[37] | 1 | /** file: rodirgues.c |
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| 2 | ** author: Andrea Vedaldi |
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| 3 | ** description: Rodrigues formulas |
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| 4 | **/ |
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| 5 | |
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| 6 | /* AUTORIGHTS |
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| 7 | Copyright (C) 2006 Andrea Vedaldi |
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| 8 | |
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| 9 | This file is part of VLUtil. |
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| 10 | |
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| 11 | VLUtil is free software; you can redistribute it and/or modify |
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| 12 | it under the terms of the GNU General Public License as published by |
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| 13 | the Free Software Foundation; either version 2, or (at your option) |
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| 14 | any later version. |
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| 15 | |
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| 16 | This program is distributed in the hope that it will be useful, |
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| 17 | but WITHOUT ANY WARRANTY; without even the implied warranty of |
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| 18 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
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| 19 | GNU General Public License for more details. |
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| 20 | |
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| 21 | You should have received a copy of the GNU General Public License |
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| 22 | along with this program; if not, write to the Free Software Foundation, |
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| 23 | Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA. |
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| 24 | */ |
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| 25 | |
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| 26 | #include"rodrigues.h" |
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| 27 | #include<math.h> |
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| 28 | #include<limits.h> |
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| 29 | |
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| 30 | /** @brief Rodrigues' formula |
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| 31 | **/ |
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| 32 | void |
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| 33 | vl_rodrigues(double* R_pt, double* dR_pt, const double* om_pt) |
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| 34 | { |
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| 35 | /* |
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| 36 | Let |
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| 37 | |
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| 38 | th = |om|, r=w/th, |
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| 39 | sth=sin(th), cth=cos(th), |
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| 40 | ^om = hat(om) |
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| 41 | |
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| 42 | Then the rodrigues formula is an expansion of the exponential |
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| 43 | function: |
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| 44 | |
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| 45 | rodrigues(om) = exp ^om = I + ^r sth + ^r^2 (1 - cth). |
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| 46 | |
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| 47 | The derivative can be computed by elementary means and |
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| 48 | results: |
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| 49 | |
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| 50 | d(vec rodrigues(om)) sth d ^r 1 - cth d (^r)^2 |
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| 51 | -------------------- = ---- ----- + ------- -------- + |
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| 52 | d om^T th d r^T th d r^T |
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| 53 | |
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| 54 | sth 1 - cth |
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| 55 | + vec^r (cth - -----) + vec^r^2 (sth - 2-------)r^T |
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| 56 | th th |
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| 57 | */ |
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| 58 | |
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| 59 | #define OM(i) om_pt[(i)] |
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| 60 | #define R(i,j) R_pt[(i)+3*(j)] |
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| 61 | #define DR(i,j) dR_pt[(i)+9*(j)] |
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| 62 | |
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| 63 | const double small = 1e-6 ; |
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| 64 | |
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| 65 | double th = sqrt( OM(0)*OM(0) + |
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| 66 | OM(1)*OM(1) + |
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| 67 | OM(2)*OM(2) ) ; |
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| 68 | |
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| 69 | if( th < small ) { |
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| 70 | R(0,0) = 1.0 ; R(0,1) = 0.0 ; R(0,2) = 0.0 ; |
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| 71 | R(1,0) = 0.0 ; R(1,1) = 1.0 ; R(1,2) = 0.0 ; |
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| 72 | R(2,0) = 0.0 ; R(2,1) = 0.0 ; R(2,2) = 1.0 ; |
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| 73 | |
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| 74 | if(dR_pt) { |
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| 75 | DR(0,0) = 0 ; DR(0,1) = 0 ; DR(0,2) = 0 ; |
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| 76 | DR(1,0) = 0 ; DR(1,1) = 0 ; DR(1,2) = 1 ; |
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| 77 | DR(2,0) = 0 ; DR(2,1) = -1 ; DR(2,2) = 0 ; |
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| 78 | |
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| 79 | DR(3,0) = 0 ; DR(3,1) = 0 ; DR(3,2) = -1 ; |
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| 80 | DR(4,0) = 0 ; DR(4,1) = 0 ; DR(4,2) = 0 ; |
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| 81 | DR(5,0) = 1 ; DR(5,1) = 0 ; DR(5,2) = 0 ; |
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| 82 | |
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| 83 | DR(6,0) = 0 ; DR(6,1) = 1 ; DR(6,2) = 0 ; |
