020 2020 ry) forms.. \n 00024df0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00024e00: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00024e10: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00024e20: 2020 2020 2028 6c69 6e65 2020 3238 290a (line 28).\n-00024e30: 2a20 7472 616e 7366 6f72 6d20 3c32 3e3a * transform <2>:\n-00024e40: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00024e50: 2020 2020 2020 2020 2053 7765 6570 732e Sweeps.\n-00024e60: 2020 2020 2020 2020 2020 2020 2020 286c (l\n-00024e70: 696e 6520 2031 3129 0a2a 2074 7261 6e73 ine 11).* trans\n-00024e80: 666f 726d 203c 333e 3a20 2020 2020 2020 form <3>: \n+00024e20: 2020 2020 2020 2020 2028 6c69 6e65 2020 (line \n+00024e30: 3238 290a 2a20 7472 616e 7366 6f72 6d20 28).* transform \n+00024e40: 3c32 3e3a 2020 2020 2020 2020 2020 2020 <2>: \n+00024e50: 2020 2020 2020 2020 2020 2020 2053 7765 Swe\n+00024e60: 6570 732e 2020 2020 2020 2020 2020 2020 eps. \n+00024e70: 2020 286c 696e 6520 2031 3129 0a2a 2074 (line 11).* t\n+00024e80: 7261 6e73 666f 726d 203c 333e 3a20 2020 ransform <3>: \n 00024e90: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00024ea0: 2020 5265 7065 6174 732e 2020 2020 2020 Repeats. \n-00024eb0: 2020 2020 2020 2028 6c69 6e65 2020 2039 (line 9\n-00024ec0: 290a 2a20 7472 616e 7366 6f72 6d20 3c34 ).* transform <4\n-00024ed0: 3e3a 2020 2020 2020 2020 2020 2020 2020 >: \n-00024ee0: 2020 2020 2020 2020 2020 204f 7665 7276 Overv\n-00024ef0: 6965 772e 2020 2020 2020 2020 2020 2020 iew. \n-00024f00: 286c 696e 6520 2020 3729 0a2a 2074 7261 (line 7).* tra\n-00024f10: 6e73 666f 726d 2064 6566 696e 6974 696f nsform definitio\n-00024f20: 6e3a 2020 2020 2020 2020 2020 2020 2020 n: \n-00024f30: 2020 2020 4f76 6572 7669 6577 2e20 2020 Overview. \n-00024f40: 2020 2020 2020 2020 2028 6c69 6e65 2020 (line \n-00024f50: 3335 290a 2a20 7472 616e 7366 6f72 6d20 35).* transform \n-00024f60: 6c69 7465 7261 6c3a 2020 2020 2020 2020 literal: \n-00024f70: 2020 2020 2020 2020 2020 2020 2054 7261 Tra\n-00024f80: 6e73 666f 726d 206c 6974 6572 616c 732e nsform literals.\n-00024f90: 2020 286c 696e 6520 2020 3629 0a2a 2074 (line 6).* t\n-00024fa0: 7261 6e73 666f 726d 2072 6566 6572 656e ransform referen\n-00024fb0: 6365 3a20 2020 2020 2020 2020 2020 2020 ce: \n-00024fc0: 2020 2020 2020 466f 726d 7320 6f66 2072 Forms of r\n-00024fd0: 6566 6572 656e 6365 732e 2028 6c69 6e65 eferences. (line\n-00024fe0: 2020 3136 290a 2a20 7472 616e 7366 6f72 16).* transfor\n-00024ff0: 6d20 7265 6665 7265 6e63 6520 3c31 3e3a m reference <1>:\n-00025000: 2020 2020 2020 2020 2020 2020 2020 204f O\n-00025010: 7665 7276 6965 772e 2020 2020 2020 2020 verview. \n-00025020: 2020 2020 286c 696e 6520 2033 3529 0a2a (line 35).*\n-00025030: 2074 7261 6e73 666f 726d 2c20 6469 7265 transform, dire\n-00025040: 6374 3a20 2020 2020 2020 2020 2020 2020 ct: \n-00025050: 2020 2020 2020 2020 5472 616e 7366 6f72 Transfor\n-00025060: 6d20 6c69 7465 7261 6c73 2e20 2028 6c69 m literals. (li\n-00025070: 6e65 2020 3437 290a 2a20 7472 616e 7366 ne 47).