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+--------+-------------------------+-------------------------+--------+--------+
|00000000| 46 72 6f 6d 20 61 6c 62 | 61 6e 79 63 73 21 6c 65 |From alb|anycs!le|
|00000010| 61 68 3a 72 73 62 35 38 | 34 20 46 72 69 20 46 65 |ah:rsb58|4 Fri Fe|
|00000020| 62 20 31 39 20 30 34 3a | 30 36 3a 33 34 20 31 39 |b 19 04:|06:34 19|
|00000030| 38 38 0a 52 65 63 65 69 | 76 65 64 3a 20 62 79 20 |88.Recei|ved: by |
|00000040| 61 6c 62 61 6e 79 63 73 | 2e 61 6c 62 61 6e 79 2e |albanycs|.albany.|
|00000050| 65 64 75 20 28 35 2e 35 | 34 2f 34 2e 38 29 0a 09 |edu (5.5|4/4.8)..|
|00000060| 69 64 20 41 41 30 30 32 | 33 33 3b 20 54 68 75 2c |id AA002|33; Thu,|
|00000070| 20 31 38 20 46 65 62 20 | 38 38 20 31 32 3a 31 31 | 18 Feb |88 12:11|
|00000080| 3a 31 30 20 45 53 54 0a | 44 61 74 65 3a 20 54 68 |:10 EST.|Date: Th|
|00000090| 75 2c 20 31 38 20 46 65 | 62 20 38 38 20 31 32 3a |u, 18 Fe|b 88 12:|
|000000a0| 31 31 3a 30 35 20 45 53 | 54 0a 46 72 6f 6d 3a 20 |11:05 ES|T.From: |
|000000b0| 61 6c 62 61 6e 79 63 73 | 21 6c 65 61 68 3a 72 73 |albanycs|!leah:rs|
|000000c0| 62 35 38 34 20 28 52 61 | 79 6d 6f 6e 64 20 53 20 |b584 (Ra|ymond S |
|000000d0| 42 72 61 6e 64 29 0a 52 | 65 63 65 69 76 65 64 3a |Brand).R|eceived:|
|000000e0| 20 62 79 20 6c 65 61 68 | 2e 41 6c 62 61 6e 79 2e | by leah|.Albany.|
|000000f0| 45 44 55 20 28 35 2e 35 | 38 2f 31 2e 31 29 0a 09 |EDU (5.5|8/1.1)..|
|00000100| 69 64 20 41 41 30 35 35 | 35 30 3b 20 54 68 75 2c |id AA055|50; Thu,|
|00000110| 20 31 38 20 46 65 62 20 | 38 38 20 31 32 3a 31 31 | 18 Feb |88 12:11|
|00000120| 3a 30 35 20 45 53 54 0a | 4d 65 73 73 61 67 65 2d |:05 EST.|Message-|
|00000130| 49 64 3a 20 3c 38 38 30 | 32 31 38 31 37 31 31 2e |Id: <880|2181711.|
|00000140| 41 41 30 35 35 35 30 40 | 6c 65 61 68 2e 41 6c 62 |AA05550@|leah.Alb|
|00000150| 61 6e 79 2e 45 44 55 3e | 0a 54 6f 3a 20 61 6c 62 |any.EDU>|.To: alb|
|00000160| 61 6e 79 63 73 3a 62 65 | 6f 77 75 6c 66 21 72 73 |anycs:be|owulf!rs|
|00000170| 62 78 0a 53 75 62 6a 65 | 63 74 3a 20 63 6f 72 64 |bx.Subje|ct: cord|
|00000180| 69 63 2e 74 78 74 0a 0a | 3e 46 72 6f 6d 20 74 75 |ic.txt..|>From tu|
|00000190| 72 6b 40 61 70 70 6c 65 | 2e 55 55 43 50 20 57 65 |rk@apple|.UUCP We|
|000001a0| 64 20 46 65 62 20 31 37 | 20 32 31 3a 31 39 3a 30 |d Feb 17| 21:19:0|
|000001b0| 36 20 31 39 38 38 0a 50 | 61 74 68 3a 20 6c 65 61 |6 1988.P|ath: lea|
|000001c0| 68 21 69 74 73 67 77 21 | 6e 79 73 65 72 6e 69 63 |h!itsgw!|nysernic|
|000001d0| 21 72 75 74 67 65 72 73 | 21 75 63 6c 61 2d 63 73 |!rutgers|!ucla-cs|
|000001e0| 21 63 69 74 2d 76 61 78 | 21 65 6c 72 6f 79 21 61 |!cit-vax|!elroy!a|
|000001f0| 6d 65 73 21 70 61 73 74 | 65 75 72 21 75 63 62 76 |mes!past|eur!ucbv|
|00000200| 61 78 21 68 70 6c 61 62 | 73 21 6e 73 63 21 76 6f |ax!hplab|s!nsc!vo|
|00000210| 64 65 72 21 61 70 70 6c | 65 21 74 75 72 6b 0a 46 |der!appl|e!turk.F|
|00000220| 72 6f 6d 3a 20 74 75 72 | 6b 40 61 70 70 6c 65 2e |rom: tur|k@apple.|
|00000230| 55 55 43 50 20 28 4b 65 | 6e 20 22 54 75 72 6b 22 |UUCP (Ke|n "Turk"|
|00000240| 20 54 75 72 6b 6f 77 73 | 6b 69 29 0a 4e 65 77 73 | Turkows|ki).News|
|00000250| 67 72 6f 75 70 73 3a 20 | 63 6f 6d 70 2e 67 72 61 |groups: |comp.gra|
|00000260| 70 68 69 63 73 0a 53 75 | 62 6a 65 63 74 3a 20 52 |phics.Su|bject: R|
|00000270| 65 3a 20 43 4f 52 44 49 | 43 53 3a 20 49 73 20 4b |e: CORDI|CS: Is K|
|00000280| 65 6e 20 54 75 72 6b 6f | 77 73 6b 69 20 74 68 65 |en Turko|wski the|
|00000290| 72 65 3f 20 28 77 61 73 | 3a 20 46 72 61 63 74 61 |re? (was|: Fracta|
|000002a0| 6c 73 2e 2e 2e 4a 69 6d | 20 43 61 74 68 65 79 20 |ls...Jim| Cathey |
|000002b0| 77 68 65 72 65 20 61 72 | 65 20 79 6f 75 3f 29 0a |where ar|e you?).|
|000002c0| 53 75 6d 6d 61 72 79 3a | 20 43 4f 52 44 49 43 20 |Summary:| CORDIC |
|000002d0| 64 65 72 69 76 61 74 69 | 6f 6e 20 61 6e 64 20 63 |derivati|on and c|
|000002e0| 6f 64 65 20 65 6e 63 6c | 6f 73 65 64 0a 4b 65 79 |ode encl|osed.Key|
|000002f0| 77 6f 72 64 73 3a 20 43 | 4f 52 44 49 43 2c 20 66 |words: C|ORDIC, f|
|00000300| 69 78 65 64 2d 70 6f 69 | 6e 74 20 74 72 69 67 6f |ixed-poi|nt trigo|
|00000310| 6e 6f 6d 65 74 72 69 63 | 20 66 75 6e 63 74 69 6f |nometric| functio|