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| 84 | DR(7,0) = -1 ; DR(7,1) = 0 ; DR(7,2) = 0 ; |
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| 85 | DR(8,0) = 0 ; DR(8,1) = 0 ; DR(8,2) = 0 ; |
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| 86 | } |
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| 87 | return ; |
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| 88 | } |
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| 89 | |
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| 90 | { |
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| 91 | double x = OM(0) / th ; |
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| 92 | double y = OM(1) / th ; |
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| 93 | double z = OM(2) / th ; |
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| 94 | |
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| 95 | double xx = x*x ; |
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| 96 | double xy = x*y ; |
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| 97 | double xz = x*z ; |
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| 98 | double yy = y*y ; |
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| 99 | double yz = y*z ; |
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| 100 | double zz = z*z ; |
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| 101 | |
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| 102 | const double yx = xy ; |
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| 103 | const double zx = xz ; |
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| 104 | const double zy = yz ; |
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| 105 | |
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| 106 | double sth = sin(th) ; |
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| 107 | double cth = cos(th) ; |
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| 108 | double mcth = 1.0 - cth ; |
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| 109 | |
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| 110 | R(0,0) = 1 - mcth * (yy+zz) ; |
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| 111 | R(1,0) = sth*z + mcth * xy ; |
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| 112 | R(2,0) = - sth*y + mcth * xz ; |
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| 113 | |
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| 114 | R(0,1) = - sth*z + mcth * yx ; |
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| 115 | R(1,1) = 1 - mcth * (zz+xx) ; |
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| 116 | R(2,1) = sth*x + mcth * yz ; |
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| 117 | |
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| 118 | R(0,2) = sth*y + mcth * xz ; |
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| 119 | R(1,2) = - sth*x + mcth * yz ; |
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| 120 | R(2,2) = 1 - mcth * (xx+yy) ; |
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| 121 | |
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| 122 | if(dR_pt) { |
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| 123 | double a = sth / th ; |
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| 124 | double b = mcth / th ; |
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| 125 | double c = cth - a ; |
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| 126 | double d = sth - 2*b ; |
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| 127 | |
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| 128 | DR(0,0) = - d * (yy+zz) * x ; |
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| 129 | DR(1,0) = b*y + c * zx + d * xy * x ; |
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| 130 | DR(2,0) = b*z - c * yx + d * xz * x ; |
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| 131 | |
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| 132 | DR(3,0) = b*y - c * zx + d * xy * x ; |
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| 133 | DR(4,0) = -2*b*x - d * (zz+xx) * x ; |
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| 134 | DR(5,0) = a + c * xx + d * yz * x ; |
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| 135 | |
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| 136 | DR(6,0) = b*z + c * yx + d * zx * x ; |
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| 137 | DR(7,0) = -a - c * xx + d * zy * x ; |
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| 138 | DR(8,0) = -2*b*x - d * (yy+xx) * x ; |
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| 139 | |
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| 140 | DR(0,1) = -2*b*y - d * (yy+zz) * y ; |
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| 141 | DR(1,1) = b*x + c * zy + d * xy * y ; |
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| 142 | DR(2,1) = -a - c * yy + d * xz * y ; |
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| 143 | |
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| 144 | DR(3,1) = b*x - c * zy + d * xy * y ; |
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| 145 | DR(4,1) = - d * (zz+xx) * y ; |
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| 146 | DR(5,1) = b*z + c * xy + d * yz * y ; |
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| 147 | |
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| 148 | DR(6,1) = a + c * yy + d * zx * y ; |
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| 149 | DR(7,1) = b*z - c * xy + d * zy * y ; |
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| 150 | DR(8,1) = -2*b*y - d * (yy+xx) * y ; |
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| 151 | |
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| 152 | DR(0,2) = -2*b*z - d * (yy+zz) * z ; |
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| 153 | DR(1,2) = a + c * zz + d * xy * z ; |
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| 154 | DR(2,2) = b*x - c * yz + d * xz * z ; |
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| 155 | |
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| 156 | DR(3,2) = -a - c * zz + d * xy * z ; |
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| 157 | DR(4,2) = -2*b*z - d * (zz+xx) * z ; |
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| 158 | DR(5,2) = b*y + c * xz + d * yz * z ; |
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| 159 | |
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| 160 | DR(6,2) = b*x + c * yz + d * zx * z ; |
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| 161 | DR(7,2) = b*y - c * xz + d * zy * z ; |
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| 162 | DR(8,2) = - d * (yy+xx) * z ; |
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| 163 | } |
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| 164 | } |
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| 165 | |
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| 166 | #undef OM |
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| 167 | #undef R |
