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Revision 93 by tim, Sun Oct 17 01:19:11 2004 UTC vs.
Revision 633 by tim, Tue Sep 27 20:02:57 2005 UTC

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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.
2 > * Copyright (c) 2005 The University of Notre Dame. All Rights Reserved.
3   *
4 + * The University of Notre Dame grants you ("Licensee") a
5 + * non-exclusive, royalty free, license to use, modify and
6 + * redistribute this software in source and binary code form, provided
7 + * that the following conditions are met:
8 + *
9 + * 1. Acknowledgement of the program authors must be made in any
10 + *    publication of scientific results based in part on use of the
11 + *    program.  An acceptable form of acknowledgement is citation of
12 + *    the article in which the program was described (Matthew
13 + *    A. Meineke, Charles F. Vardeman II, Teng Lin, Christopher
14 + *    J. Fennell and J. Daniel Gezelter, "OOPSE: An Object-Oriented
15 + *    Parallel Simulation Engine for Molecular Dynamics,"
16 + *    J. Comput. Chem. 26, pp. 252-271 (2005))
17 + *
18 + * 2. Redistributions of source code must retain the above copyright
19 + *    notice, this list of conditions and the following disclaimer.
20 + *
21 + * 3. Redistributions in binary form must reproduce the above copyright
22 + *    notice, this list of conditions and the following disclaimer in the
23 + *    documentation and/or other materials provided with the
24 + *    distribution.
25 + *
26 + * This software is provided "AS IS," without a warranty of any
27 + * kind. All express or implied conditions, representations and
28 + * warranties, including any implied warranty of merchantability,
29 + * fitness for a particular purpose or non-infringement, are hereby
30 + * excluded.  The University of Notre Dame and its licensors shall not
31 + * be liable for any damages suffered by licensee as a result of
32 + * using, modifying or distributing the software or its
33 + * derivatives. In no event will the University of Notre Dame or its
34 + * licensors be liable for any lost revenue, profit or data, or for
35 + * direct, indirect, special, consequential, incidental or punitive
36 + * damages, however caused and regardless of the theory of liability,
37 + * arising out of the use of or inability to use software, even if the
38 + * University of Notre Dame has been advised of the possibility of
39 + * such damages.
40   */
41 <
41 >
42   /**
43   * @file SquareMatrix3.hpp
44   * @author Teng Lin
45   * @date 10/11/2004
46   * @version 1.0
47   */
48 < #ifndef MATH_SQUAREMATRIX_HPP
49 < #define  MATH_SQUAREMATRIX_HPP
48 > #ifndef MATH_SQUAREMATRIX3_HPP
49 > #define  MATH_SQUAREMATRIX3_HPP
50  
51   #include "Quaternion.hpp"
52   #include "SquareMatrix.hpp"
53   #include "Vector3.hpp"
54 <
54 > #include "utils/NumericConstant.hpp"
55   namespace oopse {
56  
57 <    template<typename Real>
58 <    class SquareMatrix3 : public SquareMatrix<Real, 3> {
59 <        public:
57 >  template<typename Real>
58 >  class SquareMatrix3 : public SquareMatrix<Real, 3> {
59 >  public:
60 >
61 >    typedef Real ElemType;
62 >    typedef Real* ElemPoinerType;
63              
64 <            /** default constructor */
65 <            SquareMatrix3() : SquareMatrix<Real, 3>() {
66 <            }
64 >    /** default constructor */
65 >    SquareMatrix3() : SquareMatrix<Real, 3>() {
66 >    }
67  
68 <            /** copy  constructor */
69 <            SquareMatrix3(const SquareMatrix<Real, 3>& m)  : SquareMatrix<Real, 3>(m) {
70 <            }
68 >    /** Constructs and initializes every element of this matrix to a scalar */
