home *** CD-ROM | disk | FTP | other *** search
| Unknown | 1989-10-13 | 29.7 KB |
open in: 
MacOS 8.1
|
Win98
|
DOS
view JSON data
|
view as text
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This format is not currently supported by dexvert.
| Confidence | Program | Detection | Match Type | Support
|
|---|
| 10%
| dexvert
| Jesper Olsen Module (music/jesperOlsen)
| magic
| Supported |
| 1%
| dexvert
| TTComp Archive (archive/ttcomp)
| fallback
| Supported |
| 100%
| file
| TTComp archive data, binary, 4K dictionary
| default (weak)
| |
| 99%
| file
| data
| default
| |
| 100%
| TrID
| TTComp archive compressed (bin-4K)
| default (weak)
|
|
hex view+--------+-------------------------+-------------------------+--------+--------+
|00000000| 00 06 01 57 00 01 00 01 | 00 00 00 ff ff 00 ff fc |...W....|........|
|00000010| 00 01 00 00 4b 48 01 20 | 01 20 00 00 4f 10 00 f0 |....KH. |. ..O...|
|00000020| 00 f0 c9 00 71 08 00 08 | 00 03 00 00 00 48 00 48 |....q...|.....H.H|
|00000030| 00 00 00 00 02 da 02 28 | ff e1 ff e2 02 f9 02 46 |.......(|.......F|
|00000040| 03 47 05 28 03 fc 00 02 | 00 00 00 48 00 48 00 00 |.G.(....|...H.H..|
|00000050| 00 00 02 da 02 28 00 01 | 00 00 00 64 00 00 00 01 |.....(..|...d....|
|00000060| 00 01 01 01 00 00 00 01 | 27 0f 00 01 00 01 00 64 |........|'......d|
|00000070| ea 28 00 64 ea b4 80 23 | 00 00 00 0a 00 19 01 90 |.(.d...#|........|
|00000080| 00 00 00 00 00 40 00 00 | 00 00 00 00 00 00 00 01 |.....@..|........|
|00000090| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|000000a0| 00 01 00 00 00 01 00 00 | 00 00 00 00 00 00 46 62 |........|......Fb|
|000000b0| 00 10 00 00 01 72 00 02 | ff f2 00 1e ff f2 00 ec |.....r..|........|
|000000c0| ff f2 01 ba fe 61 00 00 | 00 00 ff ff ff ff 00 01 |.....a..|........|
|000000d0| 00 00 00 01 00 00 00 00 | 00 00 00 00 46 72 00 10 |........|....Fr..|
|000000e0| 00 00 01 74 00 02 ff f2 | 00 1e ff f2 00 ec ff f2 |...t....|........|
|000000f0| 01 ba fe 61 00 00 00 00 | ff ff ff ff 01 35 00 00 |...a....|.....5..|
|00000100| 01 35 00 00 fd 65 01 46 | 00 00 61 6e 15 70 00 00 |.5...e.F|..an.p..|
|00000110| 59 d2 07 9c ff f2 00 1e | ff f2 00 ec ff f2 01 ba |Y.......|........|
|00000120| 00 00 00 00 ff 00 0a 00 | 00 16 00 00 01 e6 03 01 |........|........|
|00000130| 00 01 0d 01 00 06 00 00 | 0c 00 00 14 00 01 0d 01 |........|........|
|00000140| 00 06 00 00 0c 00 00 14 | 00 01 0d 78 00 06 00 00 |........|...x....|
|00000150| 0c 00 00 14 00 01 0d 00 | 00 06 00 00 0c 00 00 16 |........|........|
|00000160| 00 08 09 20 20 20 3d 20 | 34 0d 00 06 00 00 0c 00 |... = |4.......|
|00000170| 00 16 00 00 00 00 0c 01 | 00 14 00 01 0d 01 00 06 |........|........|
|00000180| 00 00 0c 00 00 14 00 00 | 00 00 01 68 00 00 00 00 |........|...h....|
|00000190| 00 0c 54 68 65 20 41 6d | 61 7a 69 6e 67 0d 00 06 |..The Am|azing...|
|000001a0| 00 00 18 00 00 14 00 07 | 42 69 74 6d 61 70 0d 00 |........|Bitmap..|
|000001b0| 00 06 00 00 18 00 00 14 | 00 0f 54 72 61 6e 73 6d |........|..Transm|
|000001c0| 6f 67 72 69 66 69 65 72 | 0d 00 00 06 00 00 18 00 |ogrifier|........|
|000001d0| 00 14 00 01 0d 00 00 06 | 00 00 0c 00 00 14 00 10 |........|........|
|000001e0| 62 79 20 4b 65 69 74 68 | 20 52 6f 6c 6c 69 6e 0d |by Keith| Rollin.|
|000001f0| 00 06 00 00 0e 00 00 14 | 00 01 0d 01 00 06 00 00 |........|........|
|00000200| 0c 00 00 14 00 01 0d 00 | 00 06 00 00 0c 00 00 14 |........|........|
|00000210| 01 20 01 44 01 68 00 00 | 00 00 00 01 0d 3a 00 06 |. .D.h..|.....:..|
|00000220| 00 00 0c 00 00 14 00 01 | 0d 6c 00 06 00 00 0c 00 |........|.l......|
|00000230| 00 14 00 0d 49 6e 74 72 | 6f 64 75 63 74 69 6f 6e |....Intr|oduction|
|00000240| 0d 6c 00 06 00 00 0e 01 | 00 14 00 01 0d 20 00 06 |.l......|..... ..|
|00000250| 00 00 0c 00 00 14 00 52 | f4 ff 4b f2 c0 f7 71 f2 |.......R|..K...q.|
|00000260| 7f 76 10 8b 5f 77 30 2b | 62 0f 77 10 c6 30 f6 21 |.v.._w0+|b.w..0.!|
|00000270| 0b 63 9f 79 06 45 f7 53 | 90 60 c5 f6 de f7 52 14 |.c.y.E.S|.`....R.|
|00000280| f3 b0 35 f7 70 a1 28 04 | 16 aa f7 90 8b 5f 77 05 |..5.p.(.|....._w.|
|00000290| dd f2 1f 0d 00 06 00 00 | 0c 00 00 14 00 25 f5 3c |........|.....%.<|
|000002a0| 6a 73 f6 70 f6 2f 79 0f | 32 07 30 2b 10 f5 80 f2 |js.p./y.|2.0+....|
|000002b0| 60 f5 90 97 41 c2 75 38 | f0 d0 00 06 00 00 0c 01 |`...A.u8|........|
|000002c0| 00 03 00 13 f5 25 26 21 | 0f 75 f2 bf 76 09 1f 67 |.....%&!|.u..v..g|
|000002d0| 41 18 f0 d0 00 06 00 00 | 0c 01 00 03 00 00 01 e6 |A.......|........|
|000002e0| 01 0a 00 00 00 00 00 24 | 00 48 00 6c 00 90 00 b4 |.......$|.H.l....|
|000002f0| 00 d8 00 fc 01 20 01 44 | 01 68 02 10 02 10 00 1b |..... .D|.h......|
|00000300| 09 09 53 74 65 70 20 31 | 09 09 09 53 74 65 70 20 |..Step 1|...Step |
|00000310| 32 09 09 09 53 74 65 70 | 20 33 0d 10 00 06 00 00 |2...Step| 3......|
|00000320| 0c 01 00 14 01 57 f5 4b | 78 09 5c f7 5f 6d 13 20 |.....W.K|x.\._m. |
|00000330| 97 8c f7 58 81 80 2b 10 | 21 cb 37 f7 1f 75 18 07 |...X..+.|!.7..u..|
|00000340| 3f 76 5a f7 61 90 f7 77 | 2b 0f 62 68 7c 0f 62 72 |?vZ.a..w|+.bh|.br|
|00000350| f6 d6 e0 24 63 8d 54 f6 | d6 27 53 8f 2e 0f 54 b7 |...$c.T.|.'S...T.|
|00000360| 80 73 ca f7 59 18 02 46 | 38 a6 27 53 0f 28 f6 d5 |.s..Y..F|8.'S.(..|
|00000370| f7 67 3f 67 06 aa 02 b1 | 0e 7f 78 1a 80 60 f7 53 |.g?g....|..x..`.S|
|00000380| 7d 54 f6 d0 97 82 63 c1 | 0d 45 f6 d0 53 10 ea 6c |}T....c.|.E..S..l|
|00000390| 10 25 06 35 2b 14 0f 2d | 0a 7f 6b 10 f4 35 ef 79 |.%.5+..-|..k..5.y|
|000003a0| f6 27 28 f2 9f 2c 04 52 | 62 75 30 64 5f 75 39 06 |.'(..,.R|bu0d_u9.|
|000003b0| 08 e1 c7 d7 19 0e 57 32 | f2 c0 63 90 8c 6a 73 f6 |......W2|..c..js.|
|000003c0| 70 6a a0 e7 f7 81 a8 06 | f7 76 f7 90 d4 5f 6d 06 |pj......|.v..._m.|
|000003d0| 08 e1 c7 d7 19 0e 57 32 | f2 e0 f4 35 aa 1c 27 f7 |......W2|...5..'.|
|000003e0| 61 af 79 f2 c0 2b 18 10 | 5e 14 62 75 38 06 41 0f |a.y..+..|^.bu8.A.|
|000003f0| 6b 35 f7 73 06 80 24 63 | 8d 54 f6 d6 27 53 8f 2e |k5.s..$c|.T..'S..|
|00000400| f0 d0 00 06 00 00 0c 00 | 00 14 00 01 0d 02 00 06 |........|........|
|00000410| 00 00 0c 00 00 14 00 14 | 00 01 0d e6 00 06 00 00 |........|........|
|00000420| 0c 00 00 14 00 01 0d e7 | 00 06 00 00 0c 00 00 14 |........|........|
|00000430| 00 0b 54 68 65 20 42 61 | 73 69 63 73 0d fe 00 06 |..The Ba|sics....|
|00000440| 00 00 0e 01 00 14 00 01 | 0d 73 00 06 00 00 0c 00 |........|.s......|
|00000450| 00 14 00 61 49 6e 20 61 | 63 68 69 65 76 69 6e 67 |...aIn a|chieving|
|00000460| 20 66 61 73 74 20 61 6e | 64 20 65 66 66 65 63 74 | fast an|d effect|
|00000470| 69 76 65 20 74 72 61 6e | 73 66 6f 72 6d 61 74 69 |ive tran|sformati|
|00000480| 6f 6e 73 2c 20 79 6f 75 | 20 6d 75 73 74 20 69 6e |ons, you| must in|
|00000490| 76 6f 6c 76 65 20 73 65 | 76 65 72 61 6c 20 74 65 |volve se|veral te|
|000004a0| 63 68 6e 69 71 75 65 73 | 20 61 6e 64 20 74 72 69 |chniques| and tri|
|000004b0| 63 6b 73 2e 0d 77 00 06 | 00 00 0c 00 00 14 00 01 |cks..w..|........|
|000004c0| 0d fc 00 06 00 00 0c 00 | 00 14 01 7c 54 68 65 20 |........|...|The |
|000004d0| 62 61 73 69 63 20 74 65 | 63 68 6e 69 71 75 65 20 |basic te|chnique |
|000004e0| 69 73 20 65 78 61 6d 69 | 6e 65 20 61 20 63 6f 6d |is exami|ne a com|
|000004f0| 70 6c 65 74 65 20 62 69 | 74 6d 61 70 20 6f 6e 20 |plete bi|tmap on |
|00000500| 61 20 70 69 78 65 6c 2d | 62 79 2d 70 69 78 65 6c |a pixel-|by-pixel|
|00000510| 20 62 61 73 69 73 2c 20 | 61 6e 64 20 61 70 70 6c | basis, |and appl|
|00000520| 79 20 63 65 72 74 61 69 | 6e 20 66 6f 72 6d 75 6c |y certai|n formul|
|00000530| 61 73 20 74 6f 20 74 68 | 65 20 63 6f 6f 72 64 69 |as to th|e coordi|
|00000540| 6e 61 74 65 73 20 6f 66 | 20 65 61 63 68 20 70 6f |nates of| each po|
|00000550| 69 6e 74 2e 20 53 69 6e | 63 65 20 74 68 65 20 70 |int. Sin|ce the p|
|00000560| 72 6f 63 65 73 73 20 66 | 6f 72 20 65 61 63 68 20 |rocess f|or each |
|00000570| 70 69 78 65 6c 20 69 73 | 20 74 68 65 20 73 61 6d |pixel is| the sam|
|00000580| 65 2c 20 69 6e 20 6f 75 | 72 20 73 65 70 61 72 61 |e, in ou|r separa|
|00000590| 74 65 20 64 69 73 63 75 | 73 73 69 6f 6e 73 20 6f |te discu|ssions o|
|000005a0| 66 20 74 72 61 6e 73 6c | 61 74 69 6f 6e 2c 20 73 |f transl|ation, s|
|000005b0| 63 61 6c 69 6e 67 20 61 | 6e 64 20 72 6f 74 61 74 |caling a|nd rotat|
|000005c0| 69 6f 6e 20 62 65 6c 6f | 77 2c 20 77 65 20 77 6f |ion belo|w, we wo|