* transf\n-00025080: 6f72 6d2c 2064 6972 6563 7420 3c31 3e3a orm, direct <1>:\n-00025090: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000250a0: 2054 7261 6e73 666f 726d 206c 6974 6572 Transform liter\n-000250b0: 616c 732e 2020 286c 696e 6520 2035 3329 als. (line 53)\n-000250c0: 0a2a 2074 7261 6e73 666f 726d 2c20 726f .* transform, ro\n-000250d0: 7461 7469 6f6e 3a20 2020 2020 2020 2020 tation: \n-000250e0: 2020 2020 2020 2020 2020 5472 616e 7366 Transf\n-000250f0: 6f72 6d20 6c69 7465 7261 6c73 2e20 2028 orm literals. (\n-00025100: 6c69 6e65 2020 3130 290a 2a20 7472 616e line 10).* tran\n-00025110: 7366 6f72 6d2c 2072 6f74 6174 696f 6e20 sform, rotation \n-00025120: 3c31 3e3a 2020 2020 2020 2020 2020 2020 <1>: \n-00025130: 2020 2041 2068 6965 7261 7263 6869 6361 A hierarchica\n-00025140: 6c20 6d6f 6465 6c2e 0a20 2020 2020 2020 l model.. \n+00024ea0: 2020 2020 2020 5265 7065 6174 732e 2020 Repeats. \n+00024eb0: 2020 2020 2020 2020 2020 2028 6c69 6e65 (line\n+00024ec0: 2020 2039 290a 2a20 7472 616e 7366 6f72 9).* transfor\n+00024ed0: 6d20 3c34 3e3a 2020 2020 2020 2020 2020 m <4>: \n+00024ee0: 2020 2020 2020 2020 2020 2020 2020 204f O\n+00024ef0: 7665 7276 6965 772e 2020 2020 2020 2020 verview. \n+00024f00: 2020 2020 286c 696e 6520 2020 3729 0a2a (line 7).*\n+00024f10: 2074 7261 6e73 666f 726d 2064 6566 696e transform defin\n+00024f20: 6974 696f 6e3a 2020 2020 2020 2020 2020 ition: \n+00024f30: 2020 2020 2020 2020 4f76 6572 7669 6577 Overview\n+00024f40: 2e20 2020 2020 2020 2020 2020 2028 6c69 . (li\n+00024f50: 6e65 2020 3335 290a 2a20 7472 616e 7366 ne 35).* transf\n+00024f60: 6f72 6d20 6c69 7465 7261 6c3a 2020 2020 orm literal: \n+00024f70: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00024f80: 2054 7261 6e73 666f 726d 206c 6974 6572 Transform liter\n+00024f90: 616c 732e 2020 286c 696e 6520 2020 3629 als. (line 6)\n+00024fa0: 0a2a 2074 7261 6e73 666f 726d 2072 6566 .* transform ref\n+00024fb0: 6572 656e 6365 3a20 2020 2020 2020 2020 erence: \n+00024fc0: 2020 2020 2020 2020 2020 466f 726d 7320 Forms \n+00024fd0: 6f66 2072 6566 6572 656e 6365 732e 2028 of references. (\n+00024fe0: 6c69 6e65 2020 3136 290a 2a20 7472 616e line 16).* tran\n+00024ff0: 7366 6f72 6d20 7265 6665 7265 6e63 6520 sform reference \n+00025000: 3c31 3e3a 2020 2020 2020 2020 2020 2020 <1>: \n+00025010: 2020 204f 7665 7276 6965 772e 2020 2020 Overview. \n+00025020: 2020 2020 2020 2020 286c 696e 6520 2033 (line 3\n+00025030: 3529 0a2a 2074 7261 6e73 666f 726d 2c20 5).* transform, \n+00025040: 6469 7265 6374 3a20 2020 2020 2020 2020 direct: \n+00025050: 2020 2020 2020 2020 2020 2020 5472 616e Tran\n+00025060: 7366 6f72 6d20 6c69 7465 7261 6c73 2e20 sform literals. \n+00025070: 2028 6c69 6e65 2020 3437 290a 2a20 7472 (line 47).* tr\n+00025080: 616e 7366 6f72 6d2c 2064 6972 6563 7420 ansform, direct \n+00025090: 3c31 3e3a 2020 2020 2020 2020 2020 2020 <1>: \n+000250a0: 2020 2020 2054 7261 6e73 666f 726d 206c Transform l\n+000250b0: 6974 6572 616c 732e 2020 286c 696e 6520 iterals. (line \n+000250c0: 2035 3329 0a2a 2074 7261 6e73 666f 726d 53).