| 1 |
/* |
| 2 |
* Copyright (C) 2000-2004 Object Oriented Parallel Simulation Engine (OOPSE) project |
| 3 |
* |
| 4 |
* Contact: oopse@oopse.org |
| 5 |
* |
| 6 |
* This program is free software; you can redistribute it and/or |
| 7 |
* modify it under the terms of the GNU Lesser General Public License |
| 8 |
* as published by the Free Software Foundation; either version 2.1 |
| 9 |
* of the License, or (at your option) any later version. |
| 10 |
* All we ask is that proper credit is given for our work, which includes |
| 11 |
* - but is not limited to - adding the above copyright notice to the beginning |
| 12 |
* of your source code files, and to any copyright notice that you may distribute |
| 13 |
* with programs based on this work. |
| 14 |
* |
| 15 |
* This program is distributed in the hope that it will be useful, |
| 16 |
* but WITHOUT ANY WARRANTY; without even the implied warranty of |
| 17 |
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
| 18 |
* GNU Lesser General Public License for more details. |
| 19 |
* |
| 20 |
* You should have received a copy of the GNU Lesser General Public License |
| 21 |
* along with this program; if not, write to the Free Software |
| 22 |
* Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA. |
| 23 |
* |
| 24 |
*/ |
| 25 |
|
| 26 |
/** |
| 27 |
* @file SquareMatrix3.hpp |
| 28 |
* @author Teng Lin |
| 29 |
* @date 10/11/2004 |
| 30 |
* @version 1.0 |
| 31 |
*/ |
| 32 |
#ifndef MATH_SQUAREMATRIX3_HPP |
| 33 |
#define MATH_SQUAREMATRIX3_HPP |
| 34 |
|
| 35 |
#include "Quaternion.hpp" |
| 36 |
#include "SquareMatrix.hpp" |
| 37 |
#include "Vector3.hpp" |
| 38 |
|
| 39 |
namespace oopse { |
| 40 |
|
| 41 |
template<typename Real> |
| 42 |
class SquareMatrix3 : public SquareMatrix<Real, 3> { |
| 43 |
public: |
| 44 |
|
| 45 |
typedef Real ElemType; |
| 46 |
typedef Real* ElemPoinerType; |
| 47 |
|
| 48 |
/** default constructor */ |
| 49 |
SquareMatrix3() : SquareMatrix<Real, 3>() { |
| 50 |
} |
| 51 |
|
| 52 |
/** Constructs and initializes every element of this matrix to a scalar */ |
| 53 |
SquareMatrix3(Real s) : SquareMatrix<Real,3>(s){ |
| 54 |
} |
| 55 |
|
| 56 |
/** Constructs and initializes from an array */ |
| 57 |
SquareMatrix3(Real* array) : SquareMatrix<Real,3>(array){ |
| 58 |
} |
| 59 |
|
| 60 |
|
| 61 |
/** copy constructor */ |
| 62 |
SquareMatrix3(const SquareMatrix<Real, 3>& m) : SquareMatrix<Real, 3>(m) { |
| 63 |
} |
| 64 |
|
| 65 |
SquareMatrix3( const Vector3<Real>& eulerAngles) { |
| 66 |
setupRotMat(eulerAngles); |
| 67 |
} |
| 68 |
|
| 69 |
SquareMatrix3(Real phi, Real theta, Real psi) { |
| 70 |
setupRotMat(phi, theta, psi); |
| 71 |
} |
| 72 |
|
| 73 |
SquareMatrix3(const Quaternion<Real>& q) { |
| 74 |
setupRotMat(q); |
| 75 |
|
| 76 |
} |
| 77 |
|
| 78 |
SquareMatrix3(Real w, Real x, Real y, Real z) { |
| 79 |
setupRotMat(w, x, y, z); |
| 80 |
} |
| 81 |
|
| 82 |
/** copy assignment operator */ |
| 83 |
SquareMatrix3<Real>& operator =(const SquareMatrix<Real, 3>& m) { |
| 84 |
if (this == &m) |
| 85 |
return *this; |
| 86 |
SquareMatrix<Real, 3>::operator=(m); |
| 87 |
return *this; |
| 88 |
} |
| 89 |
|
| 90 |
/** |
| 91 |
* Sets this matrix to a rotation matrix by three euler angles |