|00000320| 6e 73 0a 4d 65 73 73 61 | 67 65 2d 49 44 3a 20 3c |ns.Messa|ge-ID: <|
|00000330| 37 34 33 33 40 61 70 70 | 6c 65 2e 55 55 43 50 3e |7433@app|le.UUCP>|
|00000340| 0a 44 61 74 65 3a 20 31 | 38 20 46 65 62 20 38 38 |.Date: 1|8 Feb 88|
|00000350| 20 30 32 3a 31 39 3a 30 | 36 20 47 4d 54 0a 52 65 | 02:19:0|6 GMT.Re|
|00000360| 66 65 72 65 6e 63 65 73 | 3a 20 3c 36 30 31 39 40 |ferences|: <6019@|
|00000370| 69 75 76 61 78 2e 55 55 | 43 50 3e 20 3c 32 34 37 |iuvax.UU|CP> <247|
|00000380| 39 40 6f 72 63 61 2e 54 | 45 4b 2e 43 4f 4d 3e 0a |9@orca.T|EK.COM>.|
|00000390| 52 65 70 6c 79 2d 54 6f | 3a 20 74 75 72 6b 40 61 |Reply-To|: turk@a|
|000003a0| 70 70 6c 65 2e 55 55 43 | 50 20 28 4b 65 6e 20 22 |pple.UUC|P (Ken "|
|000003b0| 54 75 72 6b 22 20 54 75 | 72 6b 6f 77 73 6b 69 29 |Turk" Tu|rkowski)|
|000003c0| 0a 4f 72 67 61 6e 69 7a | 61 74 69 6f 6e 3a 20 41 |.Organiz|ation: A|
|000003d0| 64 76 61 6e 63 65 64 20 | 54 65 63 68 6e 6f 6c 6f |dvanced |Technolo|
|000003e0| 67 79 20 47 72 61 70 68 | 69 63 73 2c 20 41 70 70 |gy Graph|ics, App|
|000003f0| 6c 65 20 43 6f 6d 70 75 | 74 65 72 2c 20 43 75 70 |le Compu|ter, Cup|
|00000400| 65 72 74 69 6e 6f 2c 20 | 43 41 2c 20 55 53 41 0a |ertino, |CA, USA.|
|00000410| 4c 69 6e 65 73 3a 20 33 | 37 33 0a 0a 49 6e 20 61 |Lines: 3|73..In a|
|00000420| 72 74 69 63 6c 65 20 3c | 32 34 37 39 40 6f 72 63 |rticle <|2479@orc|
|00000430| 61 2e 54 45 4b 2e 43 4f | 4d 3e 20 62 72 75 63 65 |a.TEK.CO|M> bruce|
|00000440| 63 40 6f 72 63 61 2e 55 | 55 43 50 20 28 42 72 75 |c@orca.U|UCP (Bru|
|00000450| 63 65 20 43 6f 68 65 6e | 29 20 77 72 69 74 65 73 |ce Cohen|) writes|
|00000460| 3a 0a 3e 49 6e 20 61 72 | 74 69 63 6c 65 20 3c 36 |:.>In ar|ticle <6|
|00000470| 30 31 39 40 69 75 76 61 | 78 2e 55 55 43 50 3e 20 |019@iuva|x.UUCP> |
|00000480| 76 69 6b 69 6e 67 40 69 | 75 76 61 78 20 77 72 69 |viking@i|uvax wri|
|00000490| 74 65 73 3a 0a 3e 3e 0a | 3e 3e 54 68 65 20 6d 61 |tes:.>>.|>>The ma|
|000004a0| 6a 6f 72 20 65 6e 68 61 | 6e 63 65 6d 65 6e 74 73 |jor enha|ncements|
|000004b0| 20 74 68 61 74 20 4a 69 | 6d 20 61 64 64 65 64 20 | that Ji|m added |
|000004c0| 74 6f 20 74 68 65 20 70 | 72 6f 67 72 61 6d 20 77 |to the p|rogram w|
|000004d0| 65 72 65 20 74 6f 20 72 | 65 63 6f 64 65 20 69 74 |ere to r|ecode it|
|000004e0| 20 69 6e 0a 3e 3e 43 20 | 61 6e 64 20 74 6f 20 75 | in.>>C |and to u|
|000004f0| 73 65 20 22 61 20 43 4f | 52 44 49 43 20 72 6f 74 |se "a CO|RDIC rot|
|00000500| 61 74 6f 72 20 69 6e 73 | 74 65 61 64 20 6f 66 20 |ator ins|tead of |
|00000510| 74 72 69 67 20 66 75 6e | 63 74 69 6f 6e 73 22 2e |trig fun|ctions".|
|00000520| 20 20 49 20 77 61 6e 74 | 20 74 6f 20 74 61 6c 6b |  I want| to talk|
|00000530| 0a 3e 3e 77 69 74 68 20 | 4a 69 6d 20 61 62 6f 75 |.>>with |Jim abou|
|00000540| 74 20 68 69 73 20 70 72 | 6f 67 72 61 6d 2e 2e 2e |t his pr|ogram...|
|00000550| 2e 77 68 65 72 65 20 69 | 73 20 68 65 3f 21 0a 3e |.where i|s he?!.>|
|00000560| 3e 0a 3e 3e 41 6c 74 65 | 72 6e 61 74 65 6c 79 2c |>.>>Alte|rnately,|
|00000570| 20 64 6f 65 73 20 61 6e | 79 6f 6e 65 20 6b 6e 6f | does an|yone kno|
|00000580| 77 20 61 6e 79 74 68 69 | 6e 67 20 61 62 6f 75 74 |w anythi|ng about|
|00000590| 20 22 43 4f 52 44 49 43 | 20 72 6f 74 61 74 6f 72 | "CORDIC| rotator|
|000005a0| 73 22 3f 20 20 49 27 64 | 20 6c 69 6b 65 20 74 6f |s"?  I'd| like to|
|000005b0| 0a 3e 3e 73 65 65 20 74 | 68 65 20 73 6f 75 72 63 |.>>see t|he sourc|
|000005c0| 65 20 74 6f 20 68 69 73 | 20 6d 6f 64 69 66 69 63 |e to his| modific|
|000005d0| 61 74 69 6f 6e 20 6f 66 | 20 4d 69 63 68 69 65 6c |ation of| Michiel|
|000005e0| 27 73 20 70 72 6f 67 72 | 61 6d 2e 20 20 54 68 65 |'s progr|am.  The|
|000005f0| 20 61 6c 67 6f 72 69 74 | 68 6d 20 75 73 65 64 0a | algorit|hm used.|
|00000600| 3e 3e 77 61 73 20 62 61 | 73 65 64 20 6f 6e 20 6f |>>was ba|sed on o|
|00000610| 6e 65 20 64 65 76 65 6c | 6f 70 65 64 20 62 79 20 |ne devel|oped by |
|00000620| 4c 75 63 61 73 46 69 6c | 6d 20 4c 74 64 2e 20 61 |LucasFil|m Ltd. a|
|00000630| 6e 64 20 70 75 62 6c 69 | 73 68 65 64 20 69 6e 20 |nd publi|shed in |
|00000640| 74 68 65 20 53 65 70 74 | 2e 20 27 38 34 0a 3e 3e |the Sept|. '84.>>|
|00000650| 53 63 69 65 6e 74 69 66 | 69 63 20 41 6d 65 72 69 |Scientif|ic Ameri|