* transform\n+000250d0: 2c20 726f 7461 7469 6f6e 3a20 2020 2020 , rotation: \n+000250e0: 2020 2020 2020 2020 2020 2020 2020 5472 Tr\n+000250f0: 616e 7366 6f72 6d20 6c69 7465 7261 6c73 ansform literals\n+00025100: 2e20 2028 6c69 6e65 2020 3130 290a 2a20 . (line 10).* \n+00025110: 7472 616e 7366 6f72 6d2c 2072 6f74 6174 transform, rotat\n+00025120: 696f 6e20 3c31 3e3a 2020 2020 2020 2020 ion <1>: \n+00025130: 2020 2020 2020 2041 2068 6965 7261 7263 A hierarc\n+00025140: 6869 6361 6c20 6d6f 6465 6c2e 0a20 2020 hical model.. \n 00025150: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00025160: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00025170: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00025180: 2020 2020 2020 2028 6c69 6e65 2020 3438 (line 48\n-00025190: 290a 2a20 7472 616e 7366 6f72 6d2c 2072 ).* transform, r\n-000251a0: 6f74 6174 696f 6e20 3c32 3e3a 2020 2020 otation <2>: \n-000251b0: 2020 2020 2020 2020 2020 2041 2068 6965 A hie\n-000251c0: 7261 7263 6869 6361 6c20 6d6f 6465 6c2e rarchical model.\n-000251d0: 0a20 2020 2020 2020 2020 2020 2020 2020 . \n+00025180: 2020 2020 2020 2020 2020 2028 6c69 6e65 (line\n+00025190: 2020 3438 290a 2a20 7472 616e 7366 6f72 48).* transfor\n+000251a0: 6d2c 2072 6f74 6174 696f 6e20 3c32 3e3a m, rotation <2>:\n+000251b0: 2020 2020 2020 2020 2020 2020 2020 2041 A\n+000251c0: 2068 6965 7261 7263 6869 6361 6c20 6d6f hierarchical mo\n+000251d0: 6465 6c2e 0a20 2020 2020 2020 2020 2020 del.. \n 000251e0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 000251f0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00025200: 2020 2020 2020 2020 2020 2020 2020 2028 (\n-00025210: 6c69 6e65 2020 3538 290a 2a20 7472 616e line 58).* tran\n-00025220: 7366 6f72 6d2c 2072 6f74 6174 696f 6e20 sform, rotation \n-00025230: 3c33 3e3a 2020 2020 2020 2020 2020 2020 <3>: \n-00025240: 2020 2041 2068 6965 7261 7263 6869 6361 A hierarchica\n-00025250: 6c20 6d6f 6465 6c2e 0a20 2020 2020 2020 l model.. \n+00025200: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00025210: 2020 2028 6c69 6e65 2020 3538 290a 2a20 (line 58).* \n+00025220: 7472 616e 7366 6f72 6d2c 2072 6f74 6174 transform, rotat\n+00025230: 696f 6e20 3c33 3e3a 2020 2020 2020 2020 ion <3>: \n+00025240: 2020 2020 2020 2041 2068 6965 7261 7263 A hierarc\n+00025250: 6869 6361 6c20 6d6f 6465 6c2e 0a20 2020 hical model.. \n 00025260: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00025270: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00025280: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00025290: 2020 2020 2020 2028 6c69 6e65 2020 3738 (line 78\n-000252a0: 290a 2a20 7472 616e 7366 6f72 6d2c 2072 ).* transform, r\n-000252b0: 6f74 6174 696f 6e20 3c34 3e3a 2020 2020 otation <4>: \n-000252c0: 2020 2020 2020 2020 2020 2041 2068 6965 A hie\n-000252d0: 7261 7263 6869 6361 6c20 6d6f 6465 6c2e rarchical model.\n-000252e0: 0a20 2020 2020 2020 2020 2020 2020 2020 . \n+00025290: 2020 2020 2020 2020 2020 2028 6c69 6e65 (line\n+000252a0: 2020 3738 290a 2a20 7472 616e 7366 6f72 78).