|000005d0| 72 6b 20 77 69 74 68 20 | 6f 6e 6c 79 20 61 20 73 |rk with |only a s|
|000005e0| 69 6e 67 6c 65 20 70 69 | 78 65 6c 2e 20 49 6e 20 |ingle pi|xel. In |
|000005f0| 6f 75 72 20 61 63 74 75 | 61 6c 20 70 72 6f 67 72 |our actu|al progr|
|00000600| 61 6d 2c 20 77 65 20 75 | 73 65 20 61 20 6e 65 73 |am, we u|se a nes|
|00000610| 74 65 64 20 6c 6f 6f 70 | 20 74 6f 20 70 65 72 66 |ted loop| to perf|
|00000620| 6f 72 6d 20 74 68 65 73 | 65 20 6f 70 65 72 61 74 |orm thes|e operat|
|00000630| 69 6f 6e 73 20 6f 6e 20 | 61 6c 6c 20 74 68 65 20 |ions on |all the |
|00000640| 70 69 78 65 6c 73 2e 0d | 00 06 00 00 0c 00 00 14 |pixels..|........|
|00000650| 00 01 0d 00 00 06 00 00 | 0c 00 00 14 00 f3 f2 d0 |........|........|
|00000660| f4 93 ca f7 59 10 cf 75 | f6 df 75 a6 27 f7 61 02 |....Y..u|..u.'.a.|
|00000670| 46 38 d5 4f 6d 62 75 38 | f2 e0 f5 27 f6 7b 20 35 |F8.Ombu8|...'.{ 5|
|00000680| f7 7f 2c 02 b1 08 6f 6d | ea 10 e4 5f 67 46 f6 d0 |..,...om|..._gF..|
|00000690| 6a f7 76 f7 98 0e 14 d5 | 4f 6d 80 2b 10 24 63 8d |j.v.....|Om.+.$c.|
|000006a0| 54 f6 d6 27 53 80 53 02 | b1 05 47 f6 77 36 a0 f6 |T..'S.S.|..G.w6..|
|000006b0| 27 2f 6d 6e f2 e0 f5 95 | f7 50 c6 30 f6 d5 97 df |'/mn....|.P.0....|
|000006c0| 79 07 20 85 02 b6 20 72 | 0c 63 05 e2 75 36 aa f7 |y. ... r|.c..u6..|
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|00000720| 3f 2c 0f 77 10 f6 71 2f | 3a f0 d0 00 00 06 00 00 |?,.w..q/|:.......|
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|00000770| f4 93 0f 53 21 e0 f3 2f | 2c 02 b1 0b 5f 75 81 0b |...S!../|,..._u..|
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|00000820| 00 0b 09 73 68 6f 72 74 | 09 09 69 3b 0d ba 00 06 |...short|..i;....|
|00000830| 00 00 0a 00 00 16 0c 00 | 00 14 00 01 0d ff 00 06 |........|........|
|00000840| 00 00 0c 00 00 14 02 03 | f4 f3 10 e4 5f 62 a1 f6 |........|...._b..|
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|00000950| 0c a5 81 82 0e 7f 78 1a | 07 30 2b 10 85 f7 54 c1 |......x.|.0+...T.|
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|00000970| 6f 79 01 6c b0 e7 f7 81 | a0 73 02 b1 09 18 27 36 |oy.l....|.s....'6|
|00000980| 27 53 0f 67 12 80 2b 10 | c5 38 79 14 62 75 30 2b |'S.g..+.|.8y.bu0+|
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|000009a0| 0c 00 00 14 00 34 f0 91 | f7 82 13 91 9f 09 4f 32 |.....4..|......O2|
|000009b0| f3 20 f3 d0 0f 28 f3 1f | 2f 8f 79 f2 90 f2 a0 c5 |. ...(..|/.y.....|
|000009c0| 8f 28 45 26 27 53 0f 2a | 0f 28 f5 0f 49 f2 ff 31 |.(E&'S.*|.(..I..1|
|000009d0| f3 8f 30 f2 9f 29 f3 bf | 0d 00 00 06 00 00 0a 00 |..0..)..|........|
|000009e0| 00 16 00 34 f0 91 f7 82 | 13 91 9f 09 4f 32 f3 10 |...4....|....O2..|
|000009f0| f3 d0 0f 28 f3 1f 2f 8f | 79 f2 90 f2 a0 87 3f 28 |...(../.|y.....?(|
|00000a00| 45 26 27 53 0f 2a 0f 28 | f5 0f 49 f2 ff 31 f3 8f |E&'S.*.(|..I..1..|
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|00000a20| 00 28 f0 98 b5 42 f0 9f | 09 b1 7f 67 b2 0f 3d 09 |.(...B..|...g..=.|
|00000a30| 18 2f 42 f4 df 2d f3 ef | 62 5f 75 39 8f 2e f6 25 |./B..-..|b_u9...%|
|00000a40| 22 5f 6d f3 bf 0d 00 06 | 00 00 0a 00 00 16 0c 00 |"_m.....|........|
|00000a50| 00 14 00 01 0d ff 00 06 | 00 00 0c 00 00 14 00 01 |........|........|
|00000a60| 0d ff 00 06 00 00 0c 00 | 00 14 00 0c 54 72 61 6e |........|....Tran|
|00000a70| 73 6c 61 74 69 6f 6e 0d | 00 06 00 00 0e 01 00 14 |slation.|........|
|00000a80| 00 01 0d ff 00 06 00 00 | 0c 00 00 14 00 6a 54 72 |........|.....jTr|
|00000a90| 61 6e 73 6c 61 74 69 6e | 67 20 61 20 70 6f 69 6e |anslatin|g a poin|
|00000aa0| 74 20 69 73 20 74 68 65 | 20 65 61 73 69 65 73 74 |t is the| easiest|
|00000ab0| 20 6f 66 20 61 6c 6c 2e | 20 41 6c 6c 20 79 6f 75 | of all.| All you|
|00000ac0| 20 68 61 76 65 20 74 6f | 20 64 6f 20 69 73 20 61 | have to| do is a|
|00000ad0| 64 64 20 6f 66 66 73 65 | 74 73 20 74 6f 20 79 6f |dd offse|ts to yo|
|00000ae0| 75 72 20 70 6f 69 6e 74 | 27 73 20 63 6f 6f 72 64 |ur point|'s coord|
|00000af0| 69 6e 61 74 65 73 3a 0d | 00 06 00 00 0c 00 00 14 |inates:.|........|
|00000b00| 00 01 0d ff 00 06 00 00 | 0c 00 00 14 00 0d 09 78 |........|.......x|
|00000b10| 27 20 3d 20 78 20 2b 20 | 44 78 0d ff 00 06 00 00 |' = x + |Dx......|
|00000b20| 0c 00 00 16 00 0d 09 79 | 27 20 3d 20 79 20 2b 20 |.......y|' = y + |
|00000b30| 44 79 0d ff 00 06 00 00 | 0c 00 00 16 00 01 0d ff |Dy......|........|
|00000b40| 00 06 00 00 0c 00 00 14 | 00 01 0d 06 00 06 00 00 |........|........|
|00000b50| 0c 00 00 14 00 08 53 63 | 61 6c 69 6e 67 0d 00 06 |......Sc|aling...|
|00000b60| 00 00 0e 01 00 14 00 01 | 0d ff 00 06 00 00 0c 00 |........|........|
|00000b70| 00 14 00 48 f2 d0 f4 19 | 90 63 06 88 1f 6d f6 2a |...H....|.c...m.*|
|00000b80| f7 90 f7 61 48 75 30 2b | 62 06 cc 18 81 80 2b 10 |...aHu0+|b.....+.|
|00000b90| f3 6f 38 f3 8f 38 f3 1f | 2f f3 6f 38 f3 8f 38 f3 |.o8..8..|/.o8..8.|
|00000ba0| 20 f6 d5 41 01 dd 7c 71 | 32 af 79 00 00 06 00 00 | ..A..|q|2.y.....|
|00000bb0| 0c 00 00 14 00 46 f2 d0 | f4 93 ca f7 59 10 e4 73 |.....F..|....Y..s|
|00000bc0| 27 3f 67 0c 59 1f 2e 0f | 46 54 0f 4d 6c f4 1e e0 |'?g.Y...|FT.Ml...|
|00000bd0| e4 5f 67 46 f6 d8 f2 c0 | 2b 78 0f 28 6b 1f 6d f2 |._gF....|+x.(k.m.|
|00000be0| 90 78 02 47 f7 67 6a f2 | ef 0d 00 06 00 00 0c 00 |.x.G.gj.|........|
|00000bf0| 00 14 0d ff 00 06 00 00 | 0c 00 00 14 00 01 0d ff |........|........|
|00000c00| 00 06 00 00 0c 00 00 14 | 00 0e 09 78 27 20 3d 20 |........|...x' = |
|00000c10| 28 78 29 28 53 78 29 0d | 00 06 00 00 0c 00 00 16 |(x)(Sx).|........|
|00000c20| 00 0e 09 79 27 20 3d 20 | 28 79 29 28 53 79 29 0d |...y' = |(y)(Sy).|
|00000c30| 00 06 00 00 0c 00 00 16 | 00 01 0d ff 00 06 00 00 |........|........|
|00000c40| 0c 00 00 14 00 82 50 6c | 65 61 73 65 20 6e 6f 74 |......Pl|ease not|
|00000c50| 69 63 65 20 74 68 61 74 | 20 74 68 69 73 20 73 63 |ice that| this sc|
|00000c60| 61 6c 65 73 20 77 69 74 | 68 20 72 65 73 70 65 63 |ales wit|h respec|
|00000c70| 74 20 74 6f 20 74 68 65 | 20 6f 72 69 67 69 6e 2e |t to the| origin.|
|00000c80| 20 49 6e 20 6f 74 68 65 | 72 20 77 6f 72 64 73 2c | In othe|r words,|
|00000c90| 20 74 68 65 73 65 20 65 | 71 75 61 74 69 6f 6e 73 | these e|quations|
|00000ca0| 20 77 69 6c 6c 20 77 69 | 6c 6c 20 63 61 75 73 65 | will wi|ll cause|
|00000cb0| 20 74 68 65 20 66 6f 6c | 6c 6f 77 69 6e 67 20 65 | the fol|lowing e|
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|00001400| 09 72 32 32 20 3d 20 20 | 28 31 2f 53 79 29 63 6f |.r22 = |(1/Sy)co|
|00001410| 73 28 2d 76 29 0d 00 06 | 00 00 0c 00 00 16 00 00 |s(-v)...|........|
|00001420| 00 14 09 09 09 09 53 74 | 65 70 20 34 09 09 09 53 |......St|ep 4...S|
|00001430| 74 65 70 20 35 0d 00 06 | 00 00 0c 01 00 14 00 00 |tep 5...|........|
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|00001490| 00 6c 00 90 00 b4 00 d8 | 00 fc 01 20 01 44 01 68 |.l......|... .D.h|
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|000014b0| 00 48 00 6c 00 90 00 b4 | 00 d8 00 fc 01 20 01 44 |.H.l....|..... .D|
|000014c0| 01 68 00 00 00 00 00 00 | 01 e6 00 0a 00 00 00 00 |.h......|........|
|000014d0| 00 24 00 48 00 6c 00 90 | 00 b4 00 d8 00 fc 01 20 |.$.H.l..|....... |
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|000014f0| 00 00 00 24 00 48 00 6c | 00 90 00 b4 00 d8 00 fc |...$.H.l|........|
|00001500| 01 20 01 44 01 68 00 00 | 00 00 00 00 01 e6 03 0a |. .D.h..|........|
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|00001520| 00 fc 01 20 01 44 01 68 | 00 00 00 00 00 00 01 e6 |... .D.h|........|
|00001530| 00 0a 00 00 00 00 00 24 | 00 48 00 6c 00 90 00 b4 |.......$|.H.l....|
|00001540| 00 d8 00 fc 01 20 01 44 | 01 68 00 00 00 00 00 00 |..... .D|.h......|
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|00001570| 00 00 01 e6 00 0a 00 00 | 00 00 00 24 00 48 00 6c |........|...$.H.l|
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|00001590| 00 00 00 01 0d 00 00 06 | 00 00 0c 00 00 14 29 15 |........|......).|
|000015a0| 01 b0 a0 00 97 a0 00 83 | ff ff 00 00 01 e6 03 0a |........|........|
|000015b0| 00 00 00 00 00 24 00 48 | 00 6c 00 90 00 b4 00 d8 |.....$.H|.l......|
|000015c0| 00 fc 01 20 01 44 01 68 | 00 00 00 00 00 01 0d 00 |... .D.h|........|
|000015d0| 00 06 00 00 0c 00 00 14 | 00 00 01 e6 03 02 00 00 |........|........|
