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|00000000| 00 09 72 6f 74 61 74 69 | 6f 6e 73 00 00 00 00 00 |..rotati|ons.....|
|00000010| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00000020| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00000030| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00000040| 00 54 45 58 54 4d 50 61 | 64 01 00 00 00 00 00 00 |.TEXTMPa|d.......|
|00000050| 00 00 00 00 00 06 f2 00 | 00 01 8e ab c9 d4 cc ab |........|........|
|00000060| c9 d4 cc 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00000070| 00 00 00 00 00 00 00 00 | 00 00 81 81 5d 86 00 00 |........|....]...|
|00000080| 2d 2d 20 52 6f 74 61 74 | 69 6f 6e 73 0d 2d 2d 20 |-- Rotat|ions.-- |
|00000090| 62 79 20 48 61 6e 6b 2e | 44 6f 6c 62 65 6e 40 55 |by Hank.|Dolben@U|
|000000a0| 4e 48 2e 65 64 75 20 31 | 39 39 34 20 4d 61 79 20 |NH.edu 1|994 May |
|000000b0| 31 38 0d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 2d 2d 2d 2d |18.-----|--------|
|000000c0| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 2d 2d 2d 2d |--------|--------|
|000000d0| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 2d 2d 2d 2d |--------|--------|
|000000e0| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 2d 2d 2d 2d |--------|--------|
|000000f0| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 2d 2d 2d 0d |--------|-------.|
|00000100| 2d 2d 20 61 20 69 73 20 | 61 20 76 65 63 74 6f 72 |-- a is |a vector|
|00000110| 20 69 6e 20 63 6f 6f 72 | 64 69 6e 61 74 65 20 73 | in coor|dinate s|
|00000120| 79 73 74 65 6d 20 41 2e | 0d 2d 2d 20 61 27 20 69 |ystem A.|.-- a' i|
|00000130| 73 20 74 68 65 20 76 65 | 63 74 6f 72 20 69 6e 20 |s the ve|ctor in |
|00000140| 63 6f 6f 72 64 69 6e 61 | 74 65 20 73 79 73 74 65 |coordina|te syste|
|00000150| 6d 20 41 27 2e 0d 2d 2d | 20 41 27 20 69 73 20 72 |m A'..--| A' is r|
|00000160| 6f 74 61 74 65 64 20 62 | 79 20 79 61 77 20 61 72 |otated b|y yaw ar|
|00000170| 6f 75 6e 64 20 7a 2c 20 | 70 69 74 63 68 20 61 72 |ound z, |pitch ar|
|00000180| 6f 75 6e 64 20 79 2c 20 | 61 6e 64 20 72 6f 6c 6c |ound y, |and roll|
|00000190| 20 61 72 6f 75 6e 64 20 | 78 2c 20 66 72 6f 6d 20 | around |x, from |
|000001a0| 41 2e 0d 2d 2d 20 52 20 | 3d 20 72 6f 74 61 74 69 |A..-- R |= rotati|
|000001b0| 6f 6e 28 79 61 77 2c 70 | 69 74 63 68 2c 72 6f 6c |on(yaw,p|itch,rol|
|000001c0| 6c 29 0d 2d 2d 20 61 27 | 20 3d 20 52 61 20 3d 20 |l).-- a'| = Ra = |
|000001d0| 74 72 61 6e 73 66 6f 72 | 6d 28 52 2c 61 29 0d 2d |transfor|m(R,a).-|
|000001e0| 2d 20 74 68 65 20 72 6f | 77 73 20 6f 66 20 52 20 |- the ro|ws of R |
|000001f0| 61 72 65 20 74 68 65 20 | 62 61 73 69 73 20 76 65 |are the |basis ve|
|00000200| 63 74 6f 72 73 20 6f 66 | 20 41 27 20 69 6e 20 41 |ctors of| A' in A|
|00000210| 2e 0d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 2d 2d 2d 2d |..------|--------|
|00000220| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 2d 2d 2d 2d |--------|--------|
|00000230| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 2d 2d 2d 2d |--------|--------|