69 >    SquareMatrix3(Real s) : SquareMatrix<Real,3>(s){
70 >    }
71  
72 <            SquareMatrix3( const Vector3<Real>& eulerAngles) {
73 <                setupRotMat(eulerAngles);
74 <            }
56 <            
57 <            SquareMatrix3(Real phi, Real theta, Real psi) {
58 <                setupRotMat(phi, theta, psi);
59 <            }
72 >    /** Constructs and initializes from an array */
73 >    SquareMatrix3(Real* array) : SquareMatrix<Real,3>(array){
74 >    }
75  
61 –            SquareMatrix3(const Quaternion<Real>& q) {
62 –                *this = q.toRotationMatrix3();
63 –            }
76  
77 <            SquareMatrix3(Real w, Real x, Real y, Real z) {
78 <                Quaternion<Real> q(w, x, y, z);
79 <                *this = q.toRotationMatrix3();
68 <            }
77 >    /** copy  constructor */
78 >    SquareMatrix3(const SquareMatrix<Real, 3>& m)  : SquareMatrix<Real, 3>(m) {
79 >    }
80              
81 <            /** copy assignment operator */
82 <            SquareMatrix3<Real>& operator =(const SquareMatrix<Real, 3>& m) {
83 <                if (this == &m)
84 <                    return *this;
85 <                 SquareMatrix<Real, 3>::operator=(m);
86 <            }
81 >    SquareMatrix3( const Vector3<Real>& eulerAngles) {
82 >      setupRotMat(eulerAngles);
83 >    }
84 >            
85 >    SquareMatrix3(Real phi, Real theta, Real psi) {
86 >      setupRotMat(phi, theta, psi);
87 >    }
88  
89 <            /**
90 <             * Sets this matrix to a rotation matrix by three euler angles
79 <             * @ param euler
80 <             */
81 <            void setupRotMat(const Vector3<Real>& eulerAngles) {
82 <                setupRotMat(eulerAngles[0], eulerAngles[1], eulerAngles[2]);
83 <            }
89 >    SquareMatrix3(const Quaternion<Real>& q) {
90 >      setupRotMat(q);
91  
92 <            /**
86 <             * Sets this matrix to a rotation matrix by three euler angles
87 <             * @param phi
88 <             * @param theta
89 <             * @psi theta
90 <             */
91 <            void setupRotMat(Real phi, Real theta, Real psi) {
92 <                Real sphi, stheta, spsi;
93 <                Real cphi, ctheta, cpsi;
92 >    }
93  
94 <                sphi = sin(phi);
95 <                stheta = sin(theta);
96 <                spsi = sin(psi);
97 <                cphi = cos(phi);
98 <                ctheta = cos(theta);
99 <                cpsi = cos(psi);
94 >    SquareMatrix3(Real w, Real x, Real y, Real z) {
95 >      setupRotMat(w, x, y, z);
96 >    }
97 >            
98 >    /** copy assignment operator */
99 >    SquareMatrix3<Real>& operator =(const SquareMatrix<Real, 3>& m) {
100 >      if (this == &m)
101 >        return *this;
102 >      SquareMatrix<Real, 3>::operator=(m);
103 >      return *this;
104 >    }
105  
102 –                data_[0][0] = cpsi * cphi - ctheta * sphi * spsi;
103 –                data_[0][1] = cpsi * sphi + ctheta * cphi * spsi;
104 –                data_[0][2] = spsi * stheta;
105 –                
106 –                data_[1][0] = -spsi * ctheta - ctheta * sphi * cpsi;
107 –                data_[1][1] = -spsi * stheta + ctheta * cphi * cpsi;
108 –                data_[1][2] = cpsi * stheta;
106  
107 <                data_[2][0] = stheta * sphi;
108 <                data_[2][1] = -stheta * cphi;
109 <                data_[2][2] = ctheta;
110 <            }
107 >    SquareMatrix3<Real>& operator =(const Quaternion<Real>& q) {
108 >      this->setupRotMat(q);
109 >      return *this;
110 >    }
111  
112 +    /**
113 +     * Sets this matrix to a rotation matrix by three euler angles
114 +     * @ param euler
115 +     */