|000015e0| 00 00 00 24 01 cb 00 00 | 00 00 00 00 00 00 00 00 |...$....|........|
|000015f0| 00 00 00 00 00 00 00 00 | 00 00 00 28 54 68 65 20 |........|...(The |
|00001600| 63 6f 6f 72 64 69 6e 61 | 74 65 73 20 6f 66 20 20 |coordina|tes of |
|00001610| 50 20 63 61 6e 20 62 65 | 20 65 78 70 72 65 73 73 |P can be| express|
|00001620| 65 64 3a 0d 00 12 00 00 | 0c 00 00 14 00 14 0c 00 |ed:.....|........|
|00001630| 00 16 00 15 0c 00 00 14 | 00 01 0d ff 00 06 00 00 |........|........|
|00001640| 0c 00 00 14 00 0e 09 78 | 20 3d 20 72 20 63 6f 73 |.......x| = r cos|
|00001650| 28 75 29 0d 00 06 00 00 | 0c 00 00 16 00 0e 09 79 |(u).....|.......y|
|00001660| 20 3d 20 72 20 73 69 6e | 28 75 29 0d 00 06 00 00 | = r sin|(u).....|
|00001670| 0c 00 00 16 00 01 0d ff | 00 06 00 00 0c 00 00 14 |........|........|
|00001680| 00 20 53 69 6d 69 6c 61 | 72 6c 79 2c 20 50 27 20 |. Simila|rly, P' |
|00001690| 63 61 6e 20 62 65 20 65 | 78 70 72 65 73 73 65 64 |can be e|xpressed|
|000016a0| 3a 0d 00 06 00 00 0c 00 | 00 14 00 01 0d ff 00 06 |:.......|........|
|000016b0| 00 00 0c 00 00 14 00 35 | 09 78 27 20 3d 20 72 20 |.......5|.x' = r |
|000016c0| 63 6f 73 28 75 20 2b 20 | 76 29 20 3d 20 72 20 28 |cos(u + |v) = r (|
|000016d0| 63 6f 73 28 75 29 63 6f | 73 28 76 29 20 2d 20 73 |cos(u)co|s(v) - s|
|000016e0| 69 6e 28 75 29 73 69 6e | 28 76 29 29 0d ff 00 06 |in(u)sin|(v))....|
|000016f0| 00 00 0c 00 00 16 00 35 | 09 79 27 20 3d 20 72 20 |.......5|.y' = r |
|00001700| 73 69 6e 28 75 20 2b 20 | 76 29 20 3d 20 72 20 28 |sin(u + |v) = r (|
|00001710| 63 6f 73 28 75 29 73 69 | 6e 28 76 29 20 2b 20 73 |cos(u)si|n(v) + s|
|00001720| 69 6e 28 75 29 63 6f 73 | 28 76 29 29 0d ff 00 06 |in(u)cos|(v))....|
|00001730| 00 00 0c 00 00 16 00 01 | 0d ff 00 06 00 00 0c 00 |........|........|
|00001740| 00 14 00 35 44 69 73 74 | 72 69 62 75 74 69 6e 67 |...5Dist|ributing|
|00001750| 20 72 2c 20 61 6e 64 20 | 73 75 62 73 74 69 74 75 | r, and |substitu|
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|00001770| 20 77 65 20 67 65 74 3a | 0d ff 00 2a 00 00 0c 00 | we get:|...*....|
|00001780| 00 14 00 0d 0c 00 00 16 | 00 0e 0c 00 00 14 00 24 |........|.......$|
|00001790| 0c 00 00 16 00 25 0c 00 | 00 14 00 2a 0c 00 00 16 |.....%..|...*....|
|000017a0| 00 2b 0c 00 00 14 00 01 | 0d ff 00 06 00 00 0c 00 |.+......|........|
|000017b0| 00 14 00 26 09 78 27 20 | 3d 20 72 20 63 6f 73 28 |...&.x' |= r cos(|
|000017c0| 75 29 63 6f 73 28 76 29 | 20 2d 20 72 20 73 69 6e |u)cos(v)| - r sin|
|000017d0| 28 75 29 73 69 6e 28 76 | 29 0d 00 06 00 00 0c 00 |(u)sin(v|).......|
|000017e0| 00 16 00 1a 09 20 20 20 | 3d 20 78 20 63 6f 73 28 |..... |= x cos(|
|000017f0| 76 29 20 2d 20 79 20 73 | 69 6e 28 76 29 0d 00 06 |v) - y s|in(v)...|
|00001800| 00 00 0c 00 00 16 00 26 | 09 79 27 20 3d 20 72 20 |.......&|.y' = r |
|00001810| 63 6f 73 28 75 29 73 69 | 6e 28 71 29 20 2b 20 72 |cos(u)si|n(q) + r|
|00001820| 20 73 69 6e 28 75 29 63 | 6f 73 28 76 29 0d 00 06 | sin(u)c|os(v)...|
|00001830| 00 00 0c 00 00 16 00 1a | 09 20 20 20 3d 20 78 20 |........|. = x |
|00001840| 73 69 6e 28 76 29 20 2b | 20 79 20 63 6f 73 28 76 |sin(v) +| y cos(v|
|00001850| 29 0d 00 06 00 00 0c 00 | 00 16 00 01 0d ff 00 06 |).......|........|
|00001860| 00 00 0c 00 00 14 00 01 | 0d ff 00 06 00 00 0c 00 |........|........|
|00001870| 00 14 00 18 50 75 74 74 | 69 6e 67 20 69 74 20 61 |....Putt|ing it a|
|00001880| 6c 6c 20 74 6f 67 65 74 | 68 65 72 0d 00 06 00 00 |ll toget|her.....|
|00001890| 0e 01 00 14 00 01 0d ff | 00 06 00 00 0c 00 00 14 |........|........|
|000018a0| 00 de 57 65 20 6e 6f 77 | 20 68 61 76 65 20 65 76 |..We now| have ev|
|000018b0| 65 72 79 74 68 69 6e 67 | 20 77 65 20 6e 65 65 64 |erything| we need|
|000018c0| 20 74 6f 20 70 65 72 66 | 6f 72 6d 20 67 65 6e 65 | to perf|orm gene|
|000018d0| 72 61 6c 20 74 72 61 6e | 73 66 6f 72 6d 61 74 69 |ral tran|sformati|
|000018e0| 6f 6e 73 20 6f 66 20 61 | 20 62 69 74 6d 61 70 2e |ons of a| bitmap.|
|000018f0| 20 49 6e 20 67 65 6e 65 | 72 61 6c 2c 20 77 65 20 | In gene|ral, we |
|00001900| 74 72 61 6e 73 6c 61 74 | 65 20 77 68 61 74 65 76 |translat|e whatev|
|00001910| 65 72 20 69 74 20 69 73 | 20 77 65 20 61 72 65 20 |er it is| we are |
|00001920| 74 72 61 6e 73 6d 6f 67 | 72 69 66 79 69 6e 67 20 |transmog|rifying |
|00001930| 74 6f 20 74 68 65 20 6f | 72 69 67 69 6e 2c 20 73 |to the o|rigin, s|
|00001940| 63 61 6c 65 20 69 74 2c | 20 72 6f 74 61 74 65 20 |cale it,| rotate |
|00001950| 69 74 2c 20 61 6e 64 20 | 74 68 65 6e 20 6d 6f 76 |it, and |then mov|
|00001960| 65 20 69 74 20 62 61 63 | 6b 2e 20 54 68 69 73 20 |e it bac|k. This |
|00001970| 69 73 20 73 68 6f 77 6e | 20 62 65 6c 6f 77 3a 0d |is shown| below:.|
|00001980| 00 06 00 00 0c 00 00 14 | 00 01 0d 44 00 06 00 00 |........|...D....|
|00001990| 0c 00 00 14 00 58 00 3f | 00 eb 01 aa 01 f0 ff ff |.....X.?|........|
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|000019c0| ff ff ff ff 00 92 01 6a | 71 00 1e 00 01 00 45 00 |.......j|q.....E.|
|000019d0| 0f 00 4c 00 01 00 49 00 | 0f 00 4c 00 0f 00 49 00 |..L...I.|..L...I.|
|000019e0| 0f 00 45 00 01 00 49 71 | 00 1e 00 82 00 45 00 91 |..E...Iq|.....E..|
|000019f0| 00 4c 00 91 00 49 00 82 | 00 45 00 82 00 49 00 82 |.L...I..|.E...I..|
|00001a00| 00 4c 00 91 00 49 22 00 | 0f 00 49 00 73 a0 00 8d |.L...I".|..I.s...|
|00001a10| a1 00 64 00 08 4d 44 50 | 4c 00 0a 00 00 a1 00 64 |..d..MDP|L......d|
|00001a20| 00 0a 4d 44 50 4c 00 09 | 00 01 00 03 a0 00 8c 71 |..MDPL..|.......q|
|00001a30| 00 1e 00 45 00 01 00 4c | 00 0f 00 49 00 01 00 45 |...E...L|...I...E|
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|00001a50| 1e 00 45 00 82 00 4c 00 | 91 00 49 00 91 00 4c 00 |..E...L.|..I...L.|
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|00001df0| 6d 79 53 65 74 50 69 78 | 65 6c 28 64 65 73 74 42 |mySetPix|el(destB|
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|00001f10| 32 09 57 3f 67 02 b1 81 | 01 f7 82 46 05 e1 46 27 |2.W?g...|...F..F'|
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|00001f70| 6d 62 75 30 f6 71 28 f2 | ef 0d 00 06 00 00 0c 00 |mbu0.q(.|........|
|00001f80| 00 14 00 01 0d 02 00 06 | 00 00 0c 00 00 14 0d ff |........|........|
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|00001ff0| 00 16 00 0e 00 13 00 0e | 00 0f 00 00 00 13 01 00 |........|........|
|00002000| 0a 00 00 00 00 00 00 00 | 00 22 00 00 00 13 03 0e |........|."......|
|00002010| 23 00 00 23 fd 00 23 00 | 00 23 fc 00 23 00 00 23 |#..#..#.|.#..#..#|
|00002020| 04 f2 23 00 00 84 00 0a | 00 00 00 00 00 00 00 00 |..#.....|........|
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|000020e0| a1 07 00 01 00 01 22 00 | 6c 00 01 78 00 a0 00 8d |......".|l..x....|
|000020f0| 31 00 3f 00 5b 00 44 00 | 60 38 31 00 28 00 33 00 |1.?.[.D.|`81.(.3.|
|00002100| 2d 00 38 38 22 00 6c 00 | 13 4a d6 22 00 6c 00 13 |-.88".l.|.J.".l..|
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|00002140| 0c 2b 2d 52 01 76 03 00 | 17 29 07 01 b0 a0 00 97 |.+-R.v..|.)......|
|00002150| a1 00 96 00 0c 01 00 00 | 00 02 00 00 00 00 00 00 |........|........|
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|00002180| 6b 01 50 a0 00 97 a1 00 | 96 00 0c 01 00 00 00 02 |k.P.....|........|
|00002190| 00 00 00 00 00 00 00 a1 | 00 9a 00 08 ff fc 00 00 |........|........|
|000021a0| 00 04 00 00 01 00 0a 00 | 1a 00 2a 00 27 00 34 28 |........|..*.'.4(|
|000021b0| 00 24 00 2b 02 50 27 a0 | 00 97 a1 00 96 00 0c 01 |.$.+.P'.|........|
|000021c0| 00 00 00 02 00 00 00 00 | 00 00 00 a1 00 9a 00 08 |........|........|
|000021d0| ff fc 00 00 00 05 00 00 | 01 00 0a 00 5b 00 38 00 |........|....[.8.|
|000021e0| 68 00 45 03 00 16 2b 0e | 41 01 75 03 00 17 29 07 |h.E...+.|A.u...).|
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|00002210| 00 10 00 a4 00 51 00 00 | 00 5a a1 00 96 00 0c 01 |.....Q..|.Z......|
|00002220| 00 00 00 02 00 00 00 00 | 00 00 00 a1 00 9a 00 08 |........|........|
|00002230| ff fc 00 00 00 0c 00 00 | 01 00 0a 00 2c 00 42 00 |........|....,.B.|
|00002240| 39 00 5d 03 00 16 28 00 | 36 00 43 03 75 2b 76 03 |9.]...(.|6.C.u+v.|
|00002250| 00 17 29 15 01 b0 a0 00 | 97 a0 00 83 ff 00 00 42 |..).....|.......B|
|00002260| f0 91 f7 82 13 91 9f 09 | 2f 79 f0 9f 3d 0f 2d 91 |........|/y..=.-.|
