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