* transfor\n+000252b0: 6d2c 2072 6f74 6174 696f 6e20 3c34 3e3a m, rotation <4>:\n+000252c0: 2020 2020 2020 2020 2020 2020 2020 2041 A\n+000252d0: 2068 6965 7261 7263 6869 6361 6c20 6d6f hierarchical mo\n+000252e0: 6465 6c2e 0a20 2020 2020 2020 2020 2020 del.. \n 000252f0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00025300: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00025310: 2020 2020 2020 2020 2020 2020 2020 2028 (\n-00025320: 6c69 6e65 2020 3933 290a 2a20 7472 616e line 93).* tran\n-00025330: 7366 6f72 6d2c 2073 6361 6c65 3a20 2020 sform, scale: \n-00025340: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00025350: 2020 2054 7261 6e73 666f 726d 206c 6974 Transform lit\n-00025360: 6572 616c 732e 2020 286c 696e 6520 2031 erals. (line 1\n-00025370: 3629 0a2a 2074 7261 6e73 666f 726d 2c20 6).* transform, \n-00025380: 7363 616c 6520 3c31 3e3a 2020 2020 2020 scale <1>: \n-00025390: 2020 2020 2020 2020 2020 2020 5472 616e Tran\n-000253a0: 7366 6f72 6d20 6c69 7465 7261 6c73 2e20 sform literals. \n-000253b0: 2028 6c69 6e65 2020 3137 290a 2a20 7472 (line 17).* tr\n-000253c0: 616e 7366 6f72 6d2c 2074 7261 6e73 6c61 ansform, transla\n-000253d0: 7469 6f6e 3a20 2020 2020 2020 2020 2020 tion: \n-000253e0: 2020 2020 204f 626a 6563 7420 7472 616e Object tran\n-000253f0: 7366 6f72 6d73 2e20 2020 286c 696e 6520 sforms. (line \n-00025400: 2034 3729 0a2a 2074 7261 6e73 666f 726d 47).* transform\n-00025410: 2c20 7472 616e 736c 6174 696f 6e20 3c31 , translation <1\n-00025420: 3e3a 2020 2020 2020 2020 2020 2020 5472 >: Tr\n-00025430: 616e 7366 6f72 6d20 6c69 7465 7261 6c73 ansform literals\n-00025440: 2e20 2028 6c69 6e65 2020 3135 290a 2a20 . (line 15).* \n-00025450: 7472 616e 7366 6f72 6d2c 2074 7261 6e73 transform, trans\n-00025460: 6c61 7469 6f6e 203c 323e 3a20 2020 2020 lation <2>: \n-00025470: 2020 2020 2020 2041 2068 6965 7261 7263 A hierarc\n-00025480: 6869 6361 6c20 6d6f 6465 6c2e 0a20 2020 hical model.. \n-00025490: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00025310: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00025320: 2020 2028 6c69 6e65 2020 3933 290a 2a20 (line 93).* \n+00025330: 7472 616e 7366 6f72 6d2c 2073 6361 6c65 transform, scale\n+00025340: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+00025350: 2020 2020 2020 2054 7261 6e73 666f 726d Transform\n+00025360: 206c 6974 6572 616c 732e 2020 286c 696e literals. (lin\n+00025370: 6520 2031 3629 0a2a 2074 7261 6e73 666f e 16).* transfo\n+00025380: 726d 2c20 7363 616c 6520 3c31 3e3a 2020 rm, scale <1>: \n+00025390: 2020 2020 2020 2020 2020 2020 2020 2020 \n+000253a0: 5472 616e 7366 6f72 6d20 6c69 7465 7261 Transform litera\n+000253b0: 6c73 2e20 2028 6c69 6e65 2020 3137 290a ls. (line 17).\n+000253c0: 2a20 7472 616e 7366 6f72 6d2c 2074 7261 * transform, tra\n+000253d0: 6e73 6c61 7469 6f6e 3a20 2020 2020 2020 nslation: \n+000253e0: 2020 2020 2020 2020 204f 626a 6563 7420 Object \n+000253f0: 7472 616e 7366 6f72 6d73 2e20 2020 286c transforms. (l\n+00025400: 696e 6520 2034 3729 0a2a 2074 7261 6e73 ine 47).