116 +    void setupRotMat(const Vector3<Real>& eulerAngles) {
117 +      setupRotMat(eulerAngles[0], eulerAngles[1], eulerAngles[2]);
118 +    }
119  
120 <            /**
121 <             * Sets this matrix to a rotation matrix by quaternion
122 <             * @param quat
123 <            */
124 <            void setupRotMat(const Quaternion<Real>& quat) {
125 <                *this = quat.toRotationMatrix3();
126 <            }
120 >    /**
121 >     * Sets this matrix to a rotation matrix by three euler angles
122 >     * @param phi
123 >     * @param theta
124 >     * @psi theta
125 >     */
126 >    void setupRotMat(Real phi, Real theta, Real psi) {
127 >      Real sphi, stheta, spsi;
128 >      Real cphi, ctheta, cpsi;
129  
130 <            /**
131 <             * Sets this matrix to a rotation matrix by quaternion
132 <             * @param w the first element
133 <             * @param x the second element
134 <             * @param y the third element
135 <             * @parma z the fourth element
130 <            */
131 <            void setupRotMat(Real w, Real x, Real y, Real z) {
132 <                Quaternion<Real> q(w, x, y, z);
133 <                *this = q.toRotationMatrix3();
134 <            }
130 >      sphi = sin(phi);
131 >      stheta = sin(theta);
132 >      spsi = sin(psi);
133 >      cphi = cos(phi);
134 >      ctheta = cos(theta);
135 >      cpsi = cos(psi);
136  
137 <            /**
138 <             * Returns the quaternion from this rotation matrix
139 <             * @return the quaternion from this rotation matrix
140 <             * @exception invalid rotation matrix
141 <            */            
142 <            Quaternion<Real> toQuaternion() {
143 <                Quaternion<Real> q;
143 <                Real t, s;
144 <                Real ad1, ad2, ad3;    
145 <                t = data_[0][0] + data_[1][1] + data_[2][2] + 1.0;
137 >      this->data_[0][0] = cpsi * cphi - ctheta * sphi * spsi;
138 >      this->data_[0][1] = cpsi * sphi + ctheta * cphi * spsi;
139 >      this->data_[0][2] = spsi * stheta;
140 >                
141 >      this->data_[1][0] = -spsi * ctheta - ctheta * sphi * cpsi;
142 >      this->data_[1][1] = -spsi * stheta + ctheta * cphi * cpsi;
143 >      this->data_[1][2] = cpsi * stheta;
144  
145 <                if( t > 0.0 ){
145 >      this->data_[2][0] = stheta * sphi;
146 >      this->data_[2][1] = -stheta * cphi;
147 >      this->data_[2][2] = ctheta;
148 >    }
149  
149 –                    s = 0.5 / sqrt( t );
150 –                    q[0] = 0.25 / s;
151 –                    q[1] = (data_[1][2] - data_[2][1]) * s;
152 –                    q[2] = (data_[2][0] - data_[0][2]) * s;
153 –                    q[3] = (data_[0][1] - data_[1][0]) * s;
154 –                } else {
150  
151 <                    ad1 = fabs( data_[0][0] );
152 <                    ad2 = fabs( data_[1][1] );
153 <                    ad3 = fabs( data_[2][2] );
151 >    /**
152 >     * Sets this matrix to a rotation matrix by quaternion
153 >     * @param quat
154 >     */
155 >    void setupRotMat(const Quaternion<Real>& quat) {
156 >      setupRotMat(quat.w(), quat.x(), quat.y(), quat.z());
157 >    }
158  
159 <                    if( ad1 >= ad2 && ad1 >= ad3 ){
159 >    /**
160 >     * Sets this matrix to a rotation matrix by quaternion
161 >     * @param w the first element
162 >     * @param x the second element
163 >     * @param y the third element
164 >     * @param z the fourth element
165 >     */
166 >    void setupRotMat(Real w, Real x, Real y, Real z) {
167 >      Quaternion<Real> q(w, x, y, z);
168 >      *this = q.toRotationMatrix3();
169 >    }
170  
171 <                        s = 2.0 * sqrt( 1.0 + data_[0][0] - data_[1][1] - data_[2][2] );