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| 168 | #undef DR |
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| 169 | |
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| 170 | } |
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| 171 | |
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| 172 | /** @brief Inverse Rodrigues formula |
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| 173 | ** |
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| 174 | ** This function computes the Rodrigues formula of the |
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| 175 | ** argument @a om_pt. The result is stored int the matrix |
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| 176 | ** @a R_pt. If @a dR_pt is non null, then the derivative |
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| 177 | ** of the Rodrigues formula is computed and stored into |
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| 178 | ** the matrix @a dR_pt. |
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| 179 | ** |
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| 180 | ** @param R_pt pointer to a 3x3 matrix (array of 9 doubles). |
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| 181 | ** @param dR_pt pointer to a 9x3 matrix (array of 27 doubles). |
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| 182 | ** @param om_pt pointer to a 3 vector (array of 3 dobules). |
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| 183 | **/ |
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| 184 | |
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| 185 | void vl_irodrigues(double* om_pt, double* dom_pt, const double* R_pt) |
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| 186 | { |
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| 187 | /* |
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| 188 | tr R - 1 1 [ R32 - R23 ] |
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| 189 | th = cos^{-1} --------, r = ------ [ R13 - R31 ], w = th r. |
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| 190 | 2 2 sth [ R12 - R21 ] |
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| 191 | |
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| 192 | sth = sin(th) |
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| 193 | |
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| 194 | dw th*cth-sth dw th [di3 dj2 - di2 dj3] |
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| 195 | ---- = ---------- r, ---- = ----- [di1 dj3 - di3 dj1]. |
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| 196 | dRii 2 sth^2 dRij 2 sth [di1 dj2 - di2 dj1] |
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| 197 | |
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| 198 | trace(A) < -1 only for small num. errors. |
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| 199 | */ |
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| 200 | |
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| 201 | #define OM(i) om_pt[(i)] |
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| 202 | #define DOM(i,j) dom_pt[(i)+3*(j)] |
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| 203 | #define R(i,j) R_pt[(i)+3*(j)] |
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| 204 | #define W(i,j) W_pt[(i)+3*(j)] |
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| 205 | |
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| 206 | const double small = 1e-6 ; |
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| 207 | |
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| 208 | double th = acos |
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| 209 | (0.5*(MAX(R(0,0)+R(1,1)+R(2,2),-1.0) - 1.0)) ; |
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| 210 | |
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| 211 | double sth = sin(th) ; |
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| 212 | double cth = cos(th) ; |
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| 213 | |
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| 214 | if(fabs(sth) < small && cth < 0) { |
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| 215 | /* |
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| 216 | we have this singularity when the rotation is about pi (or -pi) |
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| 217 | we use the fact that in this case |
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| 218 | |
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| 219 | hat( sqrt(1-cth) * r )^2 = W = (0.5*(R+R') - eye(3)) |
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| 220 | |
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| 221 | which gives |
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| 222 | |
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| 223 | (1-cth) rx^2 = 0.5 * (W(1,1)-W(2,2)-W(3,3)) |
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| 224 | (1-cth) ry^2 = 0.5 * (W(2,2)-W(3,3)-W(1,1)) |
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| 225 | (1-cth) rz^2 = 0.5 * (W(3,3)-W(1,1)-W(2,2)) |
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| 226 | */ |
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| 227 | |
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| 228 | double W_pt [9], x, y, z ; |
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| 229 | W_pt[0] = 0.5*( R(0,0) + R(0,0) ) - 1.0 ; |
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| 230 | |
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| 231 | W_pt[0] = 0.5*( R(1,0) + R(0,1) ) ; |
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| 232 | W_pt[0] = 0.5*( R(2,0) + R(0,2) ); |
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| 233 | |
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| 234 | W_pt[0] = 0.5*( R(0,1) + R(1,0) ); |
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| 235 | W_pt[0] = 0.5*( R(1,1) + R(1,1) ) - 1.0; |
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| 236 | W_pt[0] = 0.5*( R(2,1) + R(1,2) ); |
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| 237 | |
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| 238 | W_pt[0] = 0.5*( R(0,2) + R(2,0) ) ; |
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| 239 | W_pt[0] = 0.5*( R(1,2) + R(2,1) ) ; |
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| 240 | W_pt[0] = 0.5*( R(2,2) + R(2,2) ) - 1.0 ; |
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| 241 | |
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| 242 | /* these are only absolute values */ |
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| 243 | x = sqrt( 0.5 * (W(0,0)-W(1,1)-W(2,2)) ) ; |
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| 244 | y = sqrt( 0.5 * (W(1,1)-W(2,2)-W(0,0)) ) ; |
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| 245 | z = sqrt( 0.5 * (W(2,2)-W(0,0)-W(1,1)) ) ; |
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| 246 | |
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| 247 | /* set the biggest component to + and use the element of the |