* trans\n+00025410: 666f 726d 2c20 7472 616e 736c 6174 696f form, translatio\n+00025420: 6e20 3c31 3e3a 2020 2020 2020 2020 2020 n <1>: \n+00025430: 2020 5472 616e 7366 6f72 6d20 6c69 7465 Transform lite\n+00025440: 7261 6c73 2e20 2028 6c69 6e65 2020 3135 rals. (line 15\n+00025450: 290a 2a20 7472 616e 7366 6f72 6d2c 2074 ).* transform, t\n+00025460: 7261 6e73 6c61 7469 6f6e 203c 323e 3a20 ranslation <2>: \n+00025470: 2020 2020 2020 2020 2020 2041 2068 6965 A hie\n+00025480: 7261 7263 6869 6361 6c20 6d6f 6465 6c2e rarchical model.\n+00025490: 0a20 2020 2020 2020 2020 2020 2020 2020 . \n 000254a0: 2020 2020 2020 2020 2020 2020 2020 2020 \n 000254b0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000254c0: 2020 2020 2020 2020 2020 2028 6c69 6e65 (line\n-000254d0: 2020 3437 290a 2a20 7472 616e 7366 6f72 47).* transfor\n-000254e0: 6d2c 2076 6965 773a 2020 2020 2020 2020 m, view: \n-000254f0: 2020 2020 2020 2020 2020 2020 2020 2054 T\n-00025500: 7261 6e73 666f 726d 206c 6974 6572 616c ransform literal\n-00025510: 732e 2020 286c 696e 6520 2032 3529 0a2a s. (line 25).*\n-00025520: 2074 7261 6e73 666f 726d 2c20 7669 6577 transform, view\n-00025530: 203c 313e 3a20 2020 2020 2020 2020 2020 <1>: \n-00025540: 2020 2020 2020 2020 5472 616e 7366 6f72 Transfor\n-00025550: 6d20 6c69 7465 7261 6c73 2e20 2028 6c69 m literals. (li\n-00025560: 6e65 2020 3335 290a 2a20 7472 616e 736c ne 35).* transl\n-00025570: 6174 696f 6e20 7472 616e 7366 6f72 6d3a ation transform:\n-00025580: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00025590: 204f 626a 6563 7420 7472 616e 7366 6f72 Object transfor\n-000255a0: 6d73 2e20 2020 286c 696e 6520 2034 3729 ms. (line 47)\n-000255b0: 0a2a 2074 7261 6e73 6c61 7469 6f6e 2074 .* translation t\n-000255c0: 7261 6e73 666f 726d 203c 313e 3a20 2020 ransform <1>: \n-000255d0: 2020 2020 2020 2020 2020 5472 616e 7366 Transf\n-000255e0: 6f72 6d20 6c69 7465 7261 6c73 2e20 2028 orm literals. (\n-000255f0: 6c69 6e65 2020 3135 290a 2a20 7472 616e line 15).* tran\n-00025600: 736c 6174 696f 6e20 7472 616e 7366 6f72 slation transfor\n-00025610: 6d20 3c32 3e3a 2020 2020 2020 2020 2020 m <2>: \n-00025620: 2020 2041 2068 6965 7261 7263 6869 6361 A hierarchica\n-00025630: 6c20 6d6f 6465 6c2e 0a20 2020 2020 2020 l model.. \n+000254c0: 2020 2020 2020 2020 2020 2020 2020 2028 (\n+000254d0: 6c69 6e65 2020 3437 290a 2a20 7472 616e line 47).* tran\n+000254e0: 7366 6f72 6d2c 2076 6965 773a 2020 2020 sform, view: \n+000254f0: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00025500: 2020 2054 7261 6e73 666f 726d 206c 6974 Transform lit\n+00025510: 6572 616c 732e 2020 286c 696e 6520 2032 erals. (line 2\n+00025520: 3529 0a2a 2074 7261 6e73 666f 726d 2c20 5).* transform, \n+00025530: 7669 6577 203c 313e 3a20 2020 2020 2020 view <1>: \n+00025540: 2020 2020 2020 2020 2020 2020 5472 616e Tran\n+00025550: 7366 6f72 6d20 6c69 7465 7261 6c73 2e20 sform literals. \n+00025560: 2028 6c69 6e65 2020 3335 290a 2a20 7472 (line 35).