|00000240| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 2d 2d 2d 2d |--------|--------|
|00000250| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 2d 2d 0d 72 |--------|------.r|
|00000260| 6f 74 61 74 69 6f 6e 28 | 79 2c 70 2c 72 29 20 3d |otation(|y,p,r) =|
|00000270| 20 7b 7b 63 6f 73 28 70 | 29 2a 20 20 20 20 20 20 | {{cos(p|)*      |
|00000280| 20 63 6f 73 28 79 29 2c | 0d 20 20 20 20 20 20 20 | cos(y),|.       |
|00000290| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 63 6f 73 |        |     cos|
|000002a0| 28 70 29 2a 20 20 20 20 | 20 20 20 73 69 6e 28 79 |(p)*    |   sin(y|
|000002b0| 29 2c 0d 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |),.     |        |
|000002c0| 20 20 20 20 20 20 2d 73 | 69 6e 28 70 29 20 20 20 |      -s|in(p)   |
|000002d0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000002e0| 20 20 20 20 20 20 20 20 | 20 7d 2c 0d 20 20 20 20 |        | },.    |
|000002f0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 7b |        |       {|
|00000300| 73 69 6e 28 70 29 2a 73 | 69 6e 28 72 29 2a 63 6f |sin(p)*s|in(r)*co|
|00000310| 73 28 79 29 2d 63 6f 73 | 28 72 29 2a 73 69 6e 28 |s(y)-cos|(r)*sin(|
|00000320| 79 29 2c 0d 20 20 20 20 | 20 20 20 20 20 20 20 20 |y),.    |        |
|00000330| 20 20 20 20 20 20 20 20 | 73 69 6e 28 70 29 2a 73 |        |sin(p)*s|
|00000340| 69 6e 28 72 29 2a 73 69 | 6e 28 79 29 2b 63 6f 73 |in(r)*si|n(y)+cos|
|00000350| 28 72 29 2a 63 6f 73 28 | 79 29 2c 0d 20 20 20 20 |(r)*cos(|y),.    |
|00000360| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000370| 63 6f 73 28 70 29 2a 73 | 69 6e 28 72 29 20 20 20 |cos(p)*s|in(r)   |
|00000380| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000390| 20 20 7d 2c 0d 20 20 20 | 20 20 20 20 20 20 20 20 |  },.   |        |
|000003a0| 20 20 20 20 20 20 20 20 | 7b 73 69 6e 28 70 29 2a |        |{sin(p)*|
|000003b0| 63 6f 73 28 72 29 2a 63 | 6f 73 28 79 29 2b 73 69 |cos(r)*c|os(y)+si|
|000003c0| 6e 28 72 29 2a 73 69 6e | 28 79 29 2c 0d 20 20 20 |n(r)*sin|(y),.   |
|000003d0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000003e0| 20 73 69 6e 28 70 29 2a | 63 6f 73 28 72 29 2a 73 | sin(p)*|cos(r)*s|
|000003f0| 69 6e 28 79 29 2d 73 69 | 6e 28 72 29 2a 63 6f 73 |in(y)-si|n(r)*cos|
|00000400| 28 79 29 2c 0d 20 20 20 | 20 20 20 20 20 20 20 20 |(y),.   |        |
|00000410| 20 20 20 20 20 20 20 20 | 20 63 6f 73 28 70 29 2a |        | cos(p)*|
|00000420| 63 6f 73 28 72 29 20 20 | 20 20 20 20 20 20 20 20 |cos(r)  |        |
|00000430| 20 20 20 20 20 20 20 20 | 20 20 20 7d 7d 0d 0d 2d |        |   }}..-|
|00000440| 2d 20 77 68 65 6e 20 7c | 70 69 74 63 68 7c 20 3d |- when ||pitch| =|
|00000450| 20 39 30 20 28 63 6f 73 | 28 70 69 74 63 68 29 3d | 90 (cos|(pitch)=|
|00000460| 30 29 2c 20 74 68 65 20 | 70 61 72 74 69 74 69 6f |0), the |partitio|
|00000470| 6e 20 6f 66 20 79 61 77 | 20 61 6e 64 20 72 6f 6c |n of yaw| and rol|
|00000480| 6c 20 69 73 20 61 72 62 | 69 74 72 61 72 79 2c 0d |l is arb|itrary,.|
|00000490| 2d 2d 20 73 61 79 20 72 | 6f 6c 6c 20 3d 20 30 2c |-- say r|oll = 0,|
|000004a0| 20 74 68 65 6e 3a 0d 2d | 2d 20 20 20 52 5b 32 2c | then:.-|-   R[2,|