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| 248 | ** matrix W to determine the sign of the other components |
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| 249 | ** then the solution is either (x,y,z) or its opposite */ |
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| 250 | if( x >= y && x >= z ) { |
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| 251 | y = (W(1,0) >=0) ? y : -y ; |
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| 252 | z = (W(2,0) >=0) ? z : -z ; |
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| 253 | } else if( y >= x && y >= z ) { |
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| 254 | z = (W(2,1) >=0) ? z : -z ; |
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| 255 | x = (W(1,0) >=0) ? x : -x ; |
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| 256 | } else { |
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| 257 | x = (W(2,0) >=0) ? x : -x ; |
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| 258 | y = (W(2,1) >=0) ? y : -y ; |
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| 259 | } |
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| 260 | |
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| 261 | /* we are left to chose between (x,y,z) and (-x,-y,-z) |
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| 262 | ** unfortunately we cannot (as the rotation is too close to pi) and |
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| 263 | ** we just keep what we have. */ |
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| 264 | { |
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| 265 | double scale = th / sqrt( 1 - cth ) ; |
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| 266 | OM(0) = scale * x ; |
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| 267 | OM(1) = scale * y ; |
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| 268 | OM(2) = scale * z ; |
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| 269 | |
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| 270 | if( dom_pt ) { |
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| 271 | int k ; |
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| 272 | for(k=0; k<3*9; ++k) |
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| 273 | dom_pt[k] = NAN ; |
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| 274 | } |
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| 275 | return ; |
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| 276 | } |
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| 277 | |
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| 278 | } else { |
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| 279 | double a = (fabs(sth) < small) ? 1 : th/sin(th) ; |
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| 280 | double b ; |
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| 281 | OM(0) = 0.5*a*(R(2,1) - R(1,2)) ; |
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| 282 | OM(1) = 0.5*a*(R(0,2) - R(2,0)) ; |
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| 283 | OM(2) = 0.5*a*(R(1,0) - R(0,1)) ; |
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| 284 | |
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| 285 | if( dom_pt ) { |
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| 286 | if( fabs(sth) < small ) { |
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| 287 | a = 0.5 ; |
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| 288 | b = 0 ; |
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| 289 | } else { |
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| 290 | a = th/(2*sth) ; |
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| 291 | b = (th*cth - sth)/(2*sth*sth)/th ; |
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| 292 | } |
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| 293 | |
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| 294 | DOM(0,0) = b*OM(0) ; |
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| 295 | DOM(1,0) = b*OM(1) ; |
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| 296 | DOM(2,0) = b*OM(2) ; |
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| 297 | |
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| 298 | DOM(0,1) = 0 ; |
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| 299 | DOM(1,1) = 0 ; |
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| 300 | DOM(2,1) = a ; |
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| 301 | |
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| 302 | DOM(0,2) = 0 ; |
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| 303 | DOM(1,2) = -a ; |
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| 304 | DOM(2,2) = 0 ; |
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| 305 | |
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| 306 | DOM(0,3) = 0 ; |
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| 307 | DOM(1,3) = 0 ; |
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| 308 | DOM(2,3) = -a ; |
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| 309 | |
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| 310 | DOM(0,4) = b*OM(0) ; |
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| 311 | DOM(1,4) = b*OM(1) ; |
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| 312 | DOM(2,4) = b*OM(2) ; |
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| 313 | |
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| 314 | DOM(0,5) = a ; |
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| 315 | DOM(1,5) = 0 ; |
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| 316 | DOM(2,5) = 0 ; |
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| 317 | |
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| 318 | DOM(0,6) = 0 ; |
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| 319 | DOM(1,6) = a ; |
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| 320 | DOM(2,6) = 0 ; |
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| 321 | |
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| 322 | DOM(0,7) = -a ; |
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| 323 | DOM(1,7) = 0 ; |
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| 324 | DOM(2,7) = 0 ; |
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| 325 | |
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| 326 | DOM(0,8) = b*OM(0) ; |
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| 327 | DOM(1,8) = b*OM(1) ; |
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| 328 | DOM(2,8) = b*OM(2) ; |
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| 329 | } |
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| 330 | } |
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| 331 | |
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| 332 | #undef OM |
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| 333 | #undef DOM |
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| 334 | #undef R |
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| 335 | #undef W |
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| 336 | } |
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