172 <                        q[0] = (data_[1][2] + data_[2][1]) / s;
173 <                        q[1] = 0.5 / s;
174 <                        q[2] = (data_[0][1] + data_[1][0]) / s;
175 <                        q[3] = (data_[0][2] + data_[2][0]) / s;
176 <                    } else if ( ad2 >= ad1 && ad2 >= ad3 ) {
177 <                        s = sqrt( 1.0 + data_[1][1] - data_[0][0] - data_[2][2] ) * 2.0;
178 <                        q[0] = (data_[0][2] + data_[2][0]) / s;
179 <                        q[1] = (data_[0][1] + data_[1][0]) / s;
180 <                        q[2] = 0.5 / s;
172 <                        q[3] = (data_[1][2] + data_[2][1]) / s;
173 <                    } else {
171 >    /**
172 >     * Returns the quaternion from this rotation matrix
173 >     * @return the quaternion from this rotation matrix
174 >     * @exception invalid rotation matrix
175 >     */            
176 >    Quaternion<Real> toQuaternion() {
177 >      Quaternion<Real> q;
178 >      Real t, s;
179 >      Real ad1, ad2, ad3;    
180 >      t = this->data_[0][0] + this->data_[1][1] + this->data_[2][2] + 1.0;
181  
182 <                        s = sqrt( 1.0 + data_[2][2] - data_[0][0] - data_[1][1] ) * 2.0;
176 <                        q[0] = (data_[0][1] + data_[1][0]) / s;
177 <                        q[1] = (data_[0][2] + data_[2][0]) / s;
178 <                        q[2] = (data_[1][2] + data_[2][1]) / s;
179 <                        q[3] = 0.5 / s;
180 <                    }
181 <                }            
182 >      if( t > 0.0 ){
183  
184 <                return q;
185 <                
186 <            }
184 >        s = 0.5 / sqrt( t );
185 >        q[0] = 0.25 / s;
186 >        q[1] = (this->data_[1][2] - this->data_[2][1]) * s;
187 >        q[2] = (this->data_[2][0] - this->data_[0][2]) * s;
188 >        q[3] = (this->data_[0][1] - this->data_[1][0]) * s;
189 >      } else {
190  
191 <            /**
192 <             * Returns the euler angles from this rotation matrix
193 <             * @return the euler angles in a vector
190 <             * @exception invalid rotation matrix
191 <             * We use so-called "x-convention", which is the most common definition.
192 <             * In this convention, the rotation given by Euler angles (phi, theta, psi), where the first
193 <             * rotation is by an angle phi about the z-axis, the second is by an angle  
194 <             * theta (0 <= theta <= 180)about the x-axis, and thethird is by an angle psi about the
195 <             * z-axis (again).
196 <            */            
197 <            Vector3<Real> toEulerAngles() {
198 <                Vector<Real> myEuler;
199 <                Real phi,theta,psi,eps;
200 <                Real ctheta,stheta;
201 <                
202 <                // set the tolerance for Euler angles and rotation elements
191 >        ad1 = this->data_[0][0];
192 >        ad2 = this->data_[1][1];
193 >        ad3 = this->data_[2][2];
194  
195 <                theta = acos(min(1.0,max(-1.0,data_[2][2])));
205 <                ctheta = data_[2][2];
206 <                stheta = sqrt(1.0 - ctheta * ctheta);
195 >        if( ad1 >= ad2 && ad1 >= ad3 ){
196  
197 <                // when sin(theta) is close to 0, we need to consider singularity
198 <                // In this case, we can assign an arbitary value to phi (or psi), and then determine
199 <                // the psi (or phi) or vice-versa. We'll assume that phi always gets the rotation, and psi is 0
200 <                // in cases of singularity.  
201 <                // we use atan2 instead of atan, since atan2 will give us -Pi to Pi.