|00000660| 63 61 6e 2e 0a 0a 54 68 | 65 20 69 64 65 61 20 69 |can...Th|e idea i|
|00000670| 73 20 72 65 61 6c 6c 79 | 20 73 69 6d 70 6c 65 2e |s really| simple.|
|00000680| 20 20 41 20 72 6f 74 61 | 74 69 6f 6e 20 63 61 6e |  A rota|tion can|
|00000690| 20 62 65 20 65 78 70 72 | 65 73 73 65 64 20 61 73 | be expr|essed as|
|000006a0| 3a 0a 0a 09 5b 20 78 27 | 20 5d 20 20 20 5b 20 63 |:...[ x'| ]   [ c|
|000006b0| 6f 73 20 74 20 20 20 2d | 73 69 6e 20 79 20 5d 20 |os t   -|sin y ] |
|000006c0| 5b 20 78 20 5d 0a 09 5b | 20 79 27 20 5d 20 3d 20 |[ x ]..[| y' ] = |
|000006d0| 5b 20 73 69 6e 20 74 20 | 20 20 20 63 6f 73 20 74 |[ sin t |   cos t|
|000006e0| 20 5d 20 5b 20 79 20 5d | 0a 0a 46 61 63 74 6f 72 | ] [ y ]|..Factor|
|000006f0| 20 6f 75 74 20 63 6f 73 | 20 74 2c 20 67 69 76 69 | out cos| t, givi|
|00000700| 6e 67 3a 0a 0a 09 5b 20 | 78 27 20 5d 20 20 20 20 |ng:...[ |x' ]    |
|00000710| 20 20 20 20 20 5b 20 20 | 20 31 20 20 20 20 20 2d |     [  | 1     -|
|00000720| 74 61 6e 20 79 20 5d 20 | 5b 20 78 20 5d 0a 09 5b |tan y ] |[ x ]..[|
|00000730| 20 79 27 20 5d 20 3d 20 | 63 6f 73 20 74 20 5b 20 | y' ] = |cos t [ |
|00000740| 74 61 6e 20 74 20 20 20 | 20 20 20 31 20 20 20 5d |tan t   |   1   ]|
|00000750| 20 5b 20 79 20 5d 0a 0a | 4e 6f 77 20 6c 65 74 20 | [ y ]..|Now let |
|00000760| 74 20 62 65 20 65 78 70 | 72 65 73 73 65 64 20 61 |t be exp|ressed a|
|00000770| 73 20 61 20 73 69 67 6e | 65 64 20 73 75 6d 20 6f |s a sign|ed sum o|
|00000780| 66 20 0a 09 09 20 20 20 | 20 20 20 20 20 2d 31 20 |f ...   |     -1 |
|00000790| 20 2d 69 0a 09 74 20 3d | 20 53 75 6d 20 7b 20 64 | -i..t =| Sum { d|
|000007a0| 20 20 74 61 6e 20 20 20 | 32 20 20 20 7d 2c 20 20 |  tan   |2   },  |
|000007b0| 20 77 68 65 72 65 20 64 | 20 20 3d 20 7b 2b 31 2c | where d|  = {+1,|
|000007c0| 20 2d 31 7d 0a 09 20 20 | 20 20 20 69 20 20 20 20 | -1}..  |   i    |
|000007d0| 20 69 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 | i      |        |
|000007e0| 20 20 20 20 20 20 20 20 | 20 69 0a 0a 54 68 69 73 |        | i..This|
|000007f0| 20 79 69 65 6c 64 73 20 | 28 70 6c 65 61 73 65 20 | yields |(please |
|00000800| 66 6f 72 67 69 76 65 20 | 74 68 65 20 63 72 75 64 |forgive |the crud|
|00000810| 64 79 20 74 79 70 65 73 | 65 74 74 69 6e 67 3b 20 |dy types|etting; |
|00000820| 4d 61 74 68 74 79 70 65 | 20 64 6f 65 73 6e 27 74 |Mathtype| doesn't|
|00000830| 20 77 6f 72 6b 0a 6f 6e | 20 55 4e 49 58 29 3a 0a | work.on| UNIX):.|
|00000840| 0a 09 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |..      |        |
|00000850| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000860| 20 20 20 20 20 20 20 20 | 2d 69 0a 09 5b 20 78 27 |        |-i..[ x'|
|00000870| 20 5d 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 | ]      |        |
|00000880| 20 20 20 20 20 20 5b 20 | 31 20 20 20 20 20 2d 64 |      [ |1     -d|
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|000008a0| 20 20 20 5d 20 20 20 20 | 20 20 20 20 20 20 20 20 |   ]    |        |
|000008b0| 20 20 20 20 20 20 20 20 | 5b 20 20 20 20 20 20 20 |        |[       |
|000008c0| 20 20 69 20 20 20 20 5d | 20 5b 20 20 20 5d 0a 09 |  i    ]| [   ]..|
|000008d0| 5b 20 20 20 20 5d 20 3d | 20 53 75 6d 20 7b 20 43 |[    ] =| Sum { C|
|000008e0| 20 20 7d 20 53 75 6d 20 | 7b 20 5b 20 20 20 20 2d |  } Sum |{ [    -|
|000008f0| 69 20 20 20 20 20 20 20 | 20 5d 20 5b 20 20 20 5d |i       | ] [   ]|
|00000900| 20 7d 0a 09 5b 20 79 27 | 20 5d 20 20 20 20 69 20 | }..[ y'| ]    i |
|00000910| 20 20 20 20 69 20 20 20 | 20 69 20 20 20 20 5b 20 |    i   | i    [ |
|00000920| 64 20 32 20 20 20 20 20 | 20 31 20 20 20 5d 20 5b |d 2     | 1   ] [|
|00000930| 20 79 20 5d 0a 09 09 09 | 09 20 20 20 20 20 69 0a | y ]....|.     i.|
|00000940| 0a 54 68 65 20 66 69 72 | 73 74 20 73 75 6d 20 69 |.The fir|st sum i|
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|00000960| 68 65 20 73 65 63 6f 6e | 64 20 73 75 6d 20 69 73 |he secon|d sum is|
|00000970| 20 61 20 73 65 72 69 65 | 73 20 6f 66 20 73 68 69 | a serie|s of shi|
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|000009a0| 68 69 73 20 69 73 20 69 | 6c 6c 75 73 74 72 61 74 |his is i|llustrat|