| 92 |
* @ param euler |
| 93 |
*/ |
| 94 |
void setupRotMat(const Vector3<Real>& eulerAngles) { |
| 95 |
setupRotMat(eulerAngles[0], eulerAngles[1], eulerAngles[2]); |
| 96 |
} |
| 97 |
|
| 98 |
/** |
| 99 |
* Sets this matrix to a rotation matrix by three euler angles |
| 100 |
* @param phi |
| 101 |
* @param theta |
| 102 |
* @psi theta |
| 103 |
*/ |
| 104 |
void setupRotMat(Real phi, Real theta, Real psi) { |
| 105 |
Real sphi, stheta, spsi; |
| 106 |
Real cphi, ctheta, cpsi; |
| 107 |
|
| 108 |
sphi = sin(phi); |
| 109 |
stheta = sin(theta); |
| 110 |
spsi = sin(psi); |
| 111 |
cphi = cos(phi); |
| 112 |
ctheta = cos(theta); |
| 113 |
cpsi = cos(psi); |
| 114 |
|
| 115 |
data_[0][0] = cpsi * cphi - ctheta * sphi * spsi; |
| 116 |
data_[0][1] = cpsi * sphi + ctheta * cphi * spsi; |
| 117 |
data_[0][2] = spsi * stheta; |
| 118 |
|
| 119 |
data_[1][0] = -spsi * ctheta - ctheta * sphi * cpsi; |
| 120 |
data_[1][1] = -spsi * stheta + ctheta * cphi * cpsi; |
| 121 |
data_[1][2] = cpsi * stheta; |
| 122 |
|
| 123 |
data_[2][0] = stheta * sphi; |
| 124 |
data_[2][1] = -stheta * cphi; |
| 125 |
data_[2][2] = ctheta; |
| 126 |
} |
| 127 |
|
| 128 |
|
| 129 |
/** |
| 130 |
* Sets this matrix to a rotation matrix by quaternion |
| 131 |
* @param quat |
| 132 |
*/ |
| 133 |
void setupRotMat(const Quaternion<Real>& quat) { |
| 134 |
setupRotMat(quat.w(), quat.x(), quat.y(), quat.z()); |
| 135 |
} |
| 136 |
|
| 137 |
/** |
| 138 |
* Sets this matrix to a rotation matrix by quaternion |
| 139 |
* @param w the first element |
| 140 |
* @param x the second element |
| 141 |
* @param y the third element |
| 142 |
* @param z the fourth element |
| 143 |
*/ |
| 144 |
void setupRotMat(Real w, Real x, Real y, Real z) { |
| 145 |
Quaternion<Real> q(w, x, y, z); |
| 146 |
*this = q.toRotationMatrix3(); |
| 147 |
} |
| 148 |
|
| 149 |
/** |
| 150 |
* Returns the quaternion from this rotation matrix |
| 151 |
* @return the quaternion from this rotation matrix |
| 152 |
* @exception invalid rotation matrix |
| 153 |
*/ |
| 154 |
Quaternion<Real> toQuaternion() { |
| 155 |
Quaternion<Real> q; |
| 156 |
Real t, s; |
| 157 |
Real ad1, ad2, ad3; |
| 158 |
t = data_[0][0] + data_[1][1] + data_[2][2] + 1.0; |
| 159 |
|
| 160 |
if( t > 0.0 ){ |
| 161 |
|
| 162 |
s = 0.5 / sqrt( t ); |
| 163 |
q[0] = 0.25 / s; |
| 164 |
q[1] = (data_[1][2] - data_[2][1]) * s; |
| 165 |
q[2] = (data_[2][0] - data_[0][2]) * s; |
| 166 |
q[3] = (data_[0][1] - data_[1][0]) * s; |
| 167 |
} else { |
| 168 |
|
| 169 |
ad1 = fabs( data_[0][0] ); |
| 170 |
ad2 = fabs( data_[1][1] ); |
| 171 |
ad3 = fabs( data_[2][2] ); |
| 172 |
|
| 173 |
if( ad1 >= ad2 && ad1 >= ad3 ){ |
| 174 |
|
| 175 |
s = 2.0 * sqrt( 1.0 + data_[0][0] - data_[1][1] - data_[2][2] ); |
| 176 |
q[0] = (data_[1][2] + data_[2][1]) / s; |
| 177 |
q[1] = 0.5 / s; |
| 178 |
q[2] = (data_[0][1] + data_[1][0]) / s; |
| 179 |
q[3] = (data_[0][2] + data_[2][0]) / s; |
| 180 |
} else if ( ad2 >= ad1 && ad2 >= ad3 ) { |
| 181 |
s = sqrt( 1.0 + data_[1][1] - data_[0][0] - data_[2][2] ) * 2.0; |
| 182 |
q[0] = (data_[0][2] + data_[2][0]) / s; |
| 183 |
q[1] = (data_[0][1] + data_[1][0]) / s; |
| 184 |
q[2] = 0.5 / s; |
| 185 |
q[3] = (data_[1][2] + data_[2][1]) / s; |