|00002270| 82 73 62 75 3f 2e bf 2a | 4f 32 f3 10 f2 d0 91 82 |.sbu?..*|O2......|
|00002280| 73 62 75 3f 2e f7 6f 2a | 4f 32 f3 20 f2 b0 c1 32 |sbu?..o*|O2. ...2|
|00002290| 14 f2 ef 76 f3 bf 0d 00 | 00 06 00 00 0a 00 00 16 |...v....|........|
|000022a0| 00 01 0d 02 00 06 00 00 | 0c 00 00 14 00 01 0d 02 |........|........|
|000022b0| 00 06 00 00 0c 00 00 14 | 0d ff 00 06 00 00 0c 01 |........|........|
|000022c0| 00 14 00 00 01 e6 03 0a | 00 00 00 03 09 09 0d 00 |........|........|
|000022d0| 00 06 00 00 0a 00 00 16 | 00 02 09 0d 00 06 00 00 |........|........|
|000022e0| 0a 00 00 16 00 01 0d ff | 00 06 00 00 0c 00 00 14 |........|........|
|000022f0| 00 da 49 6e 20 20 53 74 | 65 70 20 33 20 77 65 20 |..In St|ep 3 we |
|00002300| 73 68 6f 77 20 74 68 65 | 20 68 6f 75 73 65 20 72 |show the| house r|
|00002310| 6f 74 61 74 65 64 20 61 | 62 6f 75 74 20 74 68 65 |otated a|bout the|
|00002320| 20 6f 72 69 67 69 6e 20 | 62 79 20 61 70 70 6c 79 | origin |by apply|
|00002330| 69 6e 67 20 74 68 65 20 | 66 6f 72 6d 75 6c 61 20 |ing the |formula |
|00002340| 64 65 73 63 72 69 62 65 | 64 20 69 6e 20 74 68 65 |describe|d in the|
|00002350| 20 6c 61 73 74 20 73 65 | 63 74 69 6f 6e 20 66 6f | last se|ction fo|
|00002360| 72 20 72 6f 74 61 74 69 | 6e 67 2e 20 46 69 6e 61 |r rotati|ng. Fina|
|00002370| 6c 6c 79 2c 20 74 68 65 | 20 68 6f 75 73 65 20 69 |lly, the| house i|
|00002380| 73 20 74 72 61 6e 73 6c | 61 74 65 64 20 62 61 63 |s transl|ated bac|
|00002390| 6b 20 74 6f 20 69 74 73 | 20 6f 72 69 67 69 6e 61 |k to its| origina|
|000023a0| 6c 20 70 6f 73 69 74 69 | 6f 6e 2c 20 67 69 76 69 |l positi|on, givi|
|000023b0| 6e 67 20 75 73 20 74 68 | 65 20 72 65 73 75 6c 74 |ng us th|e result|
|000023c0| 20 69 6e 20 53 74 65 70 | 20 35 2e 0d 00 06 00 00 | in Step| 5......|
|000023d0| 0c 00 00 14 00 01 0d ff | 00 06 00 00 0c 00 00 14 |........|........|
|000023e0| 00 e7 4c 65 74 27 73 20 | 6e 6f 77 20 66 6f 6c 6c |..Let's |now foll|
|000023f0| 6f 77 20 74 68 65 20 6c | 69 66 65 20 6f 66 20 6f |ow the l|ife of o|
|00002400| 6e 65 20 6f 66 20 74 68 | 65 20 63 6f 6f 72 64 69 |ne of th|e coordi|
|00002410| 6e 61 74 65 73 20 74 68 | 72 6f 75 67 68 20 74 68 |nates th|rough th|
|00002420| 69 73 20 70 72 6f 63 65 | 73 73 2e 20 49 6e 20 74 |is proce|ss. In t|
|00002430| 68 69 73 20 63 61 73 65 | 2c 20 77 65 27 6c 6c 20 |his case|, we'll |
|00002440| 6c 6f 6f 6b 20 61 74 20 | 74 68 65 20 6c 6f 77 65 |look at |the lowe|
|00002450| 72 20 6c 65 66 74 20 68 | 61 6e 64 20 63 6f 72 6e |r left h|and corn|
|00002460| 65 72 20 6f 66 20 74 68 | 65 20 68 6f 75 73 65 2c |er of th|e house,|
|00002470| 20 61 6e 64 20 61 73 73 | 75 6d 65 20 74 68 61 74 | and ass|ume that|
|00002480| 20 69 74 27 73 20 6c 6f | 63 61 74 69 6f 6e 20 69 | it's lo|cation i|
|00002490| 73 20 28 32 2c 20 33 29 | 2c 20 61 6e 64 20 74 68 |s (2, 3)|, and th|
|000024a0| 61 74 20 74 68 65 20 6f | 72 69 67 69 6e 20 6f 66 |at the o|rigin of|
|000024b0| 20 74 68 65 20 68 6f 75 | 73 65 20 69 73 20 61 74 | the hou|se is at|
|000024c0| 20 28 34 2c 20 36 29 2e | 0d ff 00 06 00 00 0c 00 | (4, 6).|........|
|000024d0| 00 14 00 01 0d ff 00 06 | 00 00 0c 00 00 14 00 d0 |........|........|
|000024e0| 54 68 65 20 66 69 72 73 | 74 20 73 74 65 70 20 69 |The firs|t step i|
|000024f0| 73 20 74 6f 20 6d 6f 76 | 65 20 74 68 65 20 63 65 |s to mov|e the ce|
|00002500| 6e 74 65 72 20 6f 66 20 | 74 68 65 20 68 6f 75 73 |nter of |the hous|
|00002510| 65 20 74 6f 20 74 68 65 | 20 6f 72 69 67 69 6e 2e |e to the| origin.|
|00002520| 20 54 68 69 73 20 69 73 | 20 64 6f 6e 65 20 62 79 | This is| done by|
|00002530| 20 61 64 64 69 6e 67 20 | 74 68 65 20 6f 66 66 73 | adding |the offs|
|00002540| 65 74 73 20 2d 34 20 61 | 6e 64 20 2d 36 20 74 6f |ets -4 a|nd -6 to|
|00002550| 20 74 68 65 20 58 20 61 | 6e 64 20 59 20 63 6f 6f | the X a|nd Y coo|
|00002560| 72 64 69 6e 61 74 65 73 | 20 72 65 73 70 65 63 74 |rdinates| respect|
|00002570| 69 76 65 6c 79 2e 20 54 | 68 69 73 20 6d 6f 76 65 |ively. T|his move|
|00002580| 73 20 6f 75 72 20 70 65 | 74 20 70 6f 69 6e 74 20 |s our pe|t point |
|00002590| 74 6f 20 28 32 20 2d 20 | 34 2c 20 33 20 2d 20 36 |to (2 - |4, 3 - 6|
|000025a0| 29 2c 20 6f 72 20 28 2d | 32 2c 20 2d 33 29 2e 0d |), or (-|2, -3)..|
|000025b0| 00 06 00 00 0c 00 00 14 | 00 01 0d ed 00 06 00 00 |........|........|
|000025c0| 0c 00 00 14 00 9a 54 68 | 65 20 6e 65 78 74 20 73 |......Th|e next s|
|000025d0| 74 65 70 20 69 6e 76 6f | 6c 76 65 73 20 73 63 61 |tep invo|lves sca|
|000025e0| 6c 69 6e 67 20 74 68 65 | 20 68 6f 75 73 65 2e 20 |ling the| house. |
|000025f0| 49 6e 20 74 68 69 73 20 | 63 61 73 65 2c 20 49 20 |In this |case, I |
|00002600| 65 78 70 61 6e 64 65 64 | 20 74 68 65 20 68 6f 75 |expanded| the hou|
|00002610| 73 65 20 62 79 20 36 2f | 34 20 68 6f 72 69 7a 6f |se by 6/|4 horizo|
|00002620| 6e 74 61 6c 6c 79 2c 20 | 61 6e 64 20 38 2f 36 20 |ntally, |and 8/6 |
|00002630| 76 65 72 74 69 63 61 6c | 6c 79 2e 20 41 70 70 6c |vertical|ly. Appl|
|00002640| 79 69 6e 67 20 74 68 69 | 73 20 74 6f 20 6f 75 72 |ying thi|s to our|
|00002650| 20 70 6f 69 6e 74 2c 20 | 77 65 20 67 65 74 3a 0d | point, |we get:.|
|00002660| 00 06 00 00 0c 00 00 14 | 00 16 f0 9f 43 a1 64 f4 |........|....C.d.|
|00002670| 27 2f 4d 6e f2 89 18 2f | 42 f4 df 29 f3 bf 0d 00 |'/Mn.../|B..)....|
|00002680| 00 06 00 00 0a 00 00 16 | 00 00 00 01 0d 69 00 06 |........|.....i..|
|00002690| 00 00 0c 00 00 14 00 13 | 09 78 27 20 3d 20 2d 32 |........|.x' = -2|
|000026a0| 28 36 2f 34 29 20 3d 20 | 2d 33 0d 53 00 06 00 00 |(6/4) = |-3.S....|
|000026b0| 0c 00 00 16 00 13 09 79 | 27 20 3d 20 2d 33 28 38 |.......y|' = -3(8|
|000026c0| 2f 36 29 20 3d 20 2d 34 | 0d 00 00 06 00 00 0c 00 |/6) = -4|........|
|000026d0| 00 16 00 01 0d 6e 00 06 | 00 00 0c 00 00 14 00 01 |.....n..|........|
|000026e0| 0d 0f 00 06 00 00 0a 00 | 00 16 00 01 0d 06 00 06 |........|........|
|000026f0| 00 00 0c 00 00 14 01 20 | 01 44 01 68 00 00 00 00 |....... |.D.h....|
|00002700| 00 66 54 68 65 20 74 68 | 69 72 64 20 73 74 65 70 |.fThe th|ird step|
|00002710| 20 69 73 20 74 6f 20 72 | 6f 74 61 74 65 20 74 68 | is to r|otate th|
|00002720| 65 20 68 6f 75 73 65 20 | 39 30 20 64 65 67 72 65 |e house |90 degre|
|00002730| 65 73 20 61 72 6f 75 6e | 64 20 74 68 65 20 6f 72 |es aroun|d the or|
|00002740| 69 67 69 6e 2e 20 55 73 | 69 6e 67 20 6f 75 72 20 |igin. Us|ing our |
|00002750| 66 6f 72 6d 75 6c 61 73 | 20 61 62 6f 76 65 2c 20 |formulas| above, |
|00002760| 77 65 20 67 65 74 3a 0d | 00 06 00 00 0c 00 00 14 |we get:.|........|
|00002770| 00 01 0d d3 00 06 00 00 | 0c 00 00 14 00 02 09 0d |........|........|
|00002780| 00 06 00 00 0a 00 00 16 | 00 01 0d ff 00 06 00 00 |........|........|
|00002790| 0a 00 00 16 01 20 01 44 | 01 68 00 00 00 00 00 1a |..... .D|.h......|
|000027a0| 09 78 27 20 3d 20 78 20 | 63 6f 73 28 76 29 20 2d |.x' = x |cos(v) -|
|000027b0| 20 79 20 73 69 6e 28 76 | 29 0d 00 06 00 00 0c 00 | y sin(v|).......|
|000027c0| 00 16 00 20 09 20 20 20 | 3d 20 28 2d 33 29 63 6f |... . |= (-3)co|
|000027d0| 73 28 39 30 29 20 2d 20 | 28 2d 34 29 73 69 6e 28 |s(90) - |(-4)sin(|
|000027e0| 39 30 29 0d 00 06 00 00 | 0c 00 00 16 00 18 09 20 |90).....|....... |
|000027f0| 20 20 3d 20 28 2d 33 29 | 28 30 29 20 2d 20 28 2d | = (-3)|(0) - (-|
|00002800| 34 29 28 31 29 0d 00 06 | 00 00 0c 00 00 16 00 07 |4)(1)...|........|
|00002810| 57 68 65 72 65 3a 0d 02 | 00 06 00 00 0c 00 00 14 |Where:..|........|
|00002820| 00 1a 09 79 27 20 3d 20 | 78 20 73 69 6e 28 76 29 |...y' = |x sin(v)|
|00002830| 20 2b 20 79 20 63 6f 73 | 28 76 29 0d 00 06 00 00 | + y cos|(v).....|
|00002840| 0c 00 00 16 00 20 09 20 | 20 20 3d 20 28 2d 33 29 |..... . | = (-3)|
|00002850| 73 69 6e 28 39 30 29 20 | 2b 20 28 2d 34 29 63 6f |sin(90) |+ (-4)co|
|00002860| 73 28 39 30 29 0d 00 06 | 00 00 0c 00 00 16 00 18 |s(90)...|........|
|00002870| 09 20 20 20 3d 20 28 2d | 33 29 28 31 29 20 2b 20 |. = (-|3)(1) + |
|00002880| 28 2d 34 29 28 30 29 0d | 00 06 00 00 0c 00 00 16 |(-4)(0).|........|
|00002890| 00 09 09 20 20 20 3d 20 | 2d 33 0d f1 00 06 00 00 |... = |-3......|
|000028a0| 0c 00 00 16 00 01 0d f1 | 00 06 00 00 0c 00 00 14 |........|........|
|000028b0| 00 d6 f2 d0 f4 35 44 1c | 2a f7 90 c6 ac f7 5a 62 |.....5D.|*.....Zb|
|000028c0| 10 2b 10 87 f7 a1 05 d0 | 2b 10 91 82 73 62 75 30 |.+......|+...sbu0|