|000004b0| 31 5d 20 3d 20 73 69 6e | 28 70 29 2a 73 69 6e 28 |1] = sin|(p)*sin(|
|000004c0| 72 29 2a 63 6f 73 28 79 | 29 2d 63 6f 73 28 72 29 |r)*cos(y|)-cos(r)|
|000004d0| 2a 73 69 6e 28 79 29 20 | 3d 20 2d 73 69 6e 28 79 |*sin(y) |= -sin(y|
|000004e0| 29 0d 2d 2d 20 20 20 52 | 5b 32 2c 32 5d 20 3d 20 |).--   R|[2,2] = |
|000004f0| 73 69 6e 28 70 29 2a 73 | 69 6e 28 72 29 2a 73 69 |sin(p)*s|in(r)*si|
|00000500| 6e 28 79 29 2b 63 6f 73 | 28 72 29 2a 63 6f 73 28 |n(y)+cos|(r)*cos(|
|00000510| 79 29 20 3d 20 20 63 6f | 73 28 79 29 0d 0d 59 61 |y) =  co|s(y)..Ya|
|00000520| 77 28 52 29 20 20 20 3d | 20 61 74 61 6e 32 28 2d |w(R)   =| atan2(-|
|00000530| 52 5b 32 2c 31 5d 2c 52 | 5b 32 2c 32 5d 29 20 77 |R[2,1],R|[2,2]) w|
|00000540| 68 65 6e 20 52 5b 31 2c | 32 5d 3d 30 20 61 6e 64 |hen R[1,|2]=0 and|
|00000550| 20 52 5b 31 2c 31 5d 3d | 30 2c 0d 20 20 20 20 20 | R[1,1]=|0,.     |
|00000560| 20 20 20 20 20 20 61 74 | 61 6e 32 28 52 5b 31 2c |      at|an2(R[1,|
|00000570| 32 5d 2c 52 5b 31 2c 31 | 5d 29 20 6f 74 68 65 72 |2],R[1,1|]) other|
|00000580| 77 69 73 65 0d 0d 50 69 | 74 63 68 28 52 29 20 3d |wise..Pi|tch(R) =|
|00000590| 20 61 73 69 6e 28 2d 52 | 5b 31 2c 33 5d 29 20 20 | asin(-R|[1,3])  |
|000005a0| 20 20 20 20 20 20 2d 2d | 20 20 20 2d 39 30 20 b2 |      --|   -90 .|
|000005b0| 20 50 69 74 63 68 28 52 | 29 20 b2 20 39 30 0d 0d | Pitch(R|) . 90..|
|000005c0| 52 6f 6c 6c 28 52 29 20 | 20 3d 20 30 20 77 68 65 |Roll(R) | = 0 whe|
|000005d0| 6e 20 52 5b 32 2c 33 5d | 3d 30 20 61 6e 64 20 52 |n R[2,3]|=0 and R|
|000005e0| 5b 33 2c 33 5d 3d 30 2c | 0d 20 20 20 20 20 20 20 |[3,3]=0,|.       |
|000005f0| 20 20 20 20 61 74 61 6e | 32 28 52 5b 32 2c 33 5d |    atan|2(R[2,3]|
|00000600| 2c 52 5b 33 2c 33 5d 29 | 20 6f 74 68 65 72 77 69 |,R[3,3])| otherwi|
|00000610| 73 65 0d 0d 72 6f 74 61 | 74 65 28 79 61 77 2c 70 |se..rota|te(yaw,p|
|00000620| 69 74 63 68 2c 72 6f 6c | 6c 2c 76 29 20 3d 20 74 |itch,rol|l,v) = t|
|00000630| 72 61 6e 73 66 6f 72 6d | 28 72 6f 74 61 74 69 6f |ransform|(rotatio|
|00000640| 6e 28 79 61 77 2c 70 69 | 74 63 68 2c 72 6f 6c 6c |n(yaw,pi|tch,roll|
|00000650| 29 2c 76 29 0d 0d 74 72 | 61 6e 73 66 6f 72 6d 28 |),v)..tr|ansform(|
|00000660| 52 2c 62 29 5b 69 5d 20 | 3d 20 64 6f 74 28 52 5b |R,b)[i] |= dot(R[|
|00000670| 69 5d 2c 62 29 20 64 69 | 6d 20 5b 63 6f 75 6e 74 |i],b) di|m [count|
|00000680| 28 52 29 5d 0d 0d 2d 2d | 20 63 6f 6d 70 6f 73 65 |(R)]..--| compose|
|00000690| 20 74 68 65 20 74 72 61 | 6e 73 66 6f 72 6d 61 74 | the tra|nsformat|
|000006a0| 69 6f 6e 73 20 73 75 63 | 68 20 74 68 61 74 20 52 |ions suc|h that R|
|000006b0| 32 20 61 70 70 6c 69 65 | 73 20 61 66 74 65 72 20 |2 applie|s after |
|000006c0| 52 31 0d 6d 75 6c 74 69 | 70 6c 79 28 52 32 2c 52 |R1.multi|ply(R2,R|
|000006d0| 31 29 5b 69 2c 6a 5d 20 | 3d 20 64 6f 74 28 52 32 |1)[i,j] |= dot(R2|
|000006e0| 5b 69 5d 2c 74 72 61 6e | 73 70 6f 73 65 28 52 31 |[i],tran|spose(R1|
|000006f0| 29 5b 6a 5d 29 20 64 69 | 6d 20 5b 63 6f 75 6e 74 |)[j]) di|m [count|
|00000700| 28 52 32 29 2c 63 6f 75 | 6e 74 28 52 31 5b 31 5d |(R2),cou|nt(R1[1]|