* tr\n+00025570: 616e 736c 6174 696f 6e20 7472 616e 7366 anslation transf\n+00025580: 6f72 6d3a 2020 2020 2020 2020 2020 2020 orm: \n+00025590: 2020 2020 204f 626a 6563 7420 7472 616e Object tran\n+000255a0: 7366 6f72 6d73 2e20 2020 286c 696e 6520 sforms. (line \n+000255b0: 2034 3729 0a2a 2074 7261 6e73 6c61 7469 47).* translati\n+000255c0: 6f6e 2074 7261 6e73 666f 726d 203c 313e on transform <1>\n+000255d0: 3a20 2020 2020 2020 2020 2020 2020 5472 : Tr\n+000255e0: 616e 7366 6f72 6d20 6c69 7465 7261 6c73 ansform literals\n+000255f0: 2e20 2028 6c69 6e65 2020 3135 290a 2a20 . (line 15).* \n+00025600: 7472 616e 736c 6174 696f 6e20 7472 616e translation tran\n+00025610: 7366 6f72 6d20 3c32 3e3a 2020 2020 2020 sform <2>: \n+00025620: 2020 2020 2020 2041 2068 6965 7261 7263 A hierarc\n+00025630: 6869 6361 6c20 6d6f 6465 6c2e 0a20 2020 hical model.. \n 00025640: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00025650: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00025660: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00025670: 2020 2020 2020 2028 6c69 6e65 2020 3437 (line 47\n-00025680: 290a 2a20 7472 616e 7370 6172 656e 6379 ).* transparency\n-00025690: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n-000256a0: 2020 2020 2020 2020 2020 2054 7261 6e73 Trans\n-000256b0: 7061 7265 6e63 792e 2020 2020 2020 2020 parency. \n-000256c0: 286c 696e 6520 2020 3629 0a2a 2074 776f (line 6).* two\n-000256d0: 2d6f 7065 7261 6e64 2066 6f72 6d3a 2020 -operand form: \n-000256e0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000256f0: 2020 2020 5477 6f2d 6f70 6572 616e 6420 Two-operand \n-00025700: 2862 696e 6172 7929 2066 6f72 6d73 2e0a (binary) forms..\n-00025710: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00025670: 2020 2020 2020 2020 2020 2028 6c69 6e65 (line\n+00025680: 2020 3437 290a 2a20 7472 616e 7370 6172 47).* transpar\n+00025690: 656e 6379 3a20 2020 2020 2020 2020 2020 ency: \n+000256a0: 2020 2020 2020 2020 2020 2020 2020 2054 T\n+000256b0: 7261 6e73 7061 7265 6e63 792e 2020 2020 ransparency. \n+000256c0: 2020 2020 286c 696e 6520 2020 3629 0a2a (line 6).*\n+000256d0: 2074 776f 2d6f 7065 7261 6e64 2066 6f72 two-operand for\n+000256e0: 6d3a 2020 2020 2020 2020 2020 2020 2020 m: \n+000256f0: 2020 2020 2020 2020 5477 6f2d 6f70 6572 Two-oper\n+00025700: 616e 6420 2862 696e 6172 7929 2066 6f72 and (binary) for\n+00025710: 6d73 2e0a 2020 2020 2020 2020 2020 2020 ms.. \n 00025720: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00025730: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00025740: 2020 2020 2020 2020 2020 2020 2020 286c (l\n-00025750: 696e 6520 2020 3629 0a2a 2075 6e61 7279 ine 6).* unary\n-00025760: 2066 6f72 6d3a 2020 2020 2020 2020 2020 form: \n+00025740: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00025750: 2020 286c 696e 6520 2020 3629 0a2a 2075 (line 6).* u\n+00025760: 6e61 7279 2066 6f72 6d3a 2020 2020 2020 nary form: \n 00025770: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00025780: 2020 556e 6172 7920 666f 726d 732e 2020 Unary forms. \n-00025790: 2020 2020 2020 2028 6c69 6e65 2020 2036 (line 6\n-000257a0: 290a 2a20 7665 6374 6f72 3a20 2020 2020 ).