|000028d0| f6 27 2f 6d 6e f2 e0 f5 | 27 f6 7b 20 35 f7 7f 2c |.'/mn...|'.{ 5..,|
|000028e0| 02 b1 08 6f 6d ea 10 e4 | 5f 67 46 f6 d0 d5 4c 18 |...om...|_gF...L.|
|000028f0| 02 b1 09 18 27 36 27 53 | 0f 62 72 f6 d6 e0 25 0f |....'6'S|.br...%.|
|00002900| 62 10 2b 10 87 f7 a1 05 | d0 60 f4 d6 cf 50 67 32 |b.+.....|.`...Pg2|
|00002910| 0e 7c 2f 75 41 f2 c0 41 | f6 76 49 a1 88 05 d0 b5 |.|/uA..A|.vI.....|
|00002920| f7 70 f6 df 75 cb 07 80 | 31 19 19 02 50 b5 a9 02 |.p..u...|1...P...|
|00002930| b1 0d 73 6a 07 f6 d6 f6 | 71 f2 ef 0d 00 06 00 00 |..sj....|q.......|
|00002940| 0c 00 00 14 6f 73 69 74 | 65 20 61 6d 6f 75 6e 74 |....osit|e amount|
|00002950| 73 20 77 65 20 75 73 65 | 64 20 74 6f 20 6d 6f 76 |s we use|d to mov|
|00002960| 65 20 69 74 20 74 6f 20 | 74 68 65 20 6f 72 69 67 |e it to |the orig|
|00002970| 69 6e 20 69 6e 20 74 68 | 65 20 66 69 72 73 74 20 |in in th|e first |
|00002980| 70 6c 61 63 65 2e 20 44 | 6f 69 6e 67 20 74 68 69 |place. D|oing thi|
|00002990| 00 a2 f5 4b 14 10 64 10 | f6 d6 3f 79 02 b7 3f 67 |...K..d.|..?y..?g|
|000029a0| 80 f7 95 f7 50 c6 30 95 | 0f 77 b1 30 ea 6f 79 73 |....P.0.|.w.0.oys|
|000029b0| f6 70 f7 77 2b 02 b1 81 | 04 5f 75 27 31 80 d5 40 |.p.w+...|._u'1..@|
|000029c0| f7 95 f7 54 05 f7 73 0e | f7 54 e5 81 8f 2e 0f 53 |...T..s.|.T.....S|
|000029d0| 5f 6d 10 5d 02 b1 81 07 | 91 68 0f 6d 6f 79 06 ee |_m.]....|.h.moy..|
|000029e0| 16 40 73 0d f7 52 f7 54 | 10 f7 61 48 75 38 05 d0 |.@s..R.T|..aHu8..|
|000029f0| 2b 10 86 f6 de a1 0e 45 | f6 74 6f 6d f2 ef 0d 00 |+......E|.tom....|
|00002a00| 00 06 00 00 0c 00 00 14 | 20 67 72 69 64 29 2c 20 |........| grid), |
|00002a10| 74 68 61 74 27 73 20 65 | 78 61 63 74 6c 79 20 77 |that's e|xactly w|
|00002a20| 68 65 72 65 20 74 68 61 | 74 20 73 75 63 6b 65 72 |here tha|t sucker|
|00002a30| 20 69 73 21 0d 00 00 06 | 00 00 0c 00 00 14 00 01 | is!....|........|
|00002a40| 0d 00 00 06 00 00 0c 00 | 00 14 00 01 0d 00 00 06 |........|........|
|00002a50| 00 00 0c 00 00 14 00 19 | 47 65 74 20 44 6f 77 6e |........|Get Down|
|00002a60| 2c 20 47 65 74 20 43 6f | 6d 70 6f 73 69 74 65 21 |, Get Co|mposite!|
|00002a70| 0d 00 00 06 00 00 0e 01 | 00 14 00 01 0d 00 00 06 |........|........|
|00002a80| 00 00 0c 00 00 14 00 40 | f0 9f 09 f0 9f 50 57 32 |.......@|.....PW2|
|00002a90| 09 18 27 36 27 53 f2 c0 | 8b 54 20 45 26 27 53 f2 |..'6'S..|.T E&'S.|
|00002aa0| c0 1f 78 21 39 19 08 f7 | 8f 2c 01 f7 82 13 91 90 |..x!9...|.,......|
|00002ab0| 8f 79 f2 9f 0d 00 00 06 | 00 00 0a 00 00 16 73 20 |.y......|......s |
|00002ac0| 72 65 61 6c 6c 79 20 69 | 6d 70 72 65 73 73 20 74 |really i|mpress t|
|00002ad0| 00 01 0d 88 00 06 00 00 | 0c 00 00 14 00 00 0c 00 |........|........|
|00002ae0| 00 14 00 01 0d 00 00 06 | 00 00 0c 00 00 14 00 51 |........|.......Q|
|00002af0| 56 65 72 79 20 62 72 69 | 65 66 6c 79 2c 20 74 68 |Very bri|efly, th|
|00002b00| 65 20 70 61 74 68 20 74 | 68 61 74 20 74 68 65 20 |e path t|hat the |
|00002b10| 58 20 63 6f 6f 72 64 69 | 6e 61 74 65 20 6f 66 20 |X coordi|nate of |
|00002b20| 6f 75 72 20 73 61 6d 70 | 6c 65 20 70 6f 69 6e 74 |our samp|le point|
|00002b30| 20 61 62 6f 76 65 20 74 | 6f 6f 6b 20 77 61 73 3a | above t|ook was:|
|00002b40| 0d 00 00 06 00 00 0c 00 | 00 14 00 01 0d 00 00 06 |........|........|
|00002b50| 00 00 0c 00 00 14 00 01 | 0d ba 00 06 00 00 0a 00 |........|........|
|00002b60| 00 16 00 0b 09 73 68 6f | 72 74 09 09 6a 3b 0d 00 |.....sho|rt..j;..|
|00002b70| 00 06 00 00 0a 00 00 16 | 00 34 09 78 27 20 3d 20 |........|.4.x' = |
|00002b80| 28 78 2d 34 29 28 36 2f | 34 29 63 6f 73 28 39 30 |(x-4)(6/|4)cos(90|
|00002b90| 29 20 2d 20 28 79 2d 36 | 29 28 38 2f 36 29 73 69 |) - (y-6|)(8/6)si|
|00002ba0| 6e 28 39 30 29 20 2b 20 | 34 09 5b 31 5d 0d 00 06 |n(90) + |4.[1]...|
|00002bb0| 00 00 0c 00 00 16 00 11 | 09 6c 6f 6e 67 09 09 2a |........|.long..*|
|00002bc0| 6c 6f 6e 67 50 74 72 3b | 0d d8 00 06 00 00 0a 00 |longPtr;|........|
|00002bd0| 00 16 01 68 00 00 00 00 | 00 01 0d 00 00 06 00 00 |...h....|........|
|00002be0| 0c 00 00 14 00 bb 45 78 | 61 6d 69 6e 69 6e 67 20 |......Ex|amining |
|00002bf0| 74 68 69 73 20 65 71 75 | 61 74 69 6f 6e 20 74 65 |this equ|ation te|
|00002c00| 6c 6c 73 20 75 73 20 61 | 20 63 6f 75 70 6c 65 20 |lls us a| couple |
|00002c10| 6f 66 20 74 68 69 6e 67 | 73 2e 20 46 6f 72 20 6f |of thing|s. For o|
|00002c20| 6e 65 2c 20 72 6f 74 61 | 74 69 6e 67 20 62 79 20 |ne, rota|ting by |
|00002c30| 39 30 20 64 65 67 72 65 | 65 73 20 69 73 20 61 20 |90 degre|es is a |
|00002c40| 73 6e 61 70 20 69 66 20 | 79 6f 75 20 72 65 6d 65 |snap if |you reme|
|00002c50| 6d 62 65 72 20 74 68 61 | 74 20 63 6f 73 28 39 30 |mber tha|t cos(90|
|00002c60| 29 20 69 73 20 61 6c 77 | 61 79 73 20 30 2c 20 61 |) is alw|ays 0, a|
|00002c70| 6e 64 20 73 69 6e 28 39 | 30 29 20 69 73 20 61 6c |nd sin(9|0) is al|
|00002c80| 77 61 79 73 20 31 2e 20 | 54 68 69 73 20 6d 61 6b |ways 1. |This mak|
|00002c90| 65 73 20 6f 75 72 20 65 | 71 75 61 74 69 6f 6e 3a |es our e|quation:|
|00002ca0| 0d 00 00 36 00 00 0c 00 | 00 14 00 74 0c 00 00 16 |...6....|...t....|
|00002cb0| 00 7b 0c 00 00 14 00 86 | 0c 00 00 16 00 87 0c 00 |.{......|........|
|00002cc0| 00 14 00 8d 0c 00 00 16 | 00 94 0c 00 00 14 00 9f |........|........|
|00002cd0| 0c 00 00 16 00 a0 0c 00 | 00 14 00 01 0d 00 00 06 |........|........|
|00002ce0| 00 00 0c 00 00 14 00 11 | f0 9f 42 55 a1 63 f0 9f |........|..BU.c..|
|00002cf0| 09 78 f5 31 2f 3b f0 d0 | 00 06 00 00 0a 00 00 16 |.x.1/;..|........|
|00002d00| 00 00 00 00 00 00 00 00 | 00 19 09 78 27 20 3d 20 |........|...x' = |
|00002d10| 28 36 2d 79 29 28 38 2f | 36 29 20 2b 20 34 09 5b |(6-y)(8/|6) + 4.[|
|00002d20| 32 5d 0d 00 00 06 00 00 | 0c 00 00 16 00 14 09 73 |2]......|.......s|
|00002d30| 68 6f 72 74 09 09 6e 65 | 77 58 2c 20 6e 65 77 59 |hort..ne|wX, newY|
|00002d40| 3b 0d 00 06 00 00 0a 00 | 00 16 00 00 00 00 00 01 |;.......|........|
|00002d50| 0d 64 00 06 00 00 0c 00 | 00 14 00 34 09 65 78 74 |.d......|...4.ext|
|00002d60| 65 6e 64 65 64 09 72 31 | 32 20 3d 20 2d 28 31 2f |ended.r1|2 = -(1/|
|00002d70| 73 78 29 20 2a 20 73 69 | 6e 28 72 6f 74 61 74 69 |sx) * si|n(rotati|
|00002d80| 6f 6e 20 2a 20 28 50 49 | 2f 31 38 30 29 29 3b 0d |on * (PI|/180));.|
|00002d90| 00 06 00 00 0a 00 00 16 | 00 34 f0 91 f7 82 13 91 |........|.4......|
|00002da0| 9f 09 4f 31 f3 10 f3 d0 | 0f 28 f3 1f 2f 8f 78 f2 |..O1....|.(../.x.|
|00002db0| 90 f2 a0 c5 8f 28 45 26 | 27 53 0f 2a 0f 28 f5 0f |.....(E&|'S.*.(..|
|00002dc0| 49 f2 ff 31 f3 8f 30 f2 | 9f 29 f3 bf 0d 00 00 06 |I..1..0.|.)......|
|00002dd0| 00 00 0a 00 00 16 00 19 | f4 71 22 73 f6 70 73 25 |........|.q"s.ps%|
|00002de0| 02 b1 0f 52 5f 75 27 31 | f0 d0 00 06 00 00 0e 01 |...R_u'1|........|
|00002df0| 00 14 90 f6 27 2f 6d 6e | f2 ef 0d 00 00 06 00 00 |....'/mn|........|
|00002e00| 0c 00 00 14 72 65 20 6f | 6e 65 20 74 6f 6f 20 6d |....re o|ne too m|
|00002e10| 61 6e 79 20 6d 75 6c 74 | 69 70 6c 69 65 73 20 61 |any mult|iplies a|
|00002e20| 6e 64 20 6f 6e 65 20 74 | 6f 6f 20 6d 61 6e 79 20 |nd one t|oo many |
|00002e30| 64 69 76 69 64 65 73 20 | 69 6e 20 74 68 65 72 65 |divides |in there|
|00002e40| 2e 20 49 6e 20 61 20 74 | 69 6d 65 20 63 72 69 74 |. In a t|ime crit|
|00002e50| 69 63 61 6c 20 6c 00 2d | 09 09 09 69 73 53 65 74 |ical l.-|...isSet|
|00002e60| 20 3d 20 6d 79 47 65 74 | 50 69 78 65 6c 28 73 6f | = myGet|Pixel(so|
|00002e70| 75 72 63 65 42 4d 2c 20 | 6e 65 77 58 2c 20 6e 65 |urceBM, |newX, ne|
|00002e80| 77 59 29 3b 0d 00 00 06 | 00 00 0a 00 00 16 00 23 |wY);....|.......#|
|00002e90| 09 09 09 6e 65 77 59 20 | 3d 20 69 2a 72 31 32 20 |...newY |= i*r12 |
|00002ea0| 2b 20 6a 2a 72 32 32 20 | 2b 20 74 79 20 2b 20 2e |+ j*r22 |+ ty + .|
|00002eb0| 35 3b 0d 00 00 06 00 00 | 0a 00 00 16 00 23 09 09 |5;......|.....#..|
|00002ec0| 09 6e 65 77 58 20 3d 20 | 69 2a 72 31 31 20 2b 20 |.newX = |i*r11 + |
|00002ed0| 6a 2a 72 32 31 20 2b 20 | 74 78 20 2b 20 2e 35 3b |j*r21 + |tx + .5;|
|00002ee0| 0d 00 00 06 00 00 0a 00 | 00 16 00 22 f0 9f 09 d5 |........|..."....|
|00002ef0| 40 f2 80 70 f3 d0 f3 0f | 3b 07 0f 3c 0f 77 79 2b |@..p....|;..<.wy+|
|00002f00| f3 b0 7f 2b f2 b0 f2 90 | f7 bf 0d 00 00 06 00 00 |...+....|........|
|00002f10| 0a 00 00 16 6e 20 00 22 | f0 9d 54 0f 28 0f 6a 0f |....n ."|..T.(.j.|
|00002f20| 3d 0f 30 f3 b0 f6 a0 f3 | c0 b1 7f 67 b2 f3 b0 f6 |=.0.....|...g....|