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|000009c0| 69 6e 67 20 77 6f 72 6b | 69 6e 67 20 63 6f 64 65 |ing work|ing code|
|000009d0| 2c 20 77 68 69 63 68 20 | 70 65 72 66 6f 72 6d 73 |, which |performs|
|000009e0| 20 61 6c 6c 20 6f 66 20 | 74 68 65 0a 66 6f 6c 6c | all of |the.foll|
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|00000a00| 3a 0a 09 72 6f 74 61 74 | 65 20 61 20 76 65 63 74 |:..rotat|e a vect|
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|00000a40| 20 74 6f 20 70 6f 6c 61 | 72 20 63 6f 6e 76 65 72 | to pola|r conver|
|00000a50| 73 69 6f 6e 0a 09 73 69 | 6d 75 6c 74 61 6e 65 6f |sion..si|multaneo|
|00000a60| 75 73 20 63 61 6c 63 75 | 6c 61 74 69 6f 6e 20 6f |us calcu|lation o|
|00000a70| 66 20 73 69 6e 65 20 61 | 6e 64 20 63 6f 73 69 6e |f sine a|nd cosin|
|00000a80| 65 2e 0a 09 61 74 61 6e | 32 28 29 0a 4e 6f 74 65 |e...atan|2().Note|
|00000a90| 20 74 68 61 74 20 74 68 | 65 20 6e 6f 72 6d 61 6c | that th|e normal|
|00000aa0| 69 7a 61 74 69 6f 6e 20 | 61 6e 64 20 64 65 6e 6f |ization |and deno|
|00000ab0| 72 6d 61 6c 69 7a 61 74 | 69 6f 6e 20 69 73 20 64 |rmalizat|ion is d|
|00000ac0| 6f 6e 65 20 6d 61 69 6e | 6c 79 20 74 6f 20 69 6e |one main|ly to in|
|00000ad0| 63 72 65 61 73 65 0a 61 | 63 63 75 72 61 63 79 3b |crease.a|ccuracy;|
|00000ae0| 20 69 66 20 79 6f 75 27 | 72 65 20 6d 6f 72 65 20 | if you'|re more |
|00000af0| 63 6f 6e 63 65 72 6e 65 | 64 20 61 62 6f 75 74 20 |concerne|d about |
|00000b00| 73 70 65 65 64 2c 20 79 | 6f 75 27 64 20 70 72 6f |speed, y|ou'd pro|
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|00000b20| 6f 75 74 20 65 6e 74 69 | 72 65 6c 79 20 6f 72 20 |out enti|rely or |
|00000b30| 6a 75 73 74 20 73 68 69 | 66 74 20 62 79 20 61 20 |just shi|ft by a |
|00000b40| 66 69 78 65 64 20 61 6d | 6f 75 6e 74 2e 0a 0a 59 |fixed am|ount...Y|
|00000b50| 6f 75 20 63 61 6e 20 64 | 6f 20 73 69 6d 69 6c 61 |ou can d|o simila|
|00000b60| 72 20 73 6f 72 74 73 20 | 6f 66 20 74 68 69 6e 67 |r sorts |of thing|
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|00000b90| 72 65 62 79 0a 62 65 69 | 6e 67 20 61 62 6c 65 20 |reby.bei|ng able |
|00000ba0| 74 6f 20 64 6f 20 65 78 | 70 6f 6e 65 6e 74 69 61 |to do ex|ponentia|
|00000bb0| 6c 20 61 6e 64 20 6c 6f | 67 61 72 69 74 68 6d 73 |l and lo|garithms|
|00000bc0| 2e 20 20 49 20 68 61 76 | 65 20 6e 65 76 65 72 20 |.  I hav|e never |
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|00000be0| 65 2c 20 61 6c 74 68 6f | 75 67 68 20 49 20 77 6f |e, altho|ugh I wo|
|00000bf0| 75 6c 64 20 6c 69 6b 65 | 20 74 6f 20 68 61 76 65 |uld like| to have|
|00000c00| 20 69 74 2c 20 65 73 70 | 65 63 69 61 6c 6c 79 20 | it, esp|ecially |
|00000c10| 74 6f 20 63 61 6c 63 75 | 6c 61 74 65 20 67 61 6d |to calcu|late gam|
|00000c20| 6d 61 0a 63 6f 72 72 65 | 63 74 69 6f 6e 20 74 61 |ma.corre|ction ta|
|00000c30| 62 6c 65 73 2e 0a 0a 2d | 2d 2d 2d 2d 2d 2d 2d 2d |bles...-|--------|
|00000c40| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 2d 2d 2d 2d |--------|--------|
|00000c50| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 2d 2d 2d 2d |--------|--------|
|00000c60| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 2d 2d 2d 2d |--------|--------|
|00000c70| 2d 2d 2d 0a 0a 23 20 64 | 65 66 69 6e 65 20 52 41 |---..# d|efine RA|
|00000c80| 44 49 41 4e 53 20 31 0a | 23 20 64 65 66 69 6e 65 |DIANS 1.|# define|
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|00000ca0| 64 31 63 09 2f 2a 20 30 | 2e 32 37 31 35 37 32 20 |d1c./* 0|.271572 |
|00000cb0| 2a 2f 0a 0a 73 74 61 74 | 69 63 20 6c 6f 6e 67 20 |*/..stat|ic long |
|00000cc0| 61 72 63 74 61 6e 74 61 | 62 5b 33 32 5d 20 3d 20 |arctanta|b[32] = |
|00000cd0| 7b 0a 23 20 69 66 64 65 | 66 20 44 45 47 52 45 45 |{.# ifde|f DEGREE|
|00000ce0| 53 09 09 2f 2a 20 4d 53 | 20 31 30 20 69 6e 74 65 |S../* MS| 10 inte|
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|00000e20| 23 20 64 65 66 69 6e 65 | 20 51 55 41 52 54 45 52 |# define| QUARTER|
|00000e30| 20 28 28 69 6e 74 29 28 | 33 2e 31 34 31 35 39 32 | ((int)(|3.141592|
|00000e40| 36 35 34 20 2f 20 32 2e | 30 20 2a 20 28 31 20 3c |654 / 2.|0 * (1 <|
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|00000ea0| 0a 09 34 31 39 33 39 36 | 33 2c 20 32 30 39 37 31 |..419396|3, 20971|
