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* @date 10/11/2004 |
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* @version 1.0 |
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*/ |
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#ifndef MATH_SQUAREMATRIX3_HPP |
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#ifndef MATH_SQUAREMATRIX3_HPP |
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#define MATH_SQUAREMATRIX3_HPP |
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#include "Quaternion.hpp" |
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m /= det; |
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return m; |
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} |
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|
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void diagonalize(SquareMatrix3<Real>& a, Vector3<Real>& w, SquareMatrix3<Real>& v) { |
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int i,j,k,maxI; |
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Real tmp, maxVal; |
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Vector3<Real> v_maxI, v_k, v_j; |
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/** |
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* Extract the eigenvalues and eigenvectors from a 3x3 matrix. |
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* The eigenvectors (the columns of V) will be normalized. |
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* The eigenvectors are aligned optimally with the x, y, and z |
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* axes respectively. |
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* @param a symmetric matrix whose eigenvectors are to be computed. On return, the matrix is |
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* overwritten |
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* @param w will contain the eigenvalues of the matrix On return of this function |
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* @param v the columns of this matrix will contain the eigenvectors. The eigenvectors are |
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* normalized and mutually orthogonal. |
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* @warning a will be overwritten |
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*/ |
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static void diagonalize(SquareMatrix3<Real>& a, Vector3<Real>& w, SquareMatrix3<Real>& v); |
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}; |
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/*========================================================================= |
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// diagonalize using Jacobi |
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jacobi(a, w, v); |
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Program: Visualization Toolkit |
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Module: $RCSfile: SquareMatrix3.hpp,v $ |
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// if all the eigenvalues are the same, return identity matrix |
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if (w[0] == w[1] && w[0] == w[2] ) { |
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v = SquareMatrix3<Real>::identity(); |
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return; |
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} |
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|
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// transpose temporarily, it makes it easier to sort the eigenvectors |
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v = v.transpose(); |
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|
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// if two eigenvalues are the same, re-orthogonalize to optimally line |
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// up the eigenvectors with the x, y, and z axes |
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for (i = 0; i < 3; i++) { |
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if (w((i+1)%3) == w((i+2)%3)) {// two eigenvalues are the same |
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// find maximum element of the independant eigenvector |
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maxVal = fabs(v(i, 0)); |
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maxI = 0; |
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for (j = 1; j < 3; j++) { |
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if (maxVal < (tmp = fabs(v(i, j)))){ |
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maxVal = tmp; |
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maxI = j; |
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} |
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} |
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|
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// swap the eigenvector into its proper position |
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if (maxI != i) { |
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tmp = w(maxI); |
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w(maxI) = w(i); |
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w(i) = tmp; |
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Copyright (c) Ken Martin, Will Schroeder, Bill Lorensen |
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All rights reserved. |
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See Copyright.txt or http://www.kitware.com/Copyright.htm for details. |
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|
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This software is distributed WITHOUT ANY WARRANTY; without even |
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the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR |
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PURPOSE. See the above copyright notice for more information. |
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v.swapRow(i, maxI); |
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} |
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// maximum element of eigenvector should be positive |
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if (v(maxI, maxI) < 0) { |
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v(maxI, 0) = -v(maxI, 0); |
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v(maxI, 1) = -v(maxI, 1); |
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v(maxI, 2) = -v(maxI, 2); |
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} |
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=========================================================================*/ |
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template<typename Real> |
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void SquareMatrix3<Real>::diagonalize(SquareMatrix3<Real>& a, Vector3<Real>& w, |
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SquareMatrix3<Real>& v) { |
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int i,j,k,maxI; |
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Real tmp, maxVal; |
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Vector3<Real> v_maxI, v_k, v_j; |
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// re-orthogonalize the other two eigenvectors |
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j = (maxI+1)%3; |
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k = (maxI+2)%3; |
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// diagonalize using Jacobi |
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jacobi(a, w, v); |
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// if all the eigenvalues are the same, return identity matrix |
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if (w[0] == w[1] && w[0] == w[2] ) { |
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v = SquareMatrix3<Real>::identity(); |
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return; |
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} |
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v(j, 0) = 0.0; |
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v(j, 1) = 0.0; |
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v(j, 2) = 0.0; |
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v(j, j) = 1.0; |
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// transpose temporarily, it makes it easier to sort the eigenvectors |
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v = v.transpose(); |
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|
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// if two eigenvalues are the same, re-orthogonalize to optimally line |
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// up the eigenvectors with the x, y, and z axes |
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for (i = 0; i < 3; i++) { |
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if (w((i+1)%3) == w((i+2)%3)) {// two eigenvalues are the same |
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// find maximum element of the independant eigenvector |
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maxVal = fabs(v(i, 0)); |
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maxI = 0; |
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for (j = 1; j < 3; j++) { |
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if (maxVal < (tmp = fabs(v(i, j)))){ |
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maxVal = tmp; |
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maxI = j; |
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} |
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} |
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|
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// swap the eigenvector into its proper position |
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if (maxI != i) { |
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tmp = w(maxI); |
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w(maxI) = w(i); |
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w(i) = tmp; |
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/** @todo */ |