|00002f30| af 2b f2 b0 f2 90 f7 bf | 0d 00 00 06 00 00 0a 00 |.+......|........|
|00002f40| 00 16 00 02 7b 0d 00 06 | 00 00 0a 00 00 16 00 00 |....{...|........|
|00002f50| 0c 00 00 14 00 01 0d 00 | 00 06 00 00 0c 00 00 14 |........|........|
|00002f60| 00 9f 54 68 65 20 66 69 | 72 73 74 20 74 68 69 6e |..The fi|rst thin|
|00002f70| 67 20 77 65 20 64 6f 20 | 69 73 20 70 75 74 20 6f |g we do |is put o|
|00002f80| 75 72 20 72 75 6c 65 73 | 20 66 6f 72 20 72 6f 74 |ur rules| for rot|
|00002f90| 61 74 69 6e 67 2c 20 73 | 63 61 6c 69 6e 67 2c 20 |ating, s|caling, |
|00002fa0| 72 65 66 6c 65 63 74 69 | 6e 67 2c 20 61 6e 64 20 |reflecti|ng, and |
|00002fb0| 74 72 61 6e 73 6c 61 74 | 69 6e 67 20 69 6e 74 6f |translat|ing into|
|00002fc0| 20 6d 61 74 72 69 78 20 | 66 6f 72 6d 2e 20 46 6f | matrix |form. Fo|
|00002fd0| 72 20 65 78 61 6d 70 6c | 65 2c 20 66 6f 72 20 72 |r exampl|e, for r|
|00002fe0| 6f 74 61 74 69 6e 67 2c | 20 77 65 20 63 61 6e 20 |otating,| we can |
|00002ff0| 65 78 70 72 65 73 73 20 | 74 68 69 73 20 61 73 3a |express |this as:|
|00003000| 0d 02 00 06 00 00 0c 00 | 00 14 00 01 0d a1 00 06 |........|........|
|00003010| 00 00 0c 00 00 14 00 27 | 09 50 27 20 3d 20 52 50 |.......'|.P' = RP|
|00003020| 2c 20 77 68 65 72 65 20 | 52 20 3d 09 5b 20 63 6f |, where |R =.[ co|
|00003030| 73 28 76 29 09 2d 73 69 | 6e 28 76 29 20 5d 0d 00 |s(v).-si|n(v) ]..|
|00003040| 00 12 00 00 0c 00 00 16 | 00 0a 0c 00 00 14 00 0f |........|........|
|00003050| 0c 00 00 16 00 18 09 09 | 09 09 09 5b 20 73 69 6e |........|...[ sin|
|00003060| 28 76 29 09 20 63 6f 73 | 28 76 29 20 5d 0d 00 06 |(v). cos|(v) ]...|
|00003070| 00 00 0c 00 00 16 00 01 | 0d 02 00 06 00 00 0a 00 |........|........|
|00003080| 00 16 00 1d 50 65 72 66 | 6f 72 6d 69 6e 67 20 74 |....Perf|orming t|
|00003090| 68 65 20 6d 61 74 68 2c | 20 77 65 20 67 65 74 3a |he math,| we get:|
|000030a0| 0d 00 00 06 00 00 0c 00 | 00 14 00 01 0d 3c 00 06 |........|.....<..|
|000030b0| 00 00 0c 00 00 14 00 23 | 09 50 27 20 3d 20 52 50 |.......#|.P' = RP|
|000030c0| 20 3d 09 5b 20 63 6f 73 | 28 76 29 09 2d 73 69 6e | =.[ cos|(v).-sin|
|000030d0| 28 76 29 20 5d 5b 20 78 | 20 5d 0d 00 00 06 00 00 |(v) ][ x| ]......|
|000030e0| 0c 00 00 16 00 1b 09 09 | 09 5b 20 73 69 6e 28 76 |........|.[ sin(v|
|000030f0| 29 09 20 63 6f 73 28 76 | 29 20 5d 5b 20 79 20 5d |). cos(v|) ][ y ]|
|00003100| 0d 00 00 06 00 00 0c 00 | 00 16 00 01 0d f0 00 06 |........|........|
|00003110| 00 00 0c 00 00 16 00 38 | 09 20 20 20 3d 20 5b 20 |.......8|. = [ |
|00003120| 28 78 20 63 6f 73 28 76 | 29 20 2d 20 79 20 73 69 |(x cos(v|) - y si|
|00003130| 6e 28 76 29 29 20 20 20 | 28 78 20 73 69 6e 28 76 |n(v)) |(x sin(v|
|00003140| 29 20 2b 20 79 20 63 6f | 73 28 76 29 29 20 5d 0d |) + y co|s(v)) ].|
|00003150| 00 06 00 00 0c 00 00 16 | 00 01 0d 00 00 06 00 00 |........|........|
|00003160| 0c 00 00 14 01 6f 57 68 | 69 63 68 20 61 67 72 65 |.....oWh|ich agre|
|00003170| 65 73 20 77 69 74 68 20 | 77 68 61 74 20 77 65 20 |es with |what we |
|00003180| 73 68 6f 77 65 64 20 61 | 74 20 74 68 65 20 62 65 |showed a|t the be|
|00003190| 67 69 6e 6e 69 6e 67 20 | 6f 66 20 74 68 69 73 20 |ginning |of this |
|000031a0| 64 6f 63 75 6d 65 6e 74 | 2e 20 55 6e 66 6f 72 74 |document|. Unfort|
|000031b0| 75 6e 61 74 65 6c 79 2c | 20 74 68 65 20 74 72 61 |unately,| the tra|
|000031c0| 6e 73 6c 61 74 69 6f 6e | 20 74 72 61 6e 73 66 6f |nslation| transfo|
|000031d0| 72 6d 61 74 69 6f 6e 20 | 63 61 6e 6e 6f 74 20 62 |rmation |cannot b|
|000031e0| 65 20 65 78 70 72 65 73 | 73 65 64 20 61 73 20 61 |e expres|sed as a|
|000031f0| 20 32 20 78 20 32 20 6d | 61 74 72 69 78 2e 20 48 | 2 x 2 m|atrix. H|
|00003200| 6f 77 65 76 65 72 2c 20 | 61 20 63 65 72 74 61 69 |owever, |a certai|
|00003210| 6e 20 74 72 69 63 6b 20 | 61 6c 6c 6f 77 73 20 75 |n trick |allows u|
|00003220| 73 20 74 6f 20 69 6e 74 | 72 6f 64 75 63 65 20 61 |s to int|roduce a|
|00003230| 20 33 20 78 20 33 20 6d | 61 74 72 69 78 20 66 75 | 3 x 3 m|atrix fu|
|00003240| 6e 63 74 69 6f 6e 20 77 | 68 69 63 68 20 70 65 72 |nction w|hich per|
|00003250| 66 6f 72 6d 73 20 74 68 | 65 20 74 72 61 6e 73 66 |forms th|e transf|
|00003260| 6f 72 6d 61 74 69 6f 6e | 2e 20 42 79 20 65 78 70 |ormation|. By exp|
|00003270| 72 65 73 73 69 6e 67 20 | 6f 75 72 20 70 6f 69 6e |ressing |our poin|
|00003280| 74 20 50 20 77 69 74 68 | 20 74 68 65 20 63 6f 6f |t P with| the coo|
|00003290| 72 64 69 6e 61 74 65 73 | 20 28 78 2c 20 79 2c 20 |rdinates| (x, y, |
|000032a0| 31 29 2c 20 77 65 20 63 | 61 6e 20 75 73 65 20 74 |1), we c|an use t|
|000032b0| 68 65 20 66 6f 6c 6c 6f | 77 69 6e 67 20 6d 61 74 |he follo|wing mat|
|000032c0| 72 69 78 20 66 6f 72 20 | 74 72 61 6e 73 6c 61 74 |rix for |translat|
|000032d0| 69 6f 6e 3a 0d 69 00 06 | 00 00 0c 00 00 14 00 01 |ion:.i..|........|
|000032e0| 0d 73 00 06 00 00 0c 00 | 00 14 00 0d 54 72 61 6e |.s......|....Tran|
|000032f0| 73 6c 61 74 69 6e 67 3a | 0d 2e 00 06 00 00 0c 00 |slating:|........|
|00003300| 00 14 00 13 09 09 09 09 | 09 5b 20 20 31 20 20 30 |........|.[ 1 0|
|00003310| 20 54 78 20 20 5d 0d 20 | 00 06 00 00 0c 00 00 16 | Tx ]. |........|
|00003320| 00 23 09 50 27 20 3d 20 | 54 50 2c 20 77 68 65 72 |.#.P' = |TP, wher|
|00003330| 65 09 54 20 3d 20 09 5b | 20 20 30 20 20 31 20 54 |e.T = .[| 0 1 T|
|00003340| 79 20 20 5d 0d 67 00 12 | 00 00 0c 00 00 16 00 0a |y ].g..|........|
|00003350| 0c 00 00 14 00 0f 0c 00 | 00 16 00 13 09 09 09 09 |........|........|
|00003360| 09 5b 20 20 30 20 20 30 | 20 20 31 20 20 5d 0d 70 |.[ 0 0| 1 ].p|
|00003370| 00 06 00 00 0c 00 00 16 | 00 01 0d 39 00 06 00 00 |........|...9....|
|00003380| 0c 00 00 14 00 2c 53 69 | 6d 69 6c 61 72 6c 79 2c |.....,Si|milarly,|
|00003390| 20 66 6f 72 20 74 68 65 | 20 6f 74 68 65 72 20 66 | for the| other f|
|000033a0| 75 6e 63 74 69 6f 6e 73 | 2c 20 77 65 20 67 65 74 |unctions|, we get|
|000033b0| 3a 0d 00 06 00 00 0c 00 | 00 14 00 01 0d 65 00 06 |:.......|.....e..|
|000033c0| 00 00 0c 00 00 14 00 0a | 52 6f 74 61 74 69 6e 67 |........|Rotating|
|000033d0| 3a 0d 00 06 00 00 0c 00 | 00 14 00 1c 09 09 09 09 |:.......|........|
|000033e0| 09 5b 20 63 6f 73 28 76 | 29 20 20 2d 73 69 6e 28 |.[ cos(v|) -sin(|
|000033f0| 76 29 20 20 30 20 5d 0d | 00 06 00 00 0c 00 00 16 |v) 0 ].|........|
|00003400| 00 2c 09 50 27 20 3d 20 | 54 50 2c 20 77 68 65 72 |.,.P' = |TP, wher|
|00003410| 65 09 54 20 3d 20 09 5b | 20 73 69 6e 28 76 29 20 |e.T = .[| sin(v) |
|00003420| 20 20 63 6f 73 28 76 29 | 20 20 30 20 5d 0d 00 12 | cos(v)| 0 ]...|
|00003430| 00 00 0c 00 00 16 00 0a | 0c 00 00 14 00 0f 0c 00 |........|........|
|00003440| 00 16 00 1c 09 09 09 09 | 09 5b 20 20 20 30 20 20 |........|.[ 0 |
|00003450| 20 20 20 20 20 20 30 20 | 20 20 20 20 31 20 5d 0d | 0 | 1 ].|
|00003460| 00 06 00 00 0c 00 00 16 | 00 01 0d 00 00 06 00 00 |........|........|
|00003470| 0c 00 00 14 00 09 53 63 | 61 6c 69 6e 67 3a 0d 00 |......Sc|aling:..|
|00003480| 00 06 00 00 0c 00 00 14 | 00 13 09 09 09 09 09 5b |........|.......[|
|00003490| 20 53 78 20 20 30 20 20 | 30 20 20 5d 0d 00 00 06 | Sx 0 |0 ]....|
|000034a0| 00 00 0c 00 00 16 00 23 | 09 50 27 20 3d 20 53 50 |.......#|.P' = SP|
|000034b0| 2c 20 77 68 65 72 65 09 | 53 20 3d 20 09 5b 20 20 |, where.|S = .[ |
|000034c0| 30 20 53 79 20 20 30 20 | 20 5d 0d 0d 00 12 00 00 |0 Sy 0 | ]......|
|000034d0| 0c 00 00 16 00 0a 0c 00 | 00 14 00 0f 0c 00 00 16 |........|........|
|000034e0| 00 13 09 09 09 09 09 5b | 20 20 30 20 20 30 20 20 |.......[| 0 0 |
|000034f0| 31 20 20 5d 0d 03 00 06 | 00 00 0c 00 00 16 00 01 |1 ]....|........|
|00003500| 0d 03 00 06 00 00 0c 00 | 00 14 01 0c 4e 6f 77 20 |........|....Now |
|00003510| 74 68 65 20 66 75 6e 20 | 72 65 61 6c 6c 79 20 62 |the fun |really b|
|00003520| 65 67 69 6e 73 21 20 57 | 69 74 68 20 74 68 65 73 |egins! W|ith thes|
|00003530| 65 20 6d 61 74 72 69 63 | 65 73 2c 20 77 65 20 63 |e matric|es, we c|
|00003540| 61 6e 20 6d 69 78 20 61 | 6e 64 20 6d 61 74 63 68 |an mix a|nd match|
|00003550| 20 69 6e 20 61 6e 79 20 | 6f 72 64 65 72 20 74 68 | in any |order th|
|00003560| 61 74 20 77 65 20 77 61 | 6e 74 20 69 6e 20 61 20 |at we wa|nt in a |
|00003570| 76 65 72 79 20 67 65 6e | 65 72 61 6c 20 66 61 73 |very gen|eral fas|
