src/math/SquareMatrix3.hpp

/*
 * Copyright (C) 2000-2004  Object Oriented Parallel Simulation Engine (OOPSE) project
 * 
 * Contact: oopse@oopse.org
 * 
 * This program is free software; you can redistribute it and/or
 * modify it under the terms of the GNU Lesser General Public License
 * as published by the Free Software Foundation; either version 2.1
 * of the License, or (at your option) any later version.
 * All we ask is that proper credit is given for our work, which includes
 * - but is not limited to - adding the above copyright notice to the beginning
 * of your source code files, and to any copyright notice that you may distribute
 * with programs based on this work.
 * 
 * This program is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 * GNU Lesser General Public License for more details.
 * 
 * You should have received a copy of the GNU Lesser General Public License
 * along with this program; if not, write to the Free Software
 * Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA  02111-1307, USA.
 *
 */

/**
 * @file SquareMatrix3.hpp
 * @author Teng Lin
 * @date 10/11/2004
 * @version 1.0
 */
 #ifndef MATH_SQUAREMATRIX3_HPP
#define  MATH_SQUAREMATRIX3_HPP

#include "Quaternion.hpp"
#include "SquareMatrix.hpp"
#include "Vector3.hpp"

namespace oopse {

    template<typename Real>
    class SquareMatrix3 : public SquareMatrix<Real, 3> {
        public:

            typedef Real ElemType;
            typedef Real* ElemPoinerType;
            
            /** default constructor */
            SquareMatrix3() : SquareMatrix<Real, 3>() {
            }

            /** Constructs and initializes every element of this matrix to a scalar */ 
            SquareMatrix3(Real s) : SquareMatrix<Real,3>(s){
            }

            /** Constructs and initializes from an array */ 
            SquareMatrix3(Real* array) : SquareMatrix<Real,3>(array){
            }


            /** copy  constructor */
            SquareMatrix3(const SquareMatrix<Real, 3>& m)  : SquareMatrix<Real, 3>(m) {
            }

            SquareMatrix3( const Vector3<Real>& eulerAngles) {
                setupRotMat(eulerAngles);
            }
            
            SquareMatrix3(Real phi, Real theta, Real psi) {
                setupRotMat(phi, theta, psi);
            }

            SquareMatrix3(const Quaternion<Real>& q) {
                setupRotMat(q);

            }

            SquareMatrix3(Real w, Real x, Real y, Real z) {
                setupRotMat(w, x, y, z);
            }
            
            /** copy assignment operator */
            SquareMatrix3<Real>& operator =(const SquareMatrix<Real, 3>& m) {
                if (this == &m)
                    return *this;
                 SquareMatrix<Real, 3>::operator=(m);
                 return *this;
            }

            /**
             * Sets this matrix to a rotation matrix by three euler angles
             * @ param euler
             */
            void setupRotMat(const Vector3<Real>& eulerAngles) {
                setupRotMat(eulerAngles[0], eulerAngles[1], eulerAngles[2]);
            }

            /**
             * Sets this matrix to a rotation matrix by three euler angles
             * @param phi
             * @param theta
             * @psi theta
             */
            void setupRotMat(Real phi, Real theta, Real psi) {
                Real sphi, stheta, spsi;
                Real cphi, ctheta, cpsi;

                sphi = sin(phi);
                stheta = sin(theta);
                spsi = sin(psi);
                cphi = cos(phi);
                ctheta = cos(theta);
                cpsi = cos(psi);

                data_[0][0] = cpsi * cphi - ctheta * sphi * spsi;
                data_[0][1] = cpsi * sphi + ctheta * cphi * spsi;
                data_[0][2] = spsi * stheta;
                
                data_[1][0] = -spsi * ctheta - ctheta * sphi * cpsi;
                data_[1][1] = -spsi * stheta + ctheta * cphi * cpsi;
                data_[1][2] = cpsi * stheta;

                data_[2][0] = stheta * sphi;
                data_[2][1] = -stheta * cphi;
                data_[2][2] = ctheta;
            }


            /**
             * Sets this matrix to a rotation matrix by quaternion
             * @param quat
            */
            void setupRotMat(const Quaternion<Real>& quat) {
                setupRotMat(quat.w(), quat.x(), quat.y(), quat.z());
            }