* vector: \n+00025780: 2020 2020 2020 556e 6172 7920 666f 726d Unary form\n+00025790: 732e 2020 2020 2020 2020 2028 6c69 6e65 s. (line\n+000257a0: 2020 2036 290a 2a20 7665 6374 6f72 3a20 6).* vector: \n 000257b0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000257c0: 2020 2020 2020 2020 2020 204f 626a 6563 Objec\n-000257d0: 7420 7472 616e 7366 6f72 6d73 2e20 2020 t transforms. \n-000257e0: 286c 696e 6520 2034 3829 0a2a 2076 6563 (line 48).* vec\n-000257f0: 746f 7220 6465 6669 6e69 7469 6f6e 3a20 tor definition: \n-00025800: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00025810: 2020 2020 506f 6c79 6c69 6e65 2073 7765 Polyline swe\n-00025820: 6570 732e 2020 2020 2028 6c69 6e65 2020 eps. (line \n-00025830: 3139 290a 2a20 7665 6374 6f72 206c 6974 19).* vector lit\n-00025840: 6572 616c 3a20 2020 2020 2020 2020 2020 eral: \n-00025850: 2020 2020 2020 2020 2020 2020 2050 6f69 Poi\n-00025860: 6e74 2061 6e64 2076 6563 746f 7220 6c69 nt and vector li\n-00025870: 7465 7261 6c73 2e0a 2020 2020 2020 2020 terals.. \n+000257c0: 2020 2020 2020 2020 2020 2020 2020 204f O\n+000257d0: 626a 6563 7420 7472 616e 7366 6f72 6d73 bject transforms\n+000257e0: 2e20 2020 286c 696e 6520 2034 3829 0a2a . (line 48).*\n+000257f0: 2076 6563 746f 7220 6465 6669 6e69 7469 vector definiti\n+00025800: 6f6e 3a20 2020 2020 2020 2020 2020 2020 on: \n+00025810: 2020 2020 2020 2020 506f 6c79 6c69 6e65 Polyline\n+00025820: 2073 7765 6570 732e 2020 2020 2028 6c69 sweeps. (li\n+00025830: 6e65 2020 3139 290a 2a20 7665 6374 6f72 ne 19).* vector\n+00025840: 206c 6974 6572 616c 3a20 2020 2020 2020 literal: \n+00025850: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00025860: 2050 6f69 6e74 2061 6e64 2076 6563 746f Point and vecto\n+00025870: 7220 6c69 7465 7261 6c73 2e0a 2020 2020 r literals.. \n 00025880: 2020 2020 2020 2020 2020 2020 2020 2020 \n 00025890: 2020 2020 2020 2020 2020 2020 2020 2020 \n 000258a0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000258b0: 2020 2020 2020 286c 696e 6520 2020 3629 (line 6)\n-000258c0: 0a2a 2076 6563 746f 7220 7265 6665 7265 .* vector refere\n-000258d0: 6e63 653a 2020 2020 2020 2020 2020 2020 nce: \n-000258e0: 2020 2020 2020 2020 2020 466f 726d 7320 Forms \n-000258f0: 6f66 2072 6566 6572 656e 6365 732e 2028 of references. (\n-00025900: 6c69 6e65 2020 3135 290a 2a20 7669 6577 line 15).* view\n-00025910: 2074 7261 6e73 666f 726d 3a20 2020 2020 transform: \n+000258b0: 2020 2020 2020 2020 2020 286c 696e 6520 (line \n+000258c0: 2020 3629 0a2a 2076 6563 746f 7220 7265 6).* vector re\n+000258d0: 6665 7265 6e63 653a 2020 2020 2020 2020 ference: \n+000258e0: 2020 2020 2020 2020 2020 2020 2020 466f Fo\n+000258f0: 726d 7320 6f66 2072 6566 6572 656e 6365 rms of reference\n+00025900: 732e 2028 6c69 6e65 2020 3135 290a 2a20 s. (line 15).