202 <                // Since 0 <= theta <= 180, sin(theta) will be always non-negative. Therefore, it never
203 <                // change the sign of both of the parameters passed to atan2.
197 >          s = 0.5 / sqrt( 1.0 + this->data_[0][0] - this->data_[1][1] - this->data_[2][2] );
198 >          q[0] = (this->data_[1][2] - this->data_[2][1]) * s;
199 >          q[1] = 0.25 / s;
200 >          q[2] = (this->data_[0][1] + this->data_[1][0]) * s;
201 >          q[3] = (this->data_[0][2] + this->data_[2][0]) * s;
202 >        } else if ( ad2 >= ad1 && ad2 >= ad3 ) {
203 >          s = 0.5 / sqrt( 1.0 + this->data_[1][1] - this->data_[0][0] - this->data_[2][2] );
204 >          q[0] = (this->data_[2][0] - this->data_[0][2] ) * s;
205 >          q[1] = (this->data_[0][1] + this->data_[1][0]) * s;
206 >          q[2] = 0.25 / s;
207 >          q[3] = (this->data_[1][2] + this->data_[2][1]) * s;
208 >        } else {
209  
210 <                if (fabs(stheta) <= oopse::epsilon){
211 <                    psi = 0.0;
212 <                    phi = atan2(-data_[1][0], data_[0][0]);  
213 <                }
214 <                // we only have one unique solution
215 <                else{    
216 <                    phi = atan2(data_[2][0], -data_[2][1]);
223 <                    psi = atan2(data_[0][2], data_[1][2]);
224 <                }
210 >          s = 0.5 / sqrt( 1.0 + this->data_[2][2] - this->data_[0][0] - this->data_[1][1] );
211 >          q[0] = (this->data_[0][1] - this->data_[1][0]) * s;
212 >          q[1] = (this->data_[0][2] + this->data_[2][0]) * s;
213 >          q[2] = (this->data_[1][2] + this->data_[2][1]) * s;
214 >          q[3] = 0.25 / s;
215 >        }
216 >      }            
217  
218 <                //wrap phi and psi, make sure they are in the range from 0 to 2*Pi
219 <                if (phi < 0)
220 <                  phi += M_PI;
218 >      return q;
219 >                
220 >    }
221  
222 <                if (psi < 0)
223 <                  psi += M_PI;
222 >    /**
223 >     * Returns the euler angles from this rotation matrix
224 >     * @return the euler angles in a vector
225 >     * @exception invalid rotation matrix
226 >     * We use so-called "x-convention", which is the most common definition.
227 >     * In this convention, the rotation given by Euler angles (phi, theta, psi), where the first
228 >     * rotation is by an angle phi about the z-axis, the second is by an angle  
229 >     * theta (0 <= theta <= 180)about the x-axis, and thethird is by an angle psi about the
230 >     * z-axis (again).
231 >     */            
232 >    Vector3<Real> toEulerAngles() {
233 >      Vector3<Real> myEuler;
234 >      Real phi;
235 >      Real theta;
236 >      Real psi;
237 >      Real ctheta;
238 >      Real stheta;
239 >                
240 >      // set the tolerance for Euler angles and rotation elements
241  
242 <                myEuler[0] = phi;
243 <                myEuler[1] = theta;
244 <                myEuler[2] = psi;
242 >      theta = acos(std::min(1.0, std::max(-1.0,this->data_[2][2])));
243 >      ctheta = this->data_[2][2];
244 >      stheta = sqrt(1.0 - ctheta * ctheta);
245  
246 <                return myEuler;
247 <            }
246 >      // when sin(theta) is close to 0, we need to consider singularity
247 >      // In this case, we can assign an arbitary value to phi (or psi), and then determine
248 >      // the psi (or phi) or vice-versa. We'll assume that phi always gets the rotation, and psi is 0
249 >      // in cases of singularity.  
250 >      // we use atan2 instead of atan, since atan2 will give us -Pi to Pi.