            /**
             * Sets this matrix to a rotation matrix by quaternion
             * @param w the first element 
             * @param x the second element
             * @param y the third element
             * @param z the fourth element
            */
            void setupRotMat(Real w, Real x, Real y, Real z) {
                Quaternion<Real> q(w, x, y, z);
                *this = q.toRotationMatrix3();
            }

            /**
             * Returns the quaternion from this rotation matrix
             * @return the quaternion from this rotation matrix
             * @exception invalid rotation matrix
            */            
            Quaternion<Real> toQuaternion() {
                Quaternion<Real> q;
                Real t, s;
                Real ad1, ad2, ad3;    
                t = data_[0][0] + data_[1][1] + data_[2][2] + 1.0;

                if( t > 0.0 ){

                    s = 0.5 / sqrt( t );
                    q[0] = 0.25 / s;
                    q[1] = (data_[1][2] - data_[2][1]) * s;
                    q[2] = (data_[2][0] - data_[0][2]) * s;
                    q[3] = (data_[0][1] - data_[1][0]) * s;
                } else {

                    ad1 = fabs( data_[0][0] );
                    ad2 = fabs( data_[1][1] );
                    ad3 = fabs( data_[2][2] );

                    if( ad1 >= ad2 && ad1 >= ad3 ){

                        s = 2.0 * sqrt( 1.0 + data_[0][0] - data_[1][1] - data_[2][2] );
                        q[0] = (data_[1][2] + data_[2][1]) / s;
                        q[1] = 0.5 / s;
                        q[2] = (data_[0][1] + data_[1][0]) / s;
                        q[3] = (data_[0][2] + data_[2][0]) / s;
                    } else if ( ad2 >= ad1 && ad2 >= ad3 ) {
                        s = sqrt( 1.0 + data_[1][1] - data_[0][0] - data_[2][2] ) * 2.0;
                        q[0] = (data_[0][2] + data_[2][0]) / s;
                        q[1] = (data_[0][1] + data_[1][0]) / s;
                        q[2] = 0.5 / s;
                        q[3] = (data_[1][2] + data_[2][1]) / s;
                    } else {

                        s = sqrt( 1.0 + data_[2][2] - data_[0][0] - data_[1][1] ) * 2.0;
                        q[0] = (data_[0][1] + data_[1][0]) / s;
                        q[1] = (data_[0][2] + data_[2][0]) / s;
                        q[2] = (data_[1][2] + data_[2][1]) / s;
                        q[3] = 0.5 / s;
                    }
                }             

                return q;
                
            }

            /**
             * Returns the euler angles from this rotation matrix
             * @return the euler angles in a vector 
             * @exception invalid rotation matrix
             * We use so-called "x-convention", which is the most common definition. 
             * In this convention, the rotation given by Euler angles (phi, theta, psi), where the first 
             * rotation is by an angle phi about the z-axis, the second is by an angle  
             * theta (0 <= theta <= 180)about the x-axis, and thethird is by an angle psi about the
             * z-axis (again). 
            */            
            Vector3<Real> toEulerAngles() {
                Vector3<Real> myEuler;
                Real phi,theta,psi,eps;
                Real ctheta,stheta;
                
                // set the tolerance for Euler angles and rotation elements

                theta = acos(std::min(1.0, std::max(-1.0,data_[2][2])));
                ctheta = data_[2][2]; 
                stheta = sqrt(1.0 - ctheta * ctheta);

                // when sin(theta) is close to 0, we need to consider singularity
                // In this case, we can assign an arbitary value to phi (or psi), and then determine 
                // the psi (or phi) or vice-versa. We'll assume that phi always gets the rotation, and psi is 0
                // in cases of singularity.  
                // we use atan2 instead of atan, since atan2 will give us -Pi to Pi. 
                // Since 0 <= theta <= 180, sin(theta) will be always non-negative. Therefore, it never
                // change the sign of both of the parameters passed to atan2.

                if (fabs(stheta) <= oopse::epsilon){
                    psi = 0.0;
                    phi = atan2(-data_[1][0], data_[0][0]);  
                }
                // we only have one unique solution
                else{    
                    phi = atan2(data_[2][0], -data_[2][1]);
                    psi = atan2(data_[0][2], data_[1][2]);
                }