|00000eb0| 30 39 2c 20 31 30 34 38 | 35 37 31 2c 20 35 32 34 |09, 1048|571, 524|
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|00000ef0| 20 34 30 39 36 2c 20 32 | 30 34 38 2c 20 31 30 32 | 4096, 2|048, 102|
|00000f00| 34 2c 20 35 31 32 2c 20 | 32 35 36 2c 20 31 32 38 |4, 512, |256, 128|
|00000f10| 2c 20 36 34 2c 20 33 32 | 2c 20 31 36 2c 20 38 2c |, 64, 32|, 16, 8,|
|00000f20| 20 34 2c 20 32 2c 20 31 | 2c 20 30 2c 20 30 2c 20 | 4, 2, 1|, 0, 0, |
|00000f30| 0a 23 20 65 6c 73 65 0a | 23 20 64 65 66 69 6e 65 |.# else.|# define|
|00000f40| 20 42 52 41 44 53 20 31 | 0a 23 20 64 65 66 69 6e | BRADS 1|.# defin|
|00000f50| 65 20 51 55 41 52 54 45 | 52 20 28 31 20 3c 3c 20 |e QUARTE|R (1 << |
|00000f60| 33 30 29 0a 09 37 35 36 | 38 30 38 34 31 38 2c 20 |30)..756|808418, |
|00000f70| 35 33 36 38 37 30 39 31 | 32 2c 20 33 31 36 39 33 |53687091|2, 31693|
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|00000fa0| 33 33 31 2c 0a 09 32 31 | 33 35 34 34 36 35 2c 20 |331,..21|354465, |
|00000fb0| 31 30 36 37 39 38 33 38 | 2c 20 35 33 34 30 32 34 |10679838|, 534024|
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|00000fd0| 30 38 37 2c 20 36 36 37 | 35 34 34 2c 20 33 33 33 |087, 667|544, 333|
|00000fe0| 37 37 32 2c 20 31 36 36 | 38 38 36 2c 0a 09 38 33 |772, 166|886,..83|
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|00001000| 31 2c 20 31 30 34 33 30 | 2c 20 35 32 31 35 2c 20 |1, 10430|, 5215, |
|00001010| 32 36 30 38 2c 20 31 33 | 30 34 2c 20 36 35 32 2c |2608, 13|04, 652,|
|00001020| 20 33 32 36 2c 20 31 36 | 33 2c 20 38 31 2c 20 34 | 326, 16|3, 81, 4|
|00001030| 31 2c 0a 09 32 30 2c 20 | 31 30 2c 20 35 2c 20 33 |1,..20, |10, 5, 3|
|00001040| 2c 20 31 2c 20 31 2c 20 | 0a 23 20 65 6e 64 69 66 |, 1, 1, |.# endif|
|00001050| 0a 23 20 65 6e 64 69 66 | 0a 7d 3b 0a 0a 0a 2f 2a |.# endif|.};.../*|
|00001060| 20 54 6f 20 72 6f 74 61 | 74 65 20 61 20 76 65 63 | To rota|te a vec|
|00001070| 74 6f 72 20 74 68 72 6f | 75 67 68 20 61 6e 20 61 |tor thro|ugh an a|
|00001080| 6e 67 6c 65 20 6f 66 20 | 74 68 65 74 61 2c 20 77 |ngle of |theta, w|
|00001090| 65 20 63 61 6c 63 75 6c | 61 74 65 3a 0a 20 2a 0a |e calcul|ate:. *.|
|000010a0| 20 2a 09 78 27 20 3d 20 | 78 20 63 6f 73 28 74 68 | *.x' = |x cos(th|
|000010b0| 65 74 61 29 20 2d 20 79 | 20 73 69 6e 28 74 68 65 |eta) - y| sin(the|
|000010c0| 74 61 29 0a 20 2a 09 79 | 27 20 3d 20 78 20 73 69 |ta). *.y|' = x si|
|000010d0| 6e 28 74 68 65 74 61 29 | 20 2b 20 79 20 63 6f 73 |n(theta)| + y cos|
|000010e0| 28 74 68 65 74 61 29 0a | 20 2a 0a 20 2a 20 54 68 |(theta).| *. * Th|
|000010f0| 65 20 72 6f 74 61 74 65 | 28 29 20 72 6f 75 74 69 |e rotate|() routi|
|00001100| 6e 65 20 70 65 72 66 6f | 72 6d 73 20 6d 75 6c 74 |ne perfo|rms mult|
|00001110| 69 70 6c 65 20 72 6f 74 | 61 74 69 6f 6e 73 20 6f |iple rot|ations o|
|00001120| 66 20 74 68 65 20 66 6f | 72 6d 3a 0a 20 2a 0a 20 |f the fo|rm:. *. |
|00001130| 2a 09 78 5b 69 2b 31 5d | 20 3d 20 63 6f 73 28 74 |*.x[i+1]| = cos(t|
|00001140| 68 65 74 61 5b 69 5d 29 | 20 7b 20 78 5b 69 5d 20 |heta[i])| { x[i] |
|00001150| 2d 20 79 5b 69 5d 20 74 | 61 6e 28 74 68 65 74 61 |- y[i] t|an(theta|
|00001160| 5b 69 5d 29 20 7d 0a 20 | 2a 09 79 5b 69 2b 31 5d |[i]) }. |*.y[i+1]|
|00001170| 20 3d 20 63 6f 73 28 74 | 68 65 74 61 5b 69 5d 29 | = cos(t|heta[i])|
|00001180| 20 7b 20 79 5b 69 5d 20 | 2b 20 78 5b 69 5d 20 74 | { y[i] |+ x[i] t|
|00001190| 61 6e 28 74 68 65 74 61 | 5b 69 5d 29 20 7d 0a 20 |an(theta|[i]) }. |
|000011a0| 2a 0a 20 2a 20 77 69 74 | 68 20 74 68 65 20 63 6f |*. * wit|h the co|
|000011b0| 6e 73 74 72 61 69 6e 74 | 20 74 68 61 74 20 74 61 |nstraint| that ta|
|000011c0| 6e 28 74 68 65 74 61 5b | 69 5d 29 20 3d 20 70 6f |n(theta[|i]) = po|
|000011d0| 77 28 32 2c 20 2d 69 29 | 2c 20 77 68 69 63 68 20 |w(2, -i)|, which |
|000011e0| 63 61 6e 20 62 65 0a 20 | 2a 20 69 6d 70 6c 65 6d |can be. |* implem|
|000011f0| 65 6e 74 65 64 20 62 79 | 20 73 68 69 66 74 69 6e |ented by| shiftin|
|00001200| 67 2e 20 57 65 20 61 6c | 77 61 79 73 20 73 68 69 |g. We al|ways shi|
|00001210| 66 74 20 62 79 20 65 69 | 74 68 65 72 20 61 20 70 |ft by ei|ther a p|
|00001220| 6f 73 69 74 69 76 65 20 | 6f 72 0a 20 2a 20 6e 65 |ositive |or. * ne|