251 >      // Since 0 <= theta <= 180, sin(theta) will be always non-negative. Therefore, it never
252 >      // change the sign of both of the parameters passed to atan2.
253 >
254 >      if (fabs(stheta) <= oopse::epsilon){
255 >        psi = 0.0;
256 >        phi = atan2(-this->data_[1][0], this->data_[0][0]);  
257 >      }
258 >      // we only have one unique solution
259 >      else{    
260 >        phi = atan2(this->data_[2][0], -this->data_[2][1]);
261 >        psi = atan2(this->data_[0][2], this->data_[1][2]);
262 >      }
263 >
264 >      //wrap phi and psi, make sure they are in the range from 0 to 2*Pi
265 >      if (phi < 0)
266 >        phi += M_PI;
267 >
268 >      if (psi < 0)
269 >        psi += M_PI;
270 >
271 >      myEuler[0] = phi;
272 >      myEuler[1] = theta;
273 >      myEuler[2] = psi;
274 >
275 >      return myEuler;
276 >    }
277              
278 <            /**
279 <             * Sets the value of this matrix to  the inversion of itself.
280 <             * @note since simple algorithm can be applied to inverse the 3 by 3 matrix, we hide the
243 <             * implementation of inverse in SquareMatrix class
244 <             */
245 <            void  inverse();
278 >    /** Returns the determinant of this matrix. */
279 >    Real determinant() const {
280 >      Real x,y,z;
281  
282 <            void diagonalize();
282 >      x = this->data_[0][0] * (this->data_[1][1] * this->data_[2][2] - this->data_[1][2] * this->data_[2][1]);
283 >      y = this->data_[0][1] * (this->data_[1][2] * this->data_[2][0] - this->data_[1][0] * this->data_[2][2]);
284 >      z = this->data_[0][2] * (this->data_[1][0] * this->data_[2][1] - this->data_[1][1] * this->data_[2][0]);
285  
286 <    };
286 >      return(x + y + z);
287 >    }            
288  
289 <    typedef template SquareMatrix3<double> Mat3x3d
290 <    typedef template SquareMatrix3<double> RotMat3x3d;
289 >    /** Returns the trace of this matrix. */
290 >    Real trace() const {
291 >      return this->data_[0][0] + this->data_[1][1] + this->data_[2][2];
292 >    }
293 >            
294 >    /**
295 >     * Sets the value of this matrix to  the inversion of itself.
296 >     * @note since simple algorithm can be applied to inverse the 3 by 3 matrix, we hide the
297 >     * implementation of inverse in SquareMatrix class
298 >     */
299 >    SquareMatrix3<Real>  inverse() const {
300 >      SquareMatrix3<Real> m;
301 >      double det = determinant();
302 >      if (fabs(det) <= oopse::epsilon) {
303 >        //"The method was called on a matrix with |determinant| <= 1e-6.",
304 >        //"This is a runtime or a programming error in your application.");
305 >      }
306  
307 +      m(0, 0) = this->data_[1][1]*this->data_[2][2] - this->data_[1][2]*this->data_[2][1];
308 +      m(1, 0) = this->data_[1][2]*this->data_[2][0] - this->data_[1][0]*this->data_[2][2];
309 +      m(2, 0) = this->data_[1][0]*this->data_[2][1] - this->data_[1][1]*this->data_[2][0];
310 +      m(0, 1) = this->data_[2][1]*this->data_[0][2] - this->data_[2][2]*this->data_[0][1];
311 +      m(1, 1) = this->data_[2][2]*this->data_[0][0] - this->data_[2][0]*this->data_[0][2];
312 +      m(2, 1) = this->data_[2][0]*this->data_[0][1] - this->data_[2][1]*this->data_[0][0];
313 +      m(0, 2) = this->data_[0][1]*this->data_[1][2] - this->data_[0][2]*this->data_[1][1];
314 +      m(1, 2) = this->data_[0][2]*this->data_[1][0] - this->data_[0][0]*this->data_[1][2];
315 +      m(2, 2) = this->data_[0][0]*this->data_[1][1] - this->data_[0][1]*this->data_[1][0];
316 +
317 +      m /= det;
318 +      return m;
319 +    }
320 +    /**
321 +     * Extract the eigenvalues and eigenvectors from a 3x3 matrix.