                //wrap phi and psi, make sure they are in the range from 0 to 2*Pi
                if (phi < 0)
                  phi += M_PI;

                if (psi < 0)
                  psi += M_PI;

                myEuler[0] = phi;
                myEuler[1] = theta;
                myEuler[2] = psi;

                return myEuler;
            }
            
            /** Returns the determinant of this matrix. */
            Real determinant() const {
                Real x,y,z;

                x = data_[0][0] * (data_[1][1] * data_[2][2] - data_[1][2] * data_[2][1]);
                y = data_[0][1] * (data_[1][2] * data_[2][0] - data_[1][0] * data_[2][2]);
                z = data_[0][2] * (data_[1][0] * data_[2][1] - data_[1][1] * data_[2][0]);

                return(x + y + z);
            }            
            
            /**
             * Sets the value of this matrix to  the inversion of itself. 
             * @note since simple algorithm can be applied to inverse the 3 by 3 matrix, we hide the 
             * implementation of inverse in SquareMatrix class
             */
            SquareMatrix3<Real>  inverse() {
                SquareMatrix3<Real> m;
                double det = determinant();
                if (fabs(det) <= oopse::epsilon) {
                //"The method was called on a matrix with |determinant| <= 1e-6.",
                //"This is a runtime or a programming error in your application.");
                }

                m(0, 0) = data_[1][1]*data_[2][2] - data_[1][2]*data_[2][1];
                m(1, 0) = data_[1][2]*data_[2][0] - data_[1][0]*data_[2][2];
                m(2, 0) = data_[1][0]*data_[2][1] - data_[1][1]*data_[2][0];
                m(0, 1) = data_[2][1]*data_[0][2] - data_[2][2]*data_[0][1];
                m(1, 1) = data_[2][2]*data_[0][0] - data_[2][0]*data_[0][2];
                m(2, 1) = data_[2][0]*data_[0][1] - data_[2][1]*data_[0][0];
                m(0, 2) = data_[0][1]*data_[1][2] - data_[0][2]*data_[1][1];
                m(1, 2) = data_[0][2]*data_[1][0] - data_[0][0]*data_[1][2];
                m(2, 2) = data_[0][0]*data_[1][1] - data_[0][1]*data_[1][0];

                m /= det;
                return m;
            }
            /**
             * Extract the eigenvalues and eigenvectors from a 3x3 matrix.
             * The eigenvectors (the columns of V) will be normalized. 
             * The eigenvectors are aligned optimally with the x, y, and z
             * axes respectively.
             * @param a symmetric matrix whose eigenvectors are to be computed. On return, the matrix is
             *     overwritten             
             * @param w will contain the eigenvalues of the matrix On return of this function
             * @param v the columns of this matrix will contain the eigenvectors. The eigenvectors are 
             *    normalized and mutually orthogonal.              
             * @warning a will be overwritten
             */
            static void diagonalize(SquareMatrix3<Real>& a, Vector3<Real>& w, SquareMatrix3<Real>& v); 
    };
/*=========================================================================

  Program:   Visualization Toolkit
  Module:    $RCSfile: SquareMatrix3.hpp,v $

  Copyright (c) Ken Martin, Will Schroeder, Bill Lorensen
  All rights reserved.
  See Copyright.txt or http://www.kitware.com/Copyright.htm for details.

     This software is distributed WITHOUT ANY WARRANTY; without even
     the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR
     PURPOSE.  See the above copyright notice for more information.

=========================================================================*/
    template<typename Real>
    void SquareMatrix3<Real>::diagonalize(SquareMatrix3<Real>& a, Vector3<Real>& w, 
                                                                           SquareMatrix3<Real>& v) {
        int i,j,k,maxI;
        Real tmp, maxVal;
        Vector3<Real> v_maxI, v_k, v_j;

        // diagonalize using Jacobi
        jacobi(a, w, v);
        // if all the eigenvalues are the same, return identity matrix
        if (w[0] == w[1] && w[0] == w[2] ) {
              v = SquareMatrix3<Real>::identity();
              return;
        }

        // transpose temporarily, it makes it easier to sort the eigenvectors
        v = v.transpose(); 
        
        // if two eigenvalues are the same, re-orthogonalize to optimally line
        // up the eigenvectors with the x, y, and z axes
        for (i = 0; i < 3; i++) {
            if (w((i+1)%3) == w((i+2)%3)) {// two eigenvalues are the same
            // find maximum element of the independant eigenvector
            maxVal = fabs(v(i, 0));
            maxI = 0;
            for (j = 1; j < 3; j++) {
                if (maxVal < (tmp = fabs(v(i, j)))){
                    maxVal = tmp;
                    maxI = j;
                }
            }
            