|00001230| 67 61 74 69 76 65 20 61 | 6e 67 6c 65 2c 20 73 6f |gative a|ngle, so|
|00001240| 20 74 68 65 20 63 6f 6e | 76 65 72 67 65 6e 63 65 | the con|vergence|
|00001250| 20 68 61 73 20 74 68 65 | 20 72 69 6e 67 69 6e 67 | has the| ringing|
|00001260| 20 70 72 6f 70 65 72 74 | 79 2e 20 53 69 6e 63 65 | propert|y. Since|
|00001270| 20 74 68 65 0a 20 2a 20 | 63 6f 73 69 6e 65 20 69 | the. * |cosine i|
|00001280| 73 20 61 6c 77 61 79 73 | 20 70 6f 73 69 74 69 76 |s always| positiv|
|00001290| 65 20 66 6f 72 20 70 6f | 73 69 74 69 76 65 20 61 |e for po|sitive a|
|000012a0| 6e 64 20 6e 65 67 61 74 | 69 76 65 20 61 6e 67 6c |nd negat|ive angl|
|000012b0| 65 73 20 62 65 74 77 65 | 65 6e 20 2d 39 30 0a 20 |es betwe|en -90. |
|000012c0| 2a 20 61 6e 64 20 39 30 | 20 64 65 67 72 65 65 73 |* and 90| degrees|
|000012d0| 2c 20 61 20 70 72 65 64 | 69 63 74 61 62 6c 65 20 |, a pred|ictable |
|000012e0| 6d 61 67 6e 69 74 75 64 | 65 20 73 63 61 6c 69 6e |magnitud|e scalin|
|000012f0| 67 20 6f 63 63 75 72 73 | 20 61 74 20 65 61 63 68 |g occurs| at each|
|00001300| 20 73 74 65 70 2c 0a 20 | 2a 20 61 6e 64 20 63 61 | step,. |* and ca|
|00001310| 6e 20 62 65 20 63 6f 6d | 70 65 6e 73 61 74 65 64 |n be com|pensated|
|00001320| 20 66 6f 72 20 69 6e 73 | 74 65 61 64 20 61 74 20 | for ins|tead at |
|00001330| 74 68 65 20 65 6e 64 20 | 6f 66 20 74 68 65 20 69 |the end |of the i|
|00001340| 74 65 72 61 74 69 6f 6e | 73 20 62 79 20 61 0a 20 |teration|s by a. |
|00001350| 2a 20 63 6f 6d 70 6f 73 | 69 74 65 20 73 63 61 6c |* compos|ite scal|
|00001360| 69 6e 67 20 6f 66 20 74 | 68 65 20 70 72 6f 64 75 |ing of t|he produ|
|00001370| 63 74 20 6f 66 20 61 6c | 6c 20 74 68 65 20 63 6f |ct of al|l the co|
|00001380| 73 28 74 68 65 74 61 5b | 69 5d 29 27 73 2e 0a 20 |s(theta[|i])'s.. |
|00001390| 2a 2f 0a 0a 73 74 61 74 | 69 63 0a 70 73 65 75 64 |*/..stat|ic.pseud|
|000013a0| 6f 72 6f 74 61 74 65 28 | 70 78 2c 20 70 79 2c 20 |orotate(|px, py, |
|000013b0| 74 68 65 74 61 29 0a 6c | 6f 6e 67 20 2a 70 78 2c |theta).l|ong *px,|
|000013c0| 20 2a 70 79 3b 0a 72 65 | 67 69 73 74 65 72 20 6c | *py;.re|gister l|
|000013d0| 6f 6e 67 20 74 68 65 74 | 61 3b 0a 7b 0a 09 72 65 |ong thet|a;.{..re|
|000013e0| 67 69 73 74 65 72 20 69 | 6e 74 20 69 3b 0a 09 72 |gister i|nt i;..r|
|000013f0| 65 67 69 73 74 65 72 20 | 6c 6f 6e 67 20 78 2c 20 |egister |long x, |
|00001400| 79 2c 20 78 74 65 6d 70 | 3b 0a 09 72 65 67 69 73 |y, xtemp|;..regis|
|00001410| 74 65 72 20 6c 6f 6e 67 | 20 2a 61 72 63 74 61 6e |ter long| *arctan|
|00001420| 70 74 72 3b 0a 0a 09 78 | 20 3d 20 2a 70 78 3b 0a |ptr;...x| = *px;.|
|00001430| 09 79 20 3d 20 2a 70 79 | 3b 0a 0a 09 2f 2a 20 47 |.y = *py|;.../* G|
|00001440| 65 74 20 61 6e 67 6c 65 | 20 62 65 74 77 65 65 6e |et angle| between|
|00001450| 20 2d 39 30 20 61 6e 64 | 20 39 30 20 64 65 67 72 | -90 and| 90 degr|
|00001460| 65 65 73 20 2a 2f 0a 09 | 77 68 69 6c 65 20 28 74 |ees */..|while (t|
|00001470| 68 65 74 61 20 3c 20 2d | 51 55 41 52 54 45 52 29 |heta < -|QUARTER)|
|00001480| 20 7b 0a 09 20 20 20 20 | 78 20 3d 20 2d 78 3b 0a | {..    |x = -x;.|
|00001490| 09 20 20 20 20 79 20 3d | 20 2d 79 3b 0a 09 20 20 |.    y =| -y;..  |
|000014a0| 20 20 74 68 65 74 61 20 | 2b 3d 20 32 20 2a 20 51 |  theta |+= 2 * Q|
|000014b0| 55 41 52 54 45 52 3b 0a | 09 7d 0a 09 77 68 69 6c |UARTER;.|.}..whil|
|000014c0| 65 20 28 74 68 65 74 61 | 20 3e 20 51 55 41 52 54 |e (theta| > QUART|
|000014d0| 45 52 29 20 7b 0a 09 20 | 20 20 20 78 20 3d 20 2d |ER) {.. |   x = -|
|000014e0| 78 3b 0a 09 20 20 20 20 | 79 20 3d 20 2d 79 3b 0a |x;..    |y = -y;.|
|000014f0| 09 20 20 20 20 74 68 65 | 74 61 20 2d 3d 20 32 20 |.    the|ta -= 2 |
|00001500| 2a 20 51 55 41 52 54 45 | 52 3b 0a 09 7d 0a 0a 09 |* QUARTE|R;..}...|
|00001510| 2f 2a 20 49 6e 69 74 69 | 61 6c 20 70 73 65 75 64 |/* Initi|al pseud|
|00001520| 6f 72 6f 74 61 74 69 6f | 6e 2c 20 77 69 74 68 20 |orotatio|n, with |
|00001530| 6c 65 66 74 20 73 68 69 | 66 74 20 2a 2f 0a 09 61 |left shi|ft */..a|
|00001540| 72 63 74 61 6e 70 74 72 | 20 3d 20 61 72 63 74 61 |rctanptr| = arcta|
|00001550| 6e 74 61 62 3b 0a 09 69 | 66 20 28 74 68 65 74 61 |ntab;..i|f (theta|