322 +     * The eigenvectors (the columns of V) will be normalized.
323 +     * The eigenvectors are aligned optimally with the x, y, and z
324 +     * axes respectively.
325 +     * @param a symmetric matrix whose eigenvectors are to be computed. On return, the matrix is
326 +     *     overwritten            
327 +     * @param w will contain the eigenvalues of the matrix On return of this function
328 +     * @param v the columns of this matrix will contain the eigenvectors. The eigenvectors are
329 +     *    normalized and mutually orthogonal.              
330 +     * @warning a will be overwritten
331 +     */
332 +    static void diagonalize(SquareMatrix3<Real>& a, Vector3<Real>& w, SquareMatrix3<Real>& v);
333 +  };
334 +  /*=========================================================================
335 +
336 +  Program:   Visualization Toolkit
337 +  Module:    $RCSfile: SquareMatrix3.hpp,v $
338 +
339 +  Copyright (c) Ken Martin, Will Schroeder, Bill Lorensen
340 +  All rights reserved.
341 +  See Copyright.txt or http://www.kitware.com/Copyright.htm for details.
342 +
343 +  This software is distributed WITHOUT ANY WARRANTY; without even
344 +  the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR
345 +  PURPOSE.  See the above copyright notice for more information.
346 +
347 +  =========================================================================*/
348 +  template<typename Real>
349 +  void SquareMatrix3<Real>::diagonalize(SquareMatrix3<Real>& a, Vector3<Real>& w,
350 +                                        SquareMatrix3<Real>& v) {
351 +    int i,j,k,maxI;
352 +    Real tmp, maxVal;
353 +    Vector3<Real> v_maxI, v_k, v_j;
354 +
355 +    // diagonalize using Jacobi
356 +    jacobi(a, w, v);
357 +    // if all the eigenvalues are the same, return identity matrix
358 +    if (w[0] == w[1] && w[0] == w[2] ) {
359 +      v = SquareMatrix3<Real>::identity();
360 +      return;
361 +    }
362 +
363 +    // transpose temporarily, it makes it easier to sort the eigenvectors
364 +    v = v.transpose();
365 +        
366 +    // if two eigenvalues are the same, re-orthogonalize to optimally line
367 +    // up the eigenvectors with the x, y, and z axes
368 +    for (i = 0; i < 3; i++) {
369 +      if (w((i+1)%3) == w((i+2)%3)) {// two eigenvalues are the same
370 +        // find maximum element of the independant eigenvector
371 +        maxVal = fabs(v(i, 0));
372 +        maxI = 0;
373 +        for (j = 1; j < 3; j++) {
374 +          if (maxVal < (tmp = fabs(v(i, j)))){
375 +            maxVal = tmp;
376 +            maxI = j;
377 +          }
378 +        }
379 +            
380 +        // swap the eigenvector into its proper position
381 +        if (maxI != i) {
382 +          tmp = w(maxI);
383 +          w(maxI) = w(i);
384 +          w(i) = tmp;
385 +
386 +          v.swapRow(i, maxI);
387 +        }
388 +        // maximum element of eigenvector should be positive
389 +        if (v(maxI, maxI) < 0) {
390 +          v(maxI, 0) = -v(maxI, 0);
391 +          v(maxI, 1) = -v(maxI, 1);
392 +          v(maxI, 2) = -v(maxI, 2);
393 +        }
394 +
395 +        // re-orthogonalize the other two eigenvectors
396 +        j = (maxI+1)%3;
397 +        k = (maxI+2)%3;
398 +
399 +        v(j, 0) = 0.0;