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v_maxI = v.getRow(maxI); |
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v_j = v.getRow(j); |
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v_k = cross(v_maxI, v_j); |
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v_k.normalize(); |
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v_j = cross(v_k, v_maxI); |
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v.setRow(j, v_j); |
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v.setRow(k, v_k); |
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v.swapRow(i, maxI); |
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} |
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// maximum element of eigenvector should be positive |
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if (v(maxI, maxI) < 0) { |
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v(maxI, 0) = -v(maxI, 0); |
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v(maxI, 1) = -v(maxI, 1); |
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v(maxI, 2) = -v(maxI, 2); |
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} |
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|
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// re-orthogonalize the other two eigenvectors |
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j = (maxI+1)%3; |
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k = (maxI+2)%3; |
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v(j, 0) = 0.0; |
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v(j, 1) = 0.0; |
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v(j, 2) = 0.0; |
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v(j, j) = 1.0; |
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// transpose vectors back to columns |
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v = v.transpose(); |
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return; |
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} |
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} |
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/** @todo */ |
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v_maxI = v.getRow(maxI); |
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v_j = v.getRow(j); |
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v_k = cross(v_maxI, v_j); |
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v_k.normalize(); |
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v_j = cross(v_k, v_maxI); |
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v.setRow(j, v_j); |
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v.setRow(k, v_k); |
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// the three eigenvalues are different, just sort the eigenvectors |
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// to align them with the x, y, and z axes |
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// find the vector with the largest x element, make that vector |
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// the first vector |
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maxVal = fabs(v(0, 0)); |
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maxI = 0; |
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for (i = 1; i < 3; i++) { |
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if (maxVal < (tmp = fabs(v(i, 0)))) { |
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maxVal = tmp; |
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maxI = i; |
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} |
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} |
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// transpose vectors back to columns |
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v = v.transpose(); |
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return; |
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} |
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} |
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// swap eigenvalue and eigenvector |
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if (maxI != 0) { |
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tmp = w(maxI); |
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w(maxI) = w(0); |
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w(0) = tmp; |
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v.swapRow(maxI, 0); |
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} |
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// do the same for the y element |
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if (fabs(v(1, 1)) < fabs(v(2, 1))) { |
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tmp = w(2); |
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w(2) = w(1); |
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w(1) = tmp; |
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v.swapRow(2, 1); |
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} |
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// the three eigenvalues are different, just sort the eigenvectors |
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// to align them with the x, y, and z axes |
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// ensure that the sign of the eigenvectors is correct |
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for (i = 0; i < 2; i++) { |
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if (v(i, i) < 0) { |
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v(i, 0) = -v(i, 0); |
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v(i, 1) = -v(i, 1); |
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v(i, 2) = -v(i, 2); |
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} |
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} |
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// find the vector with the largest x element, make that vector |
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// the first vector |
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maxVal = fabs(v(0, 0)); |
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maxI = 0; |
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for (i = 1; i < 3; i++) { |
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if (maxVal < (tmp = fabs(v(i, 0)))) { |
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maxVal = tmp; |
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maxI = i; |
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} |
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} |
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// set sign of final eigenvector to ensure that determinant is positive |
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if (v.determinant() < 0) { |
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v(2, 0) = -v(2, 0); |
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v(2, 1) = -v(2, 1); |
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v(2, 2) = -v(2, 2); |
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} |
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// swap eigenvalue and eigenvector |
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if (maxI != 0) { |
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tmp = w(maxI); |
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w(maxI) = w(0); |
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w(0) = tmp; |
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v.swapRow(maxI, 0); |
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} |
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// do the same for the y element |
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if (fabs(v(1, 1)) < fabs(v(2, 1))) { |
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tmp = w(2); |
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w(2) = w(1); |
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w(1) = tmp; |
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v.swapRow(2, 1); |
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} |
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// transpose the eigenvectors back again |
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v = v.transpose(); |
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return ; |
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// ensure that the sign of the eigenvectors is correct |
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for (i = 0; i < 2; i++) { |
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if (v(i, i) < 0) { |
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v(i, 0) = -v(i, 0); |
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v(i, 1) = -v(i, 1); |
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v(i, 2) = -v(i, 2); |
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} |
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}; |
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} |
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|
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// set sign of final eigenvector to ensure that determinant is positive |
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if (v.determinant() < 0) { |
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v(2, 0) = -v(2, 0); |
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v(2, 1) = -v(2, 1); |
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v(2, 2) = -v(2, 2); |
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} |
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|
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// transpose the eigenvectors back again |
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v = v.transpose(); |
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return ; |
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} |
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typedef SquareMatrix3<double> Mat3x3d; |
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typedef SquareMatrix3<double> RotMat3x3d; |
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|
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} //namespace oopse |
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#endif // MATH_SQUAREMATRIX_HPP |
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