|00001560| 20 3c 20 30 29 20 7b 0a | 09 20 20 20 20 78 74 65 | < 0) {.|.    xte|
|00001570| 6d 70 20 3d 20 78 20 2b | 20 28 79 20 3c 3c 20 31 |mp = x +| (y << 1|
|00001580| 29 3b 0a 09 20 20 20 20 | 79 20 20 20 20 20 3d 20 |);..    |y     = |
|00001590| 79 20 2d 20 28 78 20 3c | 3c 20 31 29 3b 0a 09 20 |y - (x <|< 1);.. |
|000015a0| 20 20 20 78 20 3d 20 78 | 74 65 6d 70 3b 0a 09 20 |   x = x|temp;.. |
|000015b0| 20 20 20 74 68 65 74 61 | 20 2b 3d 20 2a 61 72 63 |   theta| += *arc|
|000015c0| 74 61 6e 70 74 72 2b 2b | 3b 0a 09 7d 0a 09 65 6c |tanptr++|;..}..el|
|000015d0| 73 65 20 7b 0a 09 20 20 | 20 20 78 74 65 6d 70 20 |se {..  |  xtemp |
|000015e0| 3d 20 78 20 2d 20 28 79 | 20 3c 3c 20 31 29 3b 0a |= x - (y| << 1);.|
|000015f0| 09 20 20 20 20 79 20 20 | 20 20 20 3d 20 79 20 2b |.    y  |   = y +|
|00001600| 20 28 78 20 3c 3c 20 31 | 29 3b 0a 09 20 20 20 20 | (x << 1|);..    |
|00001610| 78 20 3d 20 78 74 65 6d | 70 3b 0a 09 20 20 20 20 |x = xtem|p;..    |
|00001620| 74 68 65 74 61 20 2d 3d | 20 2a 61 72 63 74 61 6e |theta -=| *arctan|
|00001630| 70 74 72 2b 2b 3b 0a 09 | 7d 0a 0a 09 2f 2a 20 53 |ptr++;..|}.../* S|
|00001640| 75 62 73 65 71 75 65 6e | 74 20 70 73 65 75 64 6f |ubsequen|t pseudo|
|00001650| 72 6f 74 61 74 69 6f 6e | 73 2c 20 77 69 74 68 20 |rotation|s, with |
|00001660| 72 69 67 68 74 20 73 68 | 69 66 74 73 20 2a 2f 0a |right sh|ifts */.|
|00001670| 09 66 6f 72 20 28 69 20 | 3d 20 30 3b 20 69 20 3c |.for (i |= 0; i <|
|00001680| 3d 20 32 38 3b 20 69 2b | 2b 29 20 7b 0a 09 20 20 |= 28; i+|+) {..  |
|00001690| 20 20 69 66 20 28 74 68 | 65 74 61 20 3c 20 30 29 |  if (th|eta < 0)|
|000016a0| 20 7b 0a 09 09 78 74 65 | 6d 70 20 3d 20 78 20 2b | {...xte|mp = x +|
|000016b0| 20 28 79 20 3e 3e 20 69 | 29 3b 0a 09 09 79 20 20 | (y >> i|);...y  |
|000016c0| 20 20 20 3d 20 79 20 2d | 20 28 78 20 3e 3e 20 69 |   = y -| (x >> i|
|000016d0| 29 3b 0a 09 09 78 20 3d | 20 78 74 65 6d 70 3b 0a |);...x =| xtemp;.|
|000016e0| 09 09 74 68 65 74 61 20 | 2b 3d 20 2a 61 72 63 74 |..theta |+= *arct|
|000016f0| 61 6e 70 74 72 2b 2b 3b | 0a 09 20 20 20 20 7d 0a |anptr++;|..    }.|
|00001700| 09 20 20 20 20 65 6c 73 | 65 20 7b 0a 09 09 78 74 |.    els|e {...xt|
|00001710| 65 6d 70 20 3d 20 78 20 | 2d 20 28 79 20 3e 3e 20 |emp = x |- (y >> |
|00001720| 69 29 3b 0a 09 09 79 20 | 20 20 20 20 3d 20 79 20 |i);...y |    = y |
|00001730| 2b 20 28 78 20 3e 3e 20 | 69 29 3b 0a 09 09 78 20 |+ (x >> |i);...x |
|00001740| 3d 20 78 74 65 6d 70 3b | 0a 09 09 74 68 65 74 61 |= xtemp;|...theta|
|00001750| 20 2d 3d 20 2a 61 72 63 | 74 61 6e 70 74 72 2b 2b | -= *arc|tanptr++|
|00001760| 3b 0a 09 20 20 20 20 7d | 0a 09 7d 0a 0a 09 2a 70 |;..    }|..}...*p|
|00001770| 78 20 3d 20 78 3b 0a 09 | 2a 70 79 20 3d 20 79 3b |x = x;..|*py = y;|
|00001780| 0a 7d 0a 0a 0a 73 74 61 | 74 69 63 0a 70 73 65 75 |.}...sta|tic.pseu|
|00001790| 64 6f 70 6f 6c 61 72 69 | 7a 65 28 61 72 67 78 2c |dopolari|ze(argx,|
|000017a0| 20 61 72 67 79 29 0a 6c | 6f 6e 67 20 2a 61 72 67 | argy).l|ong *arg|
|000017b0| 78 2c 20 2a 61 72 67 79 | 3b 0a 7b 0a 09 72 65 67 |x, *argy|;.{..reg|
|000017c0| 69 73 74 65 72 20 6c 6f | 6e 67 20 74 68 65 74 61 |ister lo|ng theta|
|000017d0| 3b 0a 09 72 65 67 69 73 | 74 65 72 20 6c 6f 6e 67 |;..regis|ter long|
|000017e0| 20 79 69 2c 20 69 3b 0a | 09 72 65 67 69 73 74 65 | yi, i;.|.registe|
|000017f0| 72 20 6c 6f 6e 67 20 78 | 2c 20 79 3b 0a 09 72 65 |r long x|, y;..re|
|00001800| 67 69 73 74 65 72 20 6c | 6f 6e 67 20 2a 61 72 63 |gister l|ong *arc|
|00001810| 74 61 6e 70 74 72 3b 0a | 0a 09 78 20 3d 20 2a 61 |tanptr;.|..x = *a|
|00001820| 72 67 78 3b 0a 09 79 20 | 3d 20 2a 61 72 67 79 3b |rgx;..y |= *argy;|
|00001830| 0a 0a 09 2f 2a 20 47 65 | 74 20 74 68 65 20 76 65 |.../* Ge|t the ve|
|00001840| 63 74 6f 72 20 69 6e 74 | 6f 20 74 68 65 20 72 69 |ctor int|o the ri|
|00001850| 67 68 74 20 68 61 6c 66 | 20 70 6c 61 6e 65 20 2a |ght half| plane *|
|00001860| 2f 0a 09 74 68 65 74 61 | 20 3d 20 30 3b 0a 09 69 |/..theta| = 0;..i|
|00001870| 66 20 28 78 20 3c 20 30 | 29 20 7b 0a 09 20 20 20 |f (x < 0|) {..   |
|00001880| 20 78 20 3d 20 2d 78 3b | 0a 09 20 20 20 20 79 20 | x = -x;|..    y |
|00001890| 3d 20 2d 79 3b 0a 09 20 | 20 20 20 74 68 65 74 61 |= -y;.. |   theta|