400 +        v(j, 1) = 0.0;
401 +        v(j, 2) = 0.0;
402 +        v(j, j) = 1.0;
403 +
404 +        /** @todo */
405 +        v_maxI = v.getRow(maxI);
406 +        v_j = v.getRow(j);
407 +        v_k = cross(v_maxI, v_j);
408 +        v_k.normalize();
409 +        v_j = cross(v_k, v_maxI);
410 +        v.setRow(j, v_j);
411 +        v.setRow(k, v_k);
412 +
413 +
414 +        // transpose vectors back to columns
415 +        v = v.transpose();
416 +        return;
417 +      }
418 +    }
419 +
420 +    // the three eigenvalues are different, just sort the eigenvectors
421 +    // to align them with the x, y, and z axes
422 +
423 +    // find the vector with the largest x element, make that vector
424 +    // the first vector
425 +    maxVal = fabs(v(0, 0));
426 +    maxI = 0;
427 +    for (i = 1; i < 3; i++) {
428 +      if (maxVal < (tmp = fabs(v(i, 0)))) {
429 +        maxVal = tmp;
430 +        maxI = i;
431 +      }
432 +    }
433 +
434 +    // swap eigenvalue and eigenvector
435 +    if (maxI != 0) {
436 +      tmp = w(maxI);
437 +      w(maxI) = w(0);
438 +      w(0) = tmp;
439 +      v.swapRow(maxI, 0);
440 +    }
441 +    // do the same for the y element
442 +    if (fabs(v(1, 1)) < fabs(v(2, 1))) {
443 +      tmp = w(2);
444 +      w(2) = w(1);
445 +      w(1) = tmp;
446 +      v.swapRow(2, 1);
447 +    }
448 +
449 +    // ensure that the sign of the eigenvectors is correct
450 +    for (i = 0; i < 2; i++) {
451 +      if (v(i, i) < 0) {
452 +        v(i, 0) = -v(i, 0);
453 +        v(i, 1) = -v(i, 1);
454 +        v(i, 2) = -v(i, 2);
455 +      }
456 +    }
457 +
458 +    // set sign of final eigenvector to ensure that determinant is positive
459 +    if (v.determinant() < 0) {
460 +      v(2, 0) = -v(2, 0);
461 +      v(2, 1) = -v(2, 1);
462 +      v(2, 2) = -v(2, 2);
463 +    }
464 +
465 +    // transpose the eigenvectors back again
466 +    v = v.transpose();
467 +    return ;
468 +  }
469 +
470 +  /**
471 +   * Return the multiplication of two matrixes  (m1 * m2).
472 +   * @return the multiplication of two matrixes
473 +   * @param m1 the first matrix
474 +   * @param m2 the second matrix
475 +   */
476 +  template<typename Real>
477 +  inline SquareMatrix3<Real> operator *(const SquareMatrix3<Real>& m1, const SquareMatrix3<Real>& m2) {
478 +    SquareMatrix3<Real> result;
479 +
480 +    for (unsigned int i = 0; i < 3; i++)
481 +      for (unsigned int j = 0; j < 3; j++)
482 +        for (unsigned int k = 0; k < 3; k++)
483 +          result(i, j)  += m1(i, k) * m2(k, j);                
484 +
485 +    return result;
486 +  }
487 +
488 +  template<typename Real>
489 +  inline SquareMatrix3<Real> outProduct(const Vector3<Real>& v1, const Vector3<Real>& v2) {
490 +    SquareMatrix3<Real> result;
491 +
492 +    for (unsigned int i = 0; i < 3; i++) {
493 +      for (unsigned int j = 0; j < 3; j++) {
494 +        result(i, j)  = v1[i] * v2[j];                
495 +      }
496 +    }
497 +            
498 +    return result;        
499 +  }
500 +
501 +    
502 +  typedef SquareMatrix3<double> Mat3x3d;
503 +  typedef SquareMatrix3<double> RotMat3x3d;
504 +
505   } //namespace oopse
506   #endif // MATH_SQUAREMATRIX_HPP
507 +

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