            // swap the eigenvector into its proper position
            if (maxI != i) {
                tmp = w(maxI);
                w(maxI) = w(i);
                w(i) = tmp;

                v.swapRow(i, maxI);
            }
            // maximum element of eigenvector should be positive
            if (v(maxI, maxI) < 0) {
                v(maxI, 0) = -v(maxI, 0);
                v(maxI, 1) = -v(maxI, 1);
                v(maxI, 2) = -v(maxI, 2);
            }

            // re-orthogonalize the other two eigenvectors
            j = (maxI+1)%3;
            k = (maxI+2)%3;

            v(j, 0) = 0.0; 
            v(j, 1) = 0.0; 
            v(j, 2) = 0.0;
            v(j, j) = 1.0;

            /** @todo */
            v_maxI = v.getRow(maxI);
            v_j = v.getRow(j);
            v_k = cross(v_maxI, v_j);
            v_k.normalize();
            v_j = cross(v_k, v_maxI);
            v.setRow(j, v_j);
            v.setRow(k, v_k);


            // transpose vectors back to columns
            v = v.transpose();
            return;
            }
        }

        // the three eigenvalues are different, just sort the eigenvectors
        // to align them with the x, y, and z axes

        // find the vector with the largest x element, make that vector
        // the first vector
        maxVal = fabs(v(0, 0));
        maxI = 0;
        for (i = 1; i < 3; i++) {
            if (maxVal < (tmp = fabs(v(i, 0)))) {
                maxVal = tmp;
                maxI = i;
            }
        }

        // swap eigenvalue and eigenvector
        if (maxI != 0) {
            tmp = w(maxI);
            w(maxI) = w(0);
            w(0) = tmp;
            v.swapRow(maxI, 0);
        }
        // do the same for the y element
        if (fabs(v(1, 1)) < fabs(v(2, 1))) {
            tmp = w(2);
            w(2) = w(1);
            w(1) = tmp;
            v.swapRow(2, 1);
        }

        // ensure that the sign of the eigenvectors is correct
        for (i = 0; i < 2; i++) {
            if (v(i, i) < 0) {
                v(i, 0) = -v(i, 0);
                v(i, 1) = -v(i, 1);
                v(i, 2) = -v(i, 2);
            }
        }

        // set sign of final eigenvector to ensure that determinant is positive
        if (v.determinant() < 0) {
            v(2, 0) = -v(2, 0);
            v(2, 1) = -v(2, 1);
            v(2, 2) = -v(2, 2);
        }

        // transpose the eigenvectors back again
        v = v.transpose();
        return ;
    }
    typedef SquareMatrix3<double> Mat3x3d;
    typedef SquareMatrix3<double> RotMat3x3d;