|00000710| 29 5d 0d 0d 74 72 61 6e | 73 70 6f 73 65 28 52 29 |)]..tran|spose(R)|
|00000720| 5b 69 2c 6a 5d 20 3d 20 | 52 5b 6a 2c 69 5d 20 64 |[i,j] = |R[j,i] d|
|00000730| 69 6d 20 5b 63 6f 75 6e | 74 28 52 5b 31 5d 29 2c |im [coun|t(R[1]),|
|00000740| 63 6f 75 6e 74 28 52 29 | 5d 0d 0d 64 6f 74 28 61 |count(R)|]..dot(a|
|00000750| 2c 62 29 20 3d 20 73 75 | 6d 28 61 5b 69 5d 2a 62 |,b) = su|m(a[i]*b|
|00000760| 5b 69 5d 2c 69 2c 31 2c | 63 6f 75 6e 74 28 61 29 |[i],i,1,|count(a)|
|00000770| 29 0d 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |).......|........|
|00000780| 00 00 01 00 00 00 01 3c | 00 00 00 3c 00 00 00 52 |.......<|...<...R|
|00000790| 00 00 00 9d 05 45 78 63 | 65 6c 02 00 00 00 41 50 |.....Exc|el....AP|
|000007a0| 50 4c 58 43 45 4c 21 40 | 00 68 ff c0 00 00 00 00 |PLXCEL!@|.h......|
|000007b0| 09 72 6f 74 61 74 69 6f | 6e 73 02 00 00 00 54 45 |.rotatio|ns....TE|
|000007c0| 58 54 4d 50 61 64 01 00 | 00 34 01 00 00 00 00 00 |XTMPad..|.4......|
|000007d0| 00 00 54 45 58 54 4d 50 | 61 64 01 00 00 34 01 00 |..TEXTMP|ad...4..|
|000007e0| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|000007f0| 00 00 a7 b8 fc 1b 00 00 | 06 f2 00 00 01 8e 00 00 |........|........|
|00000800| 0b 00 00 00 00 9d 05 46 | 65 74 63 68 02 00 00 00 |.......F|etch....|
|00000810| 41 50 50 4c 46 54 43 68 | 21 00 00 34 00 80 00 00 |APPLFTCh|!..4....|
|00000820| 00 00 23 9f 00 00 00 00 | 00 00 00 00 00 00 00 00 |..#.....|........|
|00000830| 00 03 1c c8 00 03 20 00 | a6 c9 23 f7 a6 c9 24 14 |...... .|..#...$.|
|00000840| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00000850| 00 00 23 9e 00 00 00 00 | 00 00 00 00 00 00 00 00 |..#.....|........|
|00000860| 00 00 53 40 00 01 53 6f | 00 02 53 82 00 01 00 00 |..S@..So|..S.....|
|00000870| 00 00 00 00 0b 00 00 00 | 00 9d 05 46 65 74 63 68 |........|...Fetch|
|00000880| 00 00 00 20 00 00 00 00 | 00 02 3f f9 8e fa 35 12 |... ....|..?...5.|
|00000890| 94 e9 c8 ae 01 f4 00 88 | 00 06 00 c8 01 06 01 2c |........|.......,|
|000008a0| 00 f8 00 28 00 00 00 14 | 00 04 06 4d 6f 6e 61 63 |...(....|...Monac|
|000008b0| 6f 01 39 06 4d 6f 6e 61 | 63 6f 01 39 00 00 01 00 |o.9.Mona|co.9....|
|000008c0| 00 00 01 3c 00 00 00 3c | 00 00 00 52 00 25 54 e4 |...<...<|...R.%T.|
|000008d0| 08 74 00 00 00 1c 00 46 | 00 01 50 52 65 66 00 00 |.t.....F|..PRef..|
|000008e0| 00 12 53 54 52 23 00 00 | 00 1e 00 80 ff ff 00 00 |..STR#..|........|
|000008f0| 00 00 00 25 54 cc 00 81 | 00 00 00 00 00 24 00 25 |...%T...|.....$.%|
|00000900| 54 44 0b 66 6f 6e 74 20 | 26 20 73 69 7a 65 00 00 |TD.font |& size..|
|00000910| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00000920| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00000930| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00000940| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00000950| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00000960| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00000970| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
+--------+-------------------------+-------------------------+--------+--------+