* \n+00025910: 7669 6577 2074 7261 6e73 666f 726d 3a20 view transform: \n 00025920: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00025930: 2020 2054 7261 6e73 666f 726d 206c 6974 Transform lit\n-00025940: 6572 616c 732e 2020 286c 696e 6520 2032 erals. (line 2\n-00025950: 3529 0a2a 2076 6965 7720 7472 616e 7366 5).* view transf\n-00025960: 6f72 6d20 3c31 3e3a 2020 2020 2020 2020 orm <1>: \n-00025970: 2020 2020 2020 2020 2020 2020 5472 616e Tran\n-00025980: 7366 6f72 6d20 6c69 7465 7261 6c73 2e20 sform literals. \n-00025990: 2028 6c69 6e65 2020 3335 290a 2a20 7669 (line 35).* vi\n-000259a0: 7274 7561 6c20 6361 6d65 7261 3a20 2020 rtual camera: \n-000259b0: 2020 2020 2020 2020 2020 2020 2020 2020 \n-000259c0: 2020 2020 204f 7665 7276 6965 772e 2020 Overview. \n-000259d0: 2020 2020 2020 2020 2020 286c 696e 6520 (line \n-000259e0: 2038 3129 0a2a 2077 6869 7465 2073 7061 81).* white spa\n-000259f0: 6365 3a20 2020 2020 2020 2020 2020 2020 ce: \n-00025a00: 2020 2020 2020 2020 2020 2020 2020 4472 Dr\n-00025a10: 6177 696e 6720 6120 736f 6c69 642e 2020 awing a solid. \n-00025a20: 2020 2028 6c69 6e65 2020 3235 290a 2a20 (line 25).* \n-00025a30: 7768 6974 6520 7370 6163 6520 3c31 3e3a white space <1>:\n-00025a40: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00025a50: 2020 2020 2020 2044 7261 7769 6e67 2061 Drawing a\n-00025a60: 2073 6f6c 6964 2e20 2020 2020 286c 696e solid. (lin\n-00025a70: 6520 2032 3929 0a2a 2077 6869 7465 2073 e 29).* white s\n-00025a80: 7061 6365 203c 323e 3a20 2020 2020 2020 pace <2>: \n+00025930: 2020 2020 2020 2054 7261 6e73 666f 726d Transform\n+00025940: 206c 6974 6572 616c 732e 2020 286c 696e literals. (lin\n+00025950: 6520 2032 3529 0a2a 2076 6965 7720 7472 e 25).* view tr\n+00025960: 616e 7366 6f72 6d20 3c31 3e3a 2020 2020 ansform <1>: \n+00025970: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00025980: 5472 616e 7366 6f72 6d20 6c69 7465 7261 Transform litera\n+00025990: 6c73 2e20 2028 6c69 6e65 2020 3335 290a ls. (line 35).\n+000259a0: 2a20 7669 7274 7561 6c20 6361 6d65 7261 * virtual camera\n+000259b0: 3a20 2020 2020 2020 2020 2020 2020 2020 : \n+000259c0: 2020 2020 2020 2020 204f 7665 7276 6965 Overvie\n+000259d0: 772e 2020 2020 2020 2020 2020 2020 286c w. (l\n+000259e0: 696e 6520 2038 3129 0a2a 2077 6869 7465 ine 81).* white\n+000259f0: 2073 7061 6365 3a20 2020 2020 2020 2020 space: \n+00025a00: 2020 2020 2020 2020 2020 2020 2020 2020 \n+00025a10: 2020 4472 6177 696e 6720 6120 736f 6c69 Drawing a soli\n+00025a20: 642e 2020 2020 2028 6c69 6e65 2020 3235 d. (line 25\n+00025a30: 290a 2a20 7768 6974 6520 7370 6163 6520 ).* white space \n+00025a40: 3c31 3e3a 2020 2020 2020 2020 2020 2020 <1>: \n+00025a50: 2020 2020 2020 2020 2020 2044 7261 7769 Drawi\n+00025a60: 6e67 2061 2073 6f6c 6964 2e20 2020 2020 ng a solid. \n+00025a70: 286c 696e 6520 2032 3929 0a2a 2077 6869 (line 29).* whi\n+00025a80: 7465 2073 7061 6365 203c 323e 3a20 2020 te space <2>: \n 00025a90: 2020 2020 2020 2020 2020 2020 2020 2020 \n-00025aa0: 4f62 6a65 6374 2074 7261 6e73 666f 726d Object transform\n-00025ab0: 732e 2020 2028 6c69 6e65 2020 3332 290a s. 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