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|00003620| 00 01 0d 43 00 06 00 00 | 0c 00 00 14 00 18 09 50 |...C....|.......P|
|00003630| 27 20 3d 20 28 54 32 29 | 28 52 29 28 53 29 28 54 |' = (T2)|(R)(S)(T|
|00003640| 31 29 28 50 29 0d 00 06 | 00 00 0c 00 00 16 00 01 |1)(P)...|........|
|00003650| 0d 70 00 06 00 00 0c 00 | 00 14 00 35 45 78 70 61 |.p......|...5Expa|
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|00003670| 29 28 53 29 28 54 31 29 | 20 74 6f 20 69 74 73 20 |)(S)(T1)| to its |
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|00003690| 0d 69 00 12 00 00 0c 00 | 00 14 00 0e 0c 00 00 16 |.i......|........|
|000036a0| 00 1c 0c 00 00 14 00 01 | 0d 54 00 06 00 00 0c 00 |........|.T......|
|000036b0| 00 14 00 3c 09 5b 20 31 | 20 20 30 20 44 78 20 5d |...<.[ 1| 0 Dx ]|
|000036c0| 5b 20 63 6f 73 28 76 29 | 20 20 2d 73 69 6e 28 76 |[ cos(v)| -sin(v|
|000036d0| 29 20 20 30 20 5d 5b 20 | 53 78 20 20 30 20 20 30 |) 0 ][ |Sx 0 0|
|000036e0| 20 20 5d 5b 20 31 20 20 | 30 20 2d 43 78 20 5d 0d | ][ 1 |0 -Cx ].|
|000036f0| 00 0c 00 00 0c 00 00 14 | 00 01 0c 00 00 16 00 3c |........|.......<|
|00003700| 09 5b 20 30 20 20 31 20 | 44 79 20 5d 5b 20 73 69 |.[ 0 1 |Dy ][ si|
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|00003720| 20 5d 5b 20 20 30 20 53 | 79 20 20 30 20 20 5d 5b | ][ 0 S|y 0 ][|
|00003730| 20 30 20 20 31 20 2d 43 | 79 20 5d 0d 00 06 00 00 | 0 1 -C|y ].....|
|00003740| 0c 00 00 16 00 3c 09 5b | 20 30 20 20 30 20 20 31 |.....<.[| 0 0 1|
|00003750| 20 5d 5b 20 20 20 30 20 | 20 20 20 20 20 20 20 30 | ][ 0 | 0|
|00003760| 20 20 20 20 20 31 20 5d | 5b 20 20 30 20 20 30 20 | 1 ]|[ 0 0 |
|00003770| 20 31 20 20 5d 5b 20 30 | 20 20 30 20 20 20 31 20 | 1 ][ 0| 0 1 |
|00003780| 5d 0d 00 06 00 00 0c 00 | 00 16 00 01 0d 20 00 06 |].......|..... ..|
|00003790| 00 00 0c 00 00 14 01 63 | 49 6e 20 74 68 69 73 20 |.......c|In this |
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|00003850| 6e 73 3b 20 76 20 69 73 | 20 74 68 65 20 61 6d 6f |ns; v is| the amo|
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|00003870| 20 28 44 78 2c 20 44 79 | 29 20 69 73 20 74 68 65 | (Dx, Dy|) is the|
|00003880| 20 64 65 73 74 69 6e 61 | 74 69 6f 6e 20 6f 66 20 | destina|tion of |
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|000038b0| 65 20 63 61 6e 20 62 65 | 20 74 68 65 20 73 61 6d |e can be| the sam|
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|000038d0| 75 6c 74 69 70 6c 79 69 | 6e 67 20 74 68 65 20 66 |ultiplyi|ng the f|
|000038e0| 69 72 73 74 20 74 77 6f | 20 6d 61 74 72 69 63 65 |irst two| matrice|
|000038f0| 73 2c 20 77 65 20 67 65 | 74 3a 0d 61 00 06 00 00 |s, we ge|t:.a....|
|00003900| 0c 00 00 14 00 01 0d 64 | 00 06 00 00 0c 00 00 14 |.......d|........|
|00003910| 00 31 09 5b 20 63 6f 73 | 28 76 29 20 2d 73 69 6e |.1.[ cos|(v) -sin|
|00003920| 28 76 29 20 20 44 78 20 | 5d 5b 20 53 78 20 20 30 |(v) Dx |][ Sx 0|
|00003930| 20 20 30 20 20 5d 5b 20 | 31 20 20 30 20 2d 43 78 | 0 ][ |1 0 -Cx|
|00003940| 20 5d 0d 0d 00 0c 00 00 | 0c 00 00 14 00 01 0c 00 | ]......|........|
|00003950| 00 16 00 31 09 5b 20 73 | 69 6e 28 76 29 20 20 63 |...1.[ s|in(v) c|
|00003960| 6f 73 28 76 29 20 20 44 | 79 20 5d 5b 20 20 30 20 |os(v) D|y ][ 0 |
|00003970| 53 79 20 20 30 20 20 5d | 5b 20 30 20 20 31 20 2d |Sy 0 ]|[ 0 1 -|
|00003980| 43 79 20 5d 0d 00 00 06 | 00 00 0c 00 00 16 00 31 |Cy ]....|.......1|
|00003990| 09 5b 20 20 20 30 20 20 | 20 20 20 20 20 30 20 20 |.[ 0 | 0 |
|000039a0| 20 20 20 20 31 20 5d 5b | 20 20 30 20 20 30 20 20 | 1 ][| 0 0 |
|000039b0| 31 20 20 5d 5b 20 30 20 | 20 30 20 20 20 31 20 5d |1 ][ 0 | 0 1 ]|
|000039c0| 0d ff 00 06 00 00 0c 00 | 00 16 00 01 0d 58 00 06 |........|.....X..|
|000039d0| 00 00 0c 00 00 14 00 27 | 44 6f 69 6e 67 20 74 68 |.......'|Doing th|
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|00003a00| 00 06 00 00 0c 00 00 14 | 00 01 0d 00 00 06 00 00 |........|........|
|00003a10| 0c 00 00 14 00 2d 09 5b | 20 28 53 78 29 63 6f 73 |.....-.[| (Sx)cos|
|00003a20| 28 76 29 20 20 2d 28 53 | 79 29 73 69 6e 28 76 29 |(v) -(S|y)sin(v)|
|00003a30| 20 20 44 78 20 5d 5b 20 | 31 20 20 30 20 2d 43 78 | Dx ][ |1 0 -Cx|
|00003a40| 20 5d 0d 01 00 06 00 00 | 0c 00 00 16 00 2d 09 5b | ]......|.....-.[|
|00003a50| 20 28 53 78 29 73 69 6e | 28 76 29 20 20 20 28 53 | (Sx)sin|(v) (S|
|00003a60| 79 29 63 6f 73 28 76 29 | 20 20 44 79 20 5d 5b 20 |y)cos(v)| Dy ][ |
|00003a70| 30 20 20 31 20 2d 43 79 | 20 5d 0d ff 00 06 00 00 |0 1 -Cy| ]......|
|00003a80| 0c 00 00 16 00 2d 09 5b | 20 20 20 20 20 20 30 20 |.....-.[| 0 |
|00003a90| 20 20 20 20 20 20 20 20 | 20 20 20 30 20 20 20 20 | | 0 |
|00003aa0| 20 20 20 31 20 5d 5b 20 | 30 20 20 30 20 20 20 31 | 1 ][ |0 0 1|
|00003ab0| 20 5d 0d 55 00 06 00 00 | 0c 00 00 16 00 01 0d 6a | ].U....|.......j|
|00003ac0| 00 06 00 00 0c 00 00 14 | 00 5d 54 68 69 73 20 69 |........|.]This i|
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|00003b00| 61 20 6c 69 74 74 6c 65 | 20 6e 6f 74 61 74 69 6f |a little| notatio|
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|00003b20| 2e 20 4c 65 74 3a 0d 30 | 00 06 00 00 0c 00 00 14 |. Let:.0|........|
|00003b30| 00 01 0d 70 00 06 00 00 | 0c 00 00 14 00 13 09 72 |...p....|.......r|
|00003b40| 31 31 20 3d 20 20 28 53 | 78 29 63 6f 73 28 76 29 |11 = (S|x)cos(v)|
|00003b50| 0d f0 00 06 00 00 0c 00 | 00 16 00 13 09 72 31 32 |........|.....r12|
|00003b60| 20 3d 20 2d 28 53 79 29 | 73 69 6e 28 76 29 0d 00 | = -(Sy)|sin(v)..|
|00003b70| 00 06 00 00 0c 00 00 16 | 00 13 09 72 32 31 20 3d |........|...r21 =|
|00003b80| 20 20 28 53 78 29 73 69 | 6e 28 76 29 0d 01 00 06 | (Sx)si|n(v)....|
|00003b90| 00 00 0c 00 00 16 00 13 | 09 72 32 32 20 3d 20 20 |........|.r22 = |
|00003ba0| 28 53 79 29 63 6f 73 28 | 76 29 0d 54 00 06 00 00 |(Sy)cos(|v).T....|
|00003bb0| 0c 00 00 16 00 01 0d 08 | 00 06 00 00 0c 00 00 14 |........|........|
|00003bc0| 00 0c 54 68 69 73 20 67 | 69 76 65 73 3a 0d 00 06 |..This g|ives:...|
|00003bd0| 00 00 0c 00 00 14 00 01 | 0d 56 00 06 00 00 0c 00 |........|.V......|
|00003be0| 00 14 00 1e 09 5b 20 72 | 31 31 20 20 72 31 32 20 |.....[ r|11 r12 |
|00003bf0| 20 44 78 20 5d 5b 20 31 | 20 20 30 20 2d 43 78 20 | Dx ][ 1| 0 -Cx |
|00003c00| 5d 0d 00 06 00 00 0c 00 | 00 16 00 1e 09 5b 20 72 |].......|.....[ r|
|00003c10| 32 31 20 20 72 32 32 20 | 20 44 79 20 5d 5b 20 30 |21 r22 | Dy ][ 0|
|00003c20| 20 20 31 20 2d 43 79 20 | 5d 0d 00 06 00 00 0c 00 | 1 -Cy |].......|
|00003c30| 00 16 00 1e 09 5b 20 20 | 30 20 20 20 20 30 20 20 |.....[ |0 0 |
|00003c40| 20 20 31 20 5d 5b 20 30 | 20 20 30 20 20 20 31 20 | 1 ][ 0| 0 1 |
|00003c50| 5d 0d 00 06 00 00 0c 00 | 00 16 00 01 0d 06 00 06 |].......|........|
|00003c60| 00 00 0c 00 00 14 00 26 | f0 98 b5 42 f0 9f 09 f7 |.......&|...B....|
|00003c70| 77 92 b0 f3 d0 91 82 f4 | 2f 4d f2 df 3e f6 25 f7 |w.......|/M..>.%.|
|00003c80| 53 98 f2 e4 7f 67 b2 f3 | bf 0d 00 06 00 00 0a 00 |S....g..|........|
|00003c90| 00 16 65 74 3a 0d 00 06 | 00 00 0c 00 00 14 00 01 |..et:...|........|
|00003ca0| 0d 58 00 06 00 00 0c 00 | 00 14 00 2c 09 5b 20 72 |.X......|...,.[ r|
|00003cb0| 31 31 20 20 20 20 72 31 | 32 20 20 20 20 28 2d 43 |11 r1|2 (-C|
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|00003cd0| 32 29 2b 44 78 20 5d 0d | 00 06 00 00 0c 00 00 16 |2)+Dx ].|........|
|00003ce0| 00 2c 09 5b 20 72 32 31 | 20 20 20 20 72 32 32 20 |.,.[ r21| r22 |
|00003cf0| 20 20 20 28 2d 43 78 29 | 28 72 32 31 29 2b 28 2d | (-Cx)|(r21)+(-|
|00003d00| 43 79 29 28 72 32 32 29 | 2b 44 79 20 5d 0d 00 06 |Cy)(r22)|+Dy ]...|
|00003d10| 00 00 0c 00 00 16 00 2c | 09 5b 20 20 30 20 20 20 |.......,|.[ 0 |
|00003d20| 20 20 20 30 20 20 20 20 | 20 20 20 20 20 20 20 20 | 0 | |
|00003d30| 20 20 20 20 31 20 20 20 | 20 20 20 20 20 20 20 20 | 1 | |
|00003d40| 20 20 5d 0d 00 06 00 00 | 0c 00 00 16 00 01 0d 00 | ].....|........|
|00003d50| 00 06 00 00 0c 00 00 14 | 00 54 54 68 69 73 2c 20 |........|.TThis, |
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|00003d70| 61 6c 20 6d 61 74 72 69 | 78 20 62 79 20 77 68 69 |al matri|x by whi|
|00003d80| 63 68 20 74 6f 20 6d 75 | 6c 74 69 70 6c 79 20 74 |ch to mu|ltiply t|
|00003d90| 68 65 20 63 6f 6f 72 64 | 69 6e 61 74 65 73 20 6f |he coord|inates o|
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|00003db0| 00 00 0c 00 00 14 00 01 | 0d 80 00 06 00 00 0c 00 |........|........|