|000018a0| 20 3d 20 32 20 2a 20 51 | 55 41 52 54 45 52 3b 0a | = 2 * Q|UARTER;.|
|000018b0| 09 7d 0a 0a 09 69 66 20 | 28 79 20 3e 20 30 29 0a |.}...if |(y > 0).|
|000018c0| 09 20 20 20 20 74 68 65 | 74 61 20 3d 20 2d 20 74 |.    the|ta = - t|
|000018d0| 68 65 74 61 3b 0a 09 0a | 09 61 72 63 74 61 6e 70 |heta;...|.arctanp|
|000018e0| 74 72 20 3d 20 61 72 63 | 74 61 6e 74 61 62 3b 0a |tr = arc|tantab;.|
|000018f0| 09 69 66 20 28 79 20 3c | 20 30 29 20 7b 09 2f 2a |.if (y <| 0) {./*|
|00001900| 20 52 6f 74 61 74 65 20 | 70 6f 73 69 74 69 76 65 | Rotate |positive|
|00001910| 20 2a 2f 0a 09 09 79 69 | 20 3d 20 79 20 2b 20 28 | */...yi| = y + (|
|00001920| 78 20 3c 3c 20 31 29 3b | 0a 09 09 78 20 3d 20 78 |x << 1);|...x = x|
|00001930| 20 2d 20 28 79 20 3c 3c | 20 31 29 3b 0a 09 09 79 | - (y <<| 1);...y|
|00001940| 20 3d 20 79 69 3b 0a 09 | 09 74 68 65 74 61 20 2d | = yi;..|.theta -|
|00001950| 3d 20 2a 61 72 63 74 61 | 6e 70 74 72 2b 2b 3b 09 |= *arcta|nptr++;.|
|00001960| 2f 2a 20 53 75 62 74 72 | 61 63 74 20 61 6e 67 6c |/* Subtr|act angl|
|00001970| 65 20 2a 2f 0a 09 7d 0a | 09 65 6c 73 65 20 7b 09 |e */..}.|.else {.|
|00001980| 09 2f 2a 20 52 6f 74 61 | 74 65 20 6e 65 67 61 74 |./* Rota|te negat|
|00001990| 69 76 65 20 2a 2f 0a 09 | 09 79 69 20 3d 20 79 20 |ive */..|.yi = y |
|000019a0| 2d 20 28 78 20 3c 3c 20 | 31 29 3b 0a 09 09 78 20 |- (x << |1);...x |
|000019b0| 3d 20 78 20 2b 20 28 79 | 20 3c 3c 20 31 29 3b 0a |= x + (y| << 1);.|
|000019c0| 09 09 79 20 3d 20 79 69 | 3b 0a 09 09 74 68 65 74 |..y = yi|;...thet|
|000019d0| 61 20 2b 3d 20 2a 61 72 | 63 74 61 6e 70 74 72 2b |a += *ar|ctanptr+|
|000019e0| 2b 3b 09 2f 2a 20 41 64 | 64 20 61 6e 67 6c 65 20 |+;./* Ad|d angle |
|000019f0| 2a 2f 0a 09 7d 0a 0a 09 | 66 6f 72 20 28 69 20 3d |*/..}...|for (i =|
|00001a00| 20 30 3b 20 69 20 3c 3d | 20 32 38 3b 20 69 2b 2b | 0; i <=| 28; i++|
|00001a10| 29 20 7b 0a 09 20 20 20 | 20 69 66 20 28 79 20 3c |) {..   | if (y <|
|00001a20| 20 30 29 20 7b 09 2f 2a | 20 52 6f 74 61 74 65 20 | 0) {./*| Rotate |
|00001a30| 70 6f 73 69 74 69 76 65 | 20 2a 2f 0a 09 09 79 69 |positive| */...yi|
|00001a40| 20 3d 20 79 20 2b 20 28 | 78 20 3e 3e 20 69 29 3b | = y + (|x >> i);|
|00001a50| 0a 09 09 78 20 3d 20 78 | 20 2d 20 28 79 20 3e 3e |...x = x| - (y >>|
|00001a60| 20 69 29 3b 0a 09 09 79 | 20 3d 20 79 69 3b 0a 09 | i);...y| = yi;..|
|00001a70| 09 74 68 65 74 61 20 2d | 3d 20 2a 61 72 63 74 61 |.theta -|= *arcta|
|00001a80| 6e 70 74 72 2b 2b 3b 0a | 09 20 20 20 20 7d 0a 09 |nptr++;.|.    }..|
|00001a90| 20 20 20 20 65 6c 73 65 | 20 7b 09 09 2f 2a 20 52 |    else| {../* R|
|00001aa0| 6f 74 61 74 65 20 6e 65 | 67 61 74 69 76 65 20 2a |otate ne|gative *|
|00001ab0| 2f 0a 09 09 79 69 20 3d | 20 79 20 2d 20 28 78 20 |/...yi =| y - (x |
|00001ac0| 3e 3e 20 69 29 3b 0a 09 | 09 78 20 3d 20 78 20 2b |>> i);..|.x = x +|
|00001ad0| 20 28 79 20 3e 3e 20 69 | 29 3b 0a 09 09 79 20 3d | (y >> i|);...y =|
|00001ae0| 20 79 69 3b 0a 09 09 74 | 68 65 74 61 20 2b 3d 20 | yi;...t|heta += |
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|000027e0| 0a 23 23 23 23 23 20 54 | 68 69 73 20 69 73 20 74 |.##### T|his is t|
|000027f0| 68 65 20 65 6e 64 20 6f | 66 20 6d 79 20 70 6f 73 |he end o|f my pos|
|00002800| 74 69 6e 67 20 23 23 23 | 23 23 0a 2d 2d 20 0a 4b |ting ###|##.-- .K|
|00002810| 65 6e 20 54 75 72 6b 6f | 77 73 6b 69 20 40 20 41 |en Turko|wski @ A|
|00002820| 70 70 6c 65 20 43 6f 6d | 70 75 74 65 72 2c 20 49 |pple Com|puter, I|
|00002830| 6e 63 2e 2c 20 43 75 70 | 65 72 74 69 6e 6f 2c 20 |nc., Cup|ertino, |
|00002840| 43 41 0a 55 55 43 50 3a | 20 7b 6d 74 78 69 6e 75 |CA.UUCP:| {mtxinu|
|00002850| 2c 73 75 6e 2c 64 65 63 | 77 72 6c 2c 6e 73 63 2c |,sun,dec|wrl,nsc,|
|00002860| 76 6f 64 65 72 7d 21 61 | 70 70 6c 65 21 74 75 72 |voder}!a|pple!tur|
|00002870| 6b 0a 43 53 4e 45 54 3a | 20 74 75 72 6b 40 41 70 |k.CSNET:| turk@Ap|
|00002880| 70 6c 65 2e 43 53 4e 45 | 54 0a 41 52 50 41 3a 20 |ple.CSNE|T.ARPA: |
|00002890| 74 75 72 6b 25 41 70 70 | 6c 65 40 63 73 6e 65 74 |turk%App|le@csnet|
|000028a0| 2d 72 65 6c 61 79 2e 41 | 52 50 41 0a 0a 0a       |-relay.A|RPA...  |
+--------+-------------------------+-------------------------+--------+--------+