} //namespace oopse
#endif // MATH_SQUAREMATRIX_HPP

Revision:	151
Committed:	Mon Oct 25 22:46:19 2004 UTC (20 years, 6 months ago) by tim
File size:	16738 byte(s)
Log Message:	add getArray function to RectMatrix and Vector classes
#	User	Rev	Content
1	tim	70	/*
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	tim	123	#ifndef MATH_SQUAREMATRIX3_HPP
33	tim	99	#define MATH_SQUAREMATRIX3_HPP
34	tim	70
35	tim	93	#include "Quaternion.hpp"
36	tim	70	#include "SquareMatrix.hpp"
37	tim	93	#include "Vector3.hpp"
38
39	tim	70	namespace oopse {
40
41			template<typename Real>
42			class SquareMatrix3 : public SquareMatrix<Real, 3> {
43			public:
44	tim	137
45			typedef Real ElemType;
46			typedef Real* ElemPoinerType;
47	tim	70
48			/** default constructor */
49			SquareMatrix3() : SquareMatrix<Real, 3>() {
50			}
51
52	tim	151	/** 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	tim	70	/** copy constructor */
62			SquareMatrix3(const SquareMatrix<Real, 3>& m) : SquareMatrix<Real, 3>(m) {
63			}
64
65	tim	93	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	tim	113	setupRotMat(q);
75
76	tim	93	}
77
78			SquareMatrix3(Real w, Real x, Real y, Real z) {
79	tim	113	setupRotMat(w, x, y, z);
80	tim	93	}
81
82	tim	70	/** 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	tim	101	return *this;
88	tim	70	}
89	tim	76
90			/**
91			* Sets this matrix to a rotation matrix by three euler angles
92			* @ param euler
93			*/
94	tim	93	void setupRotMat(const Vector3<Real>& eulerAngles) {
95			setupRotMat(eulerAngles[0], eulerAngles[1], eulerAngles[2]);
96			}
97	tim	76
98			/**
99			* Sets this matrix to a rotation matrix by three euler angles
100			* @param phi
101			* @param theta
102			* @psi theta
103			*/
104	tim	93	void setupRotMat(Real phi, Real theta, Real psi) {
105			Real sphi, stheta, spsi;
106			Real cphi, ctheta, cpsi;
107	tim	76
108	tim	93	sphi = sin(phi);
109			stheta = sin(theta);
110			spsi = sin(psi);
111			cphi = cos(phi);
112			ctheta = cos(theta);
113			cpsi = cos(psi);
114	tim	76
115	tim	93	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	tim	76	/**
130			* Sets this matrix to a rotation matrix by quaternion
131			* @param quat
132			*/
133	tim	93	void setupRotMat(const Quaternion<Real>& quat) {
134	tim	113	setupRotMat(quat.w(), quat.x(), quat.y(), quat.z());
135	tim	93	}
136	tim	76
137			/**
138			* Sets this matrix to a rotation matrix by quaternion
139	tim	93	* @param w the first element
140			* @param x the second element
141			* @param y the third element
142	tim	101	* @param z the fourth element
143	tim	76	*/
144	tim	93	void setupRotMat(Real w, Real x, Real y, Real z) {
145			Quaternion<Real> q(w, x, y, z);
146			*this = q.toRotationMatrix3();
147			}
148	tim	76
149			/**
150			* Returns the quaternion from this rotation matrix
151			* @return the quaternion from this rotation matrix
152			* @exception invalid rotation matrix
153			*/
154	tim	93	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	tim	76
160	tim	93	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	tim	76	/**
201			* Returns the euler angles from this rotation matrix
202	tim	93	* @return the euler angles in a vector
203	tim	76	* @exception invalid rotation matrix
204	tim	93	* 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	tim	76	*/
210	tim	93	Vector3<Real> toEulerAngles() {
211	tim	113	Vector3<Real> myEuler;
212	tim	93	Real phi,theta,psi,eps;
213			Real ctheta,stheta;
214
215			// set the tolerance for Euler angles and rotation elements
216
217	tim	113	theta = acos(std::min(1.0, std::max(-1.0,data_[2][2])));
218	tim	93	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	tim	70
253	tim	101	/** 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	tim	70	/**
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	tim	101	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	tim	70
277	tim	101	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	tim	99	}
290	tim	123	/**
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	tim	76
306	tim	123	Program: Visualization Toolkit
307			Module: $RCSfile: SquareMatrix3.hpp,v $
308	tim	99
309	tim	123	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	tim	101
313	tim	123	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	tim	101
317	tim	123	=========================================================================*/
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	tim	101
325	tim	123	// 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	tim	101
333	tim	123	// 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	tim	101
356	tim	123	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	tim	101
365	tim	123	// re-orthogonalize the other two eigenvectors
366			j = (maxI+1)%3;
367			k = (maxI+2)%3;
368	tim	101
369	tim	123	v(j, 0) = 0.0;
370			v(j, 1) = 0.0;
371			v(j, 2) = 0.0;
372			v(j, j) = 1.0;
373	tim	101
374	tim	123	/** @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	tim	101
383
384	tim	123	// transpose vectors back to columns
385			v = v.transpose();
386			return;
387			}
388			}
389	tim	101
390	tim	123	// the three eigenvalues are different, just sort the eigenvectors
391			// to align them with the x, y, and z axes
392	tim	101
393	tim	123	// 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	tim	101
404	tim	123	// 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	tim	101
419	tim	123	// 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	tim	99	}
426	tim	123	}
427	tim	70
428	tim	123	// 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	tim	99	typedef SquareMatrix3<double> Mat3x3d;
440			typedef SquareMatrix3<double> RotMat3x3d;
441	tim	93
442			} //namespace oopse
443			#endif // MATH_SQUAREMATRIX_HPP
444	tim	123