|00003dc0| 00 14 00 31 09 5b 20 72 | 31 31 20 20 20 20 72 31 |...1.[ r|11 r1|
|00003dd0| 32 20 20 20 20 28 2d 43 | 78 29 28 72 31 31 29 2b |2 (-C|x)(r11)+|
|00003de0| 28 2d 43 79 29 28 72 31 | 32 29 2b 44 78 20 5d 5b |(-Cy)(r1|2)+Dx ][|
|00003df0| 20 78 20 5d 0d 01 00 06 | 00 00 0c 00 00 16 00 31 | x ]....|.......1|
|00003e00| 09 5b 20 72 32 31 20 20 | 20 20 72 32 32 20 20 20 |.[ r21 | r22 |
|00003e10| 20 28 2d 43 78 29 28 72 | 32 31 29 2b 28 2d 43 79 | (-Cx)(r|21)+(-Cy|
|00003e20| 29 28 72 32 32 29 2b 44 | 79 20 5d 5b 20 79 20 5d |)(r22)+D|y ][ y ]|
|00003e30| 0d 08 00 06 00 00 0c 00 | 00 16 00 31 09 5b 20 20 |........|...1.[ |
|00003e40| 30 20 20 20 20 20 20 30 | 20 20 20 20 20 20 20 20 |0 0| |
|00003e50| 20 20 20 20 20 20 20 20 | 31 20 20 20 20 20 20 20 | |1 |
|00003e60| 20 20 20 20 20 20 5d 5b | 20 31 20 5d 0d 55 00 06 | ][| 1 ].U..|
|00003e70| 00 00 0c 00 00 16 00 01 | 0d ab 00 06 00 00 0c 00 |........|........|
|00003e80| 00 16 00 34 09 5b 20 28 | 72 31 31 29 28 78 29 20 |...4.[ (|r11)(x) |
|00003e90| 2b 20 28 72 31 32 29 28 | 79 29 20 2b 20 28 2d 43 |+ (r12)(|y) + (-C|
|00003ea0| 78 29 28 72 31 31 29 2b | 28 2d 43 79 29 28 72 31 |x)(r11)+|(-Cy)(r1|
|00003eb0| 32 29 2b 44 78 20 5d 0d | 00 06 00 00 0c 00 00 16 |2)+Dx ].|........|
|00003ec0| 00 37 20 20 3d 09 5b 20 | 28 72 32 31 29 28 78 29 |.7 =.[ |(r21)(x)|
|00003ed0| 20 2b 20 28 72 32 32 29 | 28 79 29 20 2b 20 28 2d | + (r22)|(y) + (-|
|00003ee0| 43 78 29 28 72 32 31 29 | 2b 28 2d 43 79 29 28 72 |Cx)(r21)|+(-Cy)(r|
|00003ef0| 32 32 29 2b 44 79 20 5d | 0d fc 00 06 00 00 0c 00 |22)+Dy ]|........|
|00003f00| 00 16 00 34 09 5b 20 20 | 20 20 20 20 20 20 20 20 |...4.[ | |
|00003f10| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 31 20 20 | | 1 |
|00003f20| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 | | |
|00003f30| 20 20 20 20 20 20 5d 0d | 00 06 00 00 0c 00 00 16 | ].|........|
|00003f40| 00 01 0d 80 00 06 00 00 | 0c 00 00 14 00 2c 57 68 |........|.....,Wh|
|00003f50| 69 63 68 20 6a 75 73 74 | 20 6d 65 61 6e 73 20 74 |ich just| means t|
|00003f60| 68 61 74 20 6f 75 72 20 | 6e 65 77 20 50 27 28 78 |hat our |new P'(x|
|00003f70| 27 2c 79 27 29 20 69 73 | 3a 0d 00 12 00 00 0c 00 |',y') is|:.......|
|00003f80| 00 14 00 1e 0c 00 00 16 | 00 27 0c 00 00 14 00 01 |........|.'......|
|00003f90| 0d aa 00 06 00 00 0c 00 | 00 14 00 35 09 78 27 20 |........|...5.x' |
|00003fa0| 3d 20 28 72 31 31 29 28 | 78 29 20 2b 20 28 72 31 |= (r11)(|x) + (r1|
|00003fb0| 32 29 28 79 29 20 2b 20 | 28 2d 43 78 29 28 72 31 |2)(y) + |(-Cx)(r1|
|00003fc0| 31 29 2b 28 2d 43 79 29 | 28 72 31 32 29 2b 44 78 |1)+(-Cy)|(r12)+Dx|
|00003fd0| 0d 28 00 06 00 00 0c 00 | 00 16 00 35 09 79 27 20 |.(......|...5.y' |
|00003fe0| 3d 20 28 72 32 31 29 28 | 78 29 20 2b 20 28 72 32 |= (r21)(|x) + (r2|
|00003ff0| 32 29 28 79 29 20 2b 20 | 28 2d 43 78 29 28 72 32 |2)(y) + |(-Cx)(r2|
|00004000| 31 29 2b 28 2d 43 79 29 | 28 72 32 32 29 2b 44 79 |1)+(-Cy)|(r22)+Dy|
|00004010| 0d 6d 00 06 00 00 0c 00 | 00 16 00 01 0d 6d 00 06 |.m......|.....m..|
|00004020| 00 00 0c 00 00 14 00 e1 | 54 68 65 20 61 73 74 75 |........|The astu|
|00004030| 74 65 20 6f 62 73 65 72 | 76 65 72 20 77 69 6c 6c |te obser|ver will|
|00004040| 20 6e 6f 74 65 2c 20 68 | 6f 77 65 76 65 72 2c 20 | note, h|owever, |
|00004050| 74 68 61 74 20 65 76 65 | 72 79 74 68 69 6e 67 20 |that eve|rything |
|00004060| 77 65 27 76 65 20 6a 75 | 73 74 20 64 6f 6e 65 20 |we've ju|st done |
|00004070| 69 73 20 75 73 65 6c 65 | 73 73 2e 20 42 61 63 6b |is usele|ss. Back|
|00004080| 20 61 74 20 74 68 65 20 | 62 65 67 69 6e 6e 69 6e | at the |beginnin|
|00004090| 67 2c 20 77 65 20 73 61 | 69 64 20 74 68 61 74 20 |g, we sa|id that |
|000040a0| 62 65 73 74 20 72 65 73 | 75 6c 74 73 20 77 65 72 |best res|ults wer|
|000040b0| 65 20 6f 62 74 61 69 6e | 65 64 20 62 79 20 72 75 |e obtain|ed by ru|
|000040c0| 6e 6e 69 6e 67 20 74 68 | 65 20 74 72 61 6e 73 66 |nning th|e transf|
|000040d0| 6f 72 6d 20 62 61 63 6b | 77 61 72 64 73 21 20 49 |orm back|wards! I|
|000040e0| 6e 20 6f 74 68 65 72 20 | 77 6f 72 64 73 2c 20 77 |n other |words, w|
|000040f0| 65 20 6e 65 65 64 65 64 | 20 73 6f 6d 65 74 68 69 |e needed| somethi|
|00004100| 6e 67 20 6c 69 6b 65 3a | 0d 00 00 06 00 00 0c 00 |ng like:|........|
|00004110| 00 14 00 01 0d 72 00 06 | 00 00 0c 00 00 14 00 1d |.....r..|........|
|00004120| 09 50 20 20 3d 20 28 54 | 31 27 29 28 53 27 29 28 |.P = (T|1')(S')(|
|00004130| 52 27 29 28 54 32 27 29 | 28 50 27 29 0d c0 00 06 |R')(T2')|(P')....|
|00004140| 00 00 0c 00 00 16 00 01 | 0d 3f 00 06 00 00 0c 00 |........|.?......|
|00004150| 00 14 00 39 45 78 70 61 | 6e 64 69 6e 67 20 6f 75 |...9Expa|nding ou|
|00004160| 74 20 28 54 31 27 29 28 | 53 27 29 28 52 27 29 28 |t (T1')(|S')(R')(|
|00004170| 54 32 27 29 20 74 6f 20 | 69 74 73 20 70 72 6f 70 |T2') to |its prop|
|00004180| 65 72 20 6d 61 74 72 69 | 63 65 73 3a 0d 80 00 12 |er matri|ces:....|
|00004190| 00 00 0c 00 00 14 00 0e | 0c 00 00 16 00 20 0c 00 |........|..... ..|
|000041a0| 00 14 00 01 0d 00 00 06 | 00 00 0c 00 00 14 00 3f |........|.......?|
|000041b0| 09 5b 20 31 20 20 30 20 | 20 43 78 20 5d 5b 20 31 |.[ 1 0 | Cx ][ 1|
|000041c0| 2f 53 78 20 20 30 20 20 | 30 20 5d 5b 20 63 6f 73 |/Sx 0 |0 ][ cos|
|000041d0| 28 2d 76 29 20 2d 73 69 | 6e 28 2d 76 29 20 20 30 |(-v) -si|n(-v) 0|
|000041e0| 20 5d 5b 20 31 20 20 30 | 20 2d 44 78 20 5d 0d 01 | ][ 1 0| -Dx ]..|
|000041f0| 00 0c 00 00 0c 00 00 14 | 00 01 0c 00 00 16 00 3f |........|.......?|
|00004200| 09 5b 20 30 20 20 31 20 | 20 43 79 20 5d 5b 20 20 |.[ 0 1 | Cy ][ |
|00004210| 30 20 20 31 2f 53 79 20 | 30 20 5d 5b 20 73 69 6e |0 1/Sy |0 ][ sin|
|00004220| 28 2d 76 29 20 20 63 6f | 73 28 2d 76 29 20 20 30 |(-v) co|s(-v) 0|
|00004230| 20 5d 5b 20 30 20 20 31 | 20 2d 44 79 20 5d 0d 00 | ][ 0 1| -Dy ]..|
|00004240| 00 06 00 00 0c 00 00 16 | 00 3f 09 5b 20 30 20 20 |........|.?.[ 0 |
|00004250| 30 20 20 20 31 20 5d 5b | 20 20 30 20 20 20 20 30 |0 1 ][| 0 0|
|00004260| 20 20 31 20 5d 5b 20 20 | 20 20 30 20 20 20 20 20 | 1 ][ | 0 |
|00004270| 20 20 20 30 20 20 20 20 | 20 31 20 5d 5b 20 30 20 | 0 | 1 ][ 0 |
|00004280| 20 30 20 20 20 31 20 5d | 0d bf 00 06 00 00 0c 00 | 0 1 ]|........|
|00004290| 00 16 00 01 0d 9b 00 06 | 00 00 0c 00 00 14 00 35 |........|.......5|
|000042a0| 09 5b 20 31 2f 53 78 20 | 20 30 20 20 43 78 20 20 |.[ 1/Sx | 0 Cx |
|000042b0| 5d 5b 20 63 6f 73 28 2d | 76 29 20 2d 73 69 6e 28 |][ cos(-|v) -sin(|
|000042c0| 2d 76 29 20 20 30 20 5d | 5b 20 31 20 20 30 20 2d |-v) 0 ]|[ 1 0 -|
|000042d0| 44 78 20 5d 0d ff 00 0c | 00 00 0c 00 00 14 00 01 |Dx ]....|........|
|000042e0| 0c 00 00 16 00 38 20 20 | 3d 09 5b 20 20 30 20 20 |.....8 |=.[ 0 |
|000042f0| 31 2f 53 79 20 43 79 20 | 20 5d 5b 20 73 69 6e 28 |1/Sy Cy | ][ sin(|
|00004300| 2d 76 29 20 20 63 6f 73 | 28 2d 76 29 20 20 30 20 |-v) cos|(-v) 0 |
|00004310| 5d 5b 20 30 20 20 31 20 | 2d 44 79 20 5d 0d 00 06 |][ 0 1 |-Dy ]...|
|00004320| 00 00 0c 00 00 16 00 35 | 09 5b 20 20 30 20 20 20 |.......5|.[ 0 |
|00004330| 20 30 20 20 20 31 20 20 | 5d 5b 20 20 20 20 30 20 | 0 1 |][ 0 |
|00004340| 20 20 20 20 20 20 20 30 | 20 20 20 20 20 31 20 5d | 0| 1 ]|
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|000043b0| 00 14 00 01 0c 00 00 16 | 00 38 20 20 3d 09 5b 20 |........|.8 =.[ |
|000043c0| 28 31 2f 53 79 29 73 69 | 6e 28 2d 76 29 20 20 20 |(1/Sy)si|n(-v) |
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|00004450| 72 31 32 20 20 43 78 20 | 20 5d 5b 20 31 20 20 30 |r12 Cx | ][ 1 0|
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|00004670| 00 00 00 00 00 00 00 1c | d8 d4 23 00 1c 36 00 22 |........|..#..6."|
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|000046e0| 00 00 0c 00 00 14 00 01 | 0d 02 00 06 00 00 0c 00 |........|........|
|000046f0| 00 14 00 6d 54 68 69 73 | 20 74 65 6c 6c 73 20 75 |...mThis| tells u|
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|00005050| 69 6e 67 20 61 6e 64 20 | 73 65 74 74 69 6e 67 20 |ing and |setting |
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|000050b0| 61 6d 65 20 74 72 61 6e | 73 66 6f 72 6d 61 74 69 |ame tran|sformati|
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|00005140| 75 63 68 20 6f 70 74 69 | 6d 69 7a 61 74 69 6f 6e |uch opti|mization|
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