| 186 |
} else { |
| 187 |
|
| 188 |
s = sqrt( 1.0 + data_[2][2] - data_[0][0] - data_[1][1] ) * 2.0; |
| 189 |
q[0] = (data_[0][1] + data_[1][0]) / s; |
| 190 |
q[1] = (data_[0][2] + data_[2][0]) / s; |
| 191 |
q[2] = (data_[1][2] + data_[2][1]) / s; |
| 192 |
q[3] = 0.5 / s; |
| 193 |
} |
| 194 |
} |
| 195 |
|
| 196 |
return q; |
| 197 |
|
| 198 |
} |
| 199 |
|
| 200 |
/** |
| 201 |
* Returns the euler angles from this rotation matrix |
| 202 |
* @return the euler angles in a vector |
| 203 |
* @exception invalid rotation matrix |
| 204 |
* We use so-called "x-convention", which is the most common definition. |
| 205 |
* In this convention, the rotation given by Euler angles (phi, theta, psi), where the first |
| 206 |
* rotation is by an angle phi about the z-axis, the second is by an angle |
| 207 |
* theta (0 <= theta <= 180)about the x-axis, and thethird is by an angle psi about the |
| 208 |
* z-axis (again). |
| 209 |
*/ |
| 210 |
Vector3<Real> toEulerAngles() { |
| 211 |
Vector3<Real> myEuler; |
| 212 |
Real phi,theta,psi,eps; |
| 213 |
Real ctheta,stheta; |
| 214 |
|
| 215 |
// set the tolerance for Euler angles and rotation elements |
| 216 |
|
| 217 |
theta = acos(std::min(1.0, std::max(-1.0,data_[2][2]))); |
| 218 |
ctheta = data_[2][2]; |
| 219 |
stheta = sqrt(1.0 - ctheta * ctheta); |
| 220 |
|
| 221 |
// when sin(theta) is close to 0, we need to consider singularity |
| 222 |
// In this case, we can assign an arbitary value to phi (or psi), and then determine |
| 223 |
// the psi (or phi) or vice-versa. We'll assume that phi always gets the rotation, and psi is 0 |
| 224 |
// in cases of singularity. |
| 225 |
// we use atan2 instead of atan, since atan2 will give us -Pi to Pi. |
| 226 |
// Since 0 <= theta <= 180, sin(theta) will be always non-negative. Therefore, it never |
| 227 |
// change the sign of both of the parameters passed to atan2. |
| 228 |
|
| 229 |
if (fabs(stheta) <= oopse::epsilon){ |
| 230 |
psi = 0.0; |
| 231 |
phi = atan2(-data_[1][0], data_[0][0]); |
| 232 |
} |
| 233 |
// we only have one unique solution |
| 234 |
else{ |
| 235 |
phi = atan2(data_[2][0], -data_[2][1]); |
| 236 |
psi = atan2(data_[0][2], data_[1][2]); |
| 237 |
} |
| 238 |
|
| 239 |
//wrap phi and psi, make sure they are in the range from 0 to 2*Pi |
| 240 |
if (phi < 0) |
| 241 |
phi += M_PI; |
| 242 |
|
| 243 |
if (psi < 0) |
| 244 |
psi += M_PI; |
| 245 |
|
| 246 |
myEuler[0] = phi; |
| 247 |
myEuler[1] = theta; |
| 248 |
myEuler[2] = psi; |
| 249 |
|
| 250 |
return myEuler; |
| 251 |
} |
| 252 |
|
| 253 |
/** Returns the determinant of this matrix. */ |
| 254 |
Real determinant() const { |
| 255 |
Real x,y,z; |
| 256 |
|
| 257 |
x = data_[0][0] * (data_[1][1] * data_[2][2] - data_[1][2] * data_[2][1]); |
| 258 |
y = data_[0][1] * (data_[1][2] * data_[2][0] - data_[1][0] * data_[2][2]); |
| 259 |
z = data_[0][2] * (data_[1][0] * data_[2][1] - data_[1][1] * data_[2][0]); |
| 260 |
|
| 261 |
return(x + y + z); |
| 262 |
} |
| 263 |
|
| 264 |
/** |
| 265 |
* Sets the value of this matrix to the inversion of itself. |
| 266 |
* @note since simple algorithm can be applied to inverse the 3 by 3 matrix, we hide the |
| 267 |
* implementation of inverse in SquareMatrix class |
| 268 |
*/ |
| 269 |
SquareMatrix3<Real> inverse() { |
| 270 |
SquareMatrix3<Real> m; |
| 271 |
double det = determinant(); |
| 272 |
if (fabs(det) <= oopse::epsilon) { |
| 273 |
//"The method was called on a matrix with |determinant| <= 1e-6.", |
| 274 |
//"This is a runtime or a programming error in your application."); |
| 275 |
} |
| 276 |
|
| 277 |
m(0, 0) = data_[1][1]*data_[2][2] - data_[1][2]*data_[2][1]; |
| 278 |
m(1, 0) = data_[1][2]*data_[2][0] - data_[1][0]*data_[2][2]; |
| 279 |
m(2, 0) = data_[1][0]*data_[2][1] - data_[1][1]*data_[2][0]; |
| 280 |
m(0, 1) = data_[2][1]*data_[0][2] - data_[2][2]*data_[0][1]; |
| 281 |
m(1, 1) = data_[2][2]*data_[0][0] - data_[2][0]*data_[0][2]; |
| 282 |
m(2, 1) = data_[2][0]*data_[0][1] - data_[2][1]*data_[0][0]; |
| 283 |
m(0, 2) = data_[0][1]*data_[1][2] - data_[0][2]*data_[1][1]; |
| 284 |
m(1, 2) = data_[0][2]*data_[1][0] - data_[0][0]*data_[1][2]; |
| 285 |
m(2, 2) = data_[0][0]*data_[1][1] - data_[0][1]*data_[1][0]; |
| 286 |
|
| 287 |
m /= det; |
| 288 |
return m; |
| 289 |
} |
| 290 |
/** |
| 291 |
* Extract the eigenvalues and eigenvectors from a 3x3 matrix. |
| 292 |
* The eigenvectors (the columns of V) will be normalized. |
| 293 |
* The eigenvectors are aligned optimally with the x, y, and z |
| 294 |
* axes respectively. |
| 295 |
* @param a symmetric matrix whose eigenvectors are to be computed. On return, the matrix is |
| 296 |
* overwritten |
| 297 |
* @param w will contain the eigenvalues of the matrix On return of this function |
| 298 |
* @param v the columns of this matrix will contain the eigenvectors. The eigenvectors are |
| 299 |
* normalized and mutually orthogonal. |
| 300 |
* @warning a will be overwritten |
| 301 |
*/ |
| 302 |
static void diagonalize(SquareMatrix3<Real>& a, Vector3<Real>& w, SquareMatrix3<Real>& v); |
| 303 |
}; |
| 304 |
/*========================================================================= |
| 305 |
|
| 306 |
Program: Visualization Toolkit |
| 307 |
Module: $RCSfile: SquareMatrix3.hpp,v $ |
| 308 |
|
| 309 |
Copyright (c) Ken Martin, Will Schroeder, Bill Lorensen |
| 310 |
All rights reserved. |
| 311 |
See Copyright.txt or http://www.kitware.com/Copyright.htm for details. |
| 312 |
|
| 313 |
This software is distributed WITHOUT ANY WARRANTY; without even |
| 314 |
the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR |
| 315 |
PURPOSE. See the above copyright notice for more information. |
| 316 |
|
| 317 |
=========================================================================*/ |
| 318 |
template<typename Real> |
| 319 |
void SquareMatrix3<Real>::diagonalize(SquareMatrix3<Real>& a, Vector3<Real>& w, |
| 320 |
SquareMatrix3<Real>& v) { |
| 321 |
int i,j,k,maxI; |
| 322 |
Real tmp, maxVal; |
| 323 |
Vector3<Real> v_maxI, v_k, v_j; |
| 324 |
|
| 325 |
// diagonalize using Jacobi |
| 326 |
jacobi(a, w, v); |
| 327 |
// if all the eigenvalues are the same, return identity matrix |
| 328 |
if (w[0] == w[1] && w[0] == w[2] ) { |
| 329 |
v = SquareMatrix3<Real>::identity(); |
| 330 |
return; |
| 331 |
} |
| 332 |
|
| 333 |
// transpose temporarily, it makes it easier to sort the eigenvectors |
| 334 |
v = v.transpose(); |
| 335 |
|
| 336 |
// if two eigenvalues are the same, re-orthogonalize to optimally line |
| 337 |
// up the eigenvectors with the x, y, and z axes |
| 338 |
for (i = 0; i < 3; i++) { |
| 339 |
if (w((i+1)%3) == w((i+2)%3)) {// two eigenvalues are the same |
| 340 |
// find maximum element of the independant eigenvector |
| 341 |
maxVal = fabs(v(i, 0)); |
| 342 |
maxI = 0; |
| 343 |
for (j = 1; j < 3; j++) { |
| 344 |
if (maxVal < (tmp = fabs(v(i, j)))){ |
| 345 |
maxVal = tmp; |
| 346 |
maxI = j; |
| 347 |
} |
| 348 |
} |
| 349 |
|
| 350 |
// swap the eigenvector into its proper position |
| 351 |
if (maxI != i) { |
| 352 |
tmp = w(maxI); |
| 353 |
w(maxI) = w(i); |
| 354 |
w(i) = tmp; |
| 355 |
|
| 356 |
v.swapRow(i, maxI); |
| 357 |
} |
| 358 |
// maximum element of eigenvector should be positive |
| 359 |
if (v(maxI, maxI) < 0) { |
| 360 |
v(maxI, 0) = -v(maxI, 0); |
| 361 |
v(maxI, 1) = -v(maxI, 1); |
| 362 |
v(maxI, 2) = -v(maxI, 2); |
| 363 |
} |
| 364 |
|
| 365 |
// re-orthogonalize the other two eigenvectors |
| 366 |
j = (maxI+1)%3; |
| 367 |
k = (maxI+2)%3; |
| 368 |
|
| 369 |
v(j, 0) = 0.0; |
| 370 |
v(j, 1) = 0.0; |
| 371 |
v(j, 2) = 0.0; |
| 372 |
v(j, j) = 1.0; |
| 373 |
|
| 374 |
/** @todo */ |
| 375 |
v_maxI = v.getRow(maxI); |
| 376 |
v_j = v.getRow(j); |
| 377 |
v_k = cross(v_maxI, v_j); |
| 378 |
v_k.normalize(); |
| 379 |
v_j = cross(v_k, v_maxI); |
| 380 |
v.setRow(j, v_j); |
| 381 |
v.setRow(k, v_k); |
| 382 |
|
| 383 |
|
| 384 |
// transpose vectors back to columns |
| 385 |
v = v.transpose(); |
| 386 |
return; |
| 387 |
} |
| 388 |
} |
| 389 |
|
| 390 |
// the three eigenvalues are different, just sort the eigenvectors |
| 391 |
// to align them with the x, y, and z axes |
| 392 |
|
| 393 |
// find the vector with the largest x element, make that vector |
| 394 |
// the first vector |
| 395 |
maxVal = fabs(v(0, 0)); |
| 396 |
maxI = 0; |
| 397 |
for (i = 1; i < 3; i++) { |
| 398 |
if (maxVal < (tmp = fabs(v(i, 0)))) { |
| 399 |
maxVal = tmp; |
| 400 |
maxI = i; |
| 401 |
} |
| 402 |
} |
| 403 |
|
| 404 |
// swap eigenvalue and eigenvector |
| 405 |
if (maxI != 0) { |
| 406 |
tmp = w(maxI); |
| 407 |
w(maxI) = w(0); |
| 408 |
w(0) = tmp; |
| 409 |
v.swapRow(maxI, 0); |
| 410 |
} |
| 411 |
// do the same for the y element |
| 412 |
if (fabs(v(1, 1)) < fabs(v(2, 1))) { |
| 413 |
tmp = w(2); |
| 414 |
w(2) = w(1); |
| 415 |
w(1) = tmp; |
| 416 |
v.swapRow(2, 1); |
| 417 |
} |
| 418 |
|
| 419 |
// ensure that the sign of the eigenvectors is correct |
| 420 |
for (i = 0; i < 2; i++) { |
| 421 |
if (v(i, i) < 0) { |
| 422 |
v(i, 0) = -v(i, 0); |
| 423 |
v(i, 1) = -v(i, 1); |
| 424 |
v(i, 2) = -v(i, 2); |
| 425 |
} |
| 426 |
} |
| 427 |
|
| 428 |
// set sign of final eigenvector to ensure that determinant is positive |
| 429 |
if (v.determinant() < 0) { |
| 430 |
v(2, 0) = -v(2, 0); |
| 431 |
v(2, 1) = -v(2, 1); |
| 432 |
v(2, 2) = -v(2, 2); |
| 433 |
} |
| 434 |
|
| 435 |
// transpose the eigenvectors back again |
| 436 |
v = v.transpose(); |
| 437 |
return ; |
| 438 |
} |
| 439 |
typedef SquareMatrix3<double> Mat3x3d; |
| 440 |
typedef SquareMatrix3<double> RotMat3x3d; |
| 441 |
|
| 442 |
} //namespace oopse |
| 443 |
#endif // MATH_SQUAREMATRIX_HPP |
| 444 |
|
