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/* |
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* Copyright (C) 2000-2004 Object Oriented Parallel Simulation Engine (OOPSE) project |
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* |
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* Contact: oopse@oopse.org |
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* |
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* This program is free software; you can redistribute it and/or |
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* modify it under the terms of the GNU Lesser General Public License |
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* as published by the Free Software Foundation; either version 2.1 |
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* of the License, or (at your option) any later version. |
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* All we ask is that proper credit is given for our work, which includes |
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* - but is not limited to - adding the above copyright notice to the beginning |
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* of your source code files, and to any copyright notice that you may distribute |
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* with programs based on this work. |
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* |
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* This program is distributed in the hope that it will be useful, |
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* but WITHOUT ANY WARRANTY; without even the implied warranty of |
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* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
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* GNU Lesser General Public License for more details. |
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* |
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* You should have received a copy of the GNU Lesser General Public License |
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* along with this program; if not, write to the Free Software |
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* Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA. |
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* |
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*/ |
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|
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/** |
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* @file SquareMatrix3.hpp |
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* @author Teng Lin |
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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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#define MATH_SQUAREMATRIX3_HPP |
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|
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#include "Quaternion.hpp" |
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#include "SquareMatrix.hpp" |
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#include "Vector3.hpp" |
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|
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namespace oopse { |
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|
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template<typename Real> |
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class SquareMatrix3 : public SquareMatrix<Real, 3> { |
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public: |
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|
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typedef Real ElemType; |
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typedef Real* ElemPoinerType; |
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|
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/** default constructor */ |
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SquareMatrix3() : SquareMatrix<Real, 3>() { |
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} |
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|
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/** Constructs and initializes every element of this matrix to a scalar */ |
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SquareMatrix3(Real s) : SquareMatrix<Real,3>(s){ |
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} |
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|
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/** Constructs and initializes from an array */ |
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SquareMatrix3(Real* array) : SquareMatrix<Real,3>(array){ |
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} |
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|
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|
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/** copy constructor */ |
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SquareMatrix3(const SquareMatrix<Real, 3>& m) : SquareMatrix<Real, 3>(m) { |
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} |
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|
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SquareMatrix3( const Vector3<Real>& eulerAngles) { |
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setupRotMat(eulerAngles); |
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} |
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|
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SquareMatrix3(Real phi, Real theta, Real psi) { |
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setupRotMat(phi, theta, psi); |
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} |
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|
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SquareMatrix3(const Quaternion<Real>& q) { |
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setupRotMat(q); |
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|
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} |
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|
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SquareMatrix3(Real w, Real x, Real y, Real z) { |
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setupRotMat(w, x, y, z); |
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} |
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|
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/** copy assignment operator */ |
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SquareMatrix3<Real>& operator =(const SquareMatrix<Real, 3>& m) { |
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if (this == &m) |
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return *this; |
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SquareMatrix<Real, 3>::operator=(m); |
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return *this; |
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} |
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|
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/** |
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* Sets this matrix to a rotation matrix by three euler angles |
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* @ param euler |
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*/ |
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void setupRotMat(const Vector3<Real>& eulerAngles) { |
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setupRotMat(eulerAngles[0], eulerAngles[1], eulerAngles[2]); |
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} |
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|
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/** |
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* Sets this matrix to a rotation matrix by three euler angles |
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* @param phi |
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* @param theta |
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* @psi theta |
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*/ |
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void setupRotMat(Real phi, Real theta, Real psi) { |
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Real sphi, stheta, spsi; |
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Real cphi, ctheta, cpsi; |
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|
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sphi = sin(phi); |
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stheta = sin(theta); |
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spsi = sin(psi); |
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cphi = cos(phi); |
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ctheta = cos(theta); |
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cpsi = cos(psi); |
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|
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data_[0][0] = cpsi * cphi - ctheta * sphi * spsi; |
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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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|
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data_[2][0] = stheta * sphi; |
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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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|
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|
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/** |
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* Sets this matrix to a rotation matrix by quaternion |
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* @param quat |
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*/ |
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void setupRotMat(const Quaternion<Real>& quat) { |
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setupRotMat(quat.w(), quat.x(), quat.y(), quat.z()); |
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} |
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|
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/** |
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* Sets this matrix to a rotation matrix by quaternion |
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* @param w the first element |
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* @param x the second element |
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* @param y the third element |
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* @param z the fourth element |
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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); |
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*this = q.toRotationMatrix3(); |
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} |
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|
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/** |
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* Returns the quaternion from this rotation matrix |
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* @return the quaternion from this rotation matrix |
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* @exception invalid rotation matrix |
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*/ |
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Quaternion<Real> toQuaternion() { |
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Quaternion<Real> q; |
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Real t, s; |
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Real ad1, ad2, ad3; |
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t = data_[0][0] + data_[1][1] + data_[2][2] + 1.0; |
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|
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if( t > 0.0 ){ |
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|
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s = 0.5 / sqrt( t ); |
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q[0] = 0.25 / s; |
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q[1] = (data_[1][2] - data_[2][1]) * s; |
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q[2] = (data_[2][0] - data_[0][2]) * s; |
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q[3] = (data_[0][1] - data_[1][0]) * s; |
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} else { |
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|
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ad1 = fabs( data_[0][0] ); |
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ad2 = fabs( data_[1][1] ); |
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ad3 = fabs( data_[2][2] ); |
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|
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if( ad1 >= ad2 && ad1 >= ad3 ){ |
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|
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s = 2.0 * sqrt( 1.0 + data_[0][0] - data_[1][1] - data_[2][2] ); |
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q[0] = (data_[1][2] + data_[2][1]) / s; |
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q[1] = 0.5 / s; |
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q[2] = (data_[0][1] + data_[1][0]) / s; |
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q[3] = (data_[0][2] + data_[2][0]) / s; |
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} else if ( ad2 >= ad1 && ad2 >= ad3 ) { |
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s = sqrt( 1.0 + data_[1][1] - data_[0][0] - data_[2][2] ) * 2.0; |
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q[0] = (data_[0][2] + data_[2][0]) / s; |
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q[1] = (data_[0][1] + data_[1][0]) / s; |
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q[2] = 0.5 / s; |
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q[3] = (data_[1][2] + data_[2][1]) / s; |
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} else { |
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|
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s = sqrt( 1.0 + data_[2][2] - data_[0][0] - data_[1][1] ) * 2.0; |
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q[0] = (data_[0][1] + data_[1][0]) / s; |
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q[1] = (data_[0][2] + data_[2][0]) / s; |
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q[2] = (data_[1][2] + data_[2][1]) / s; |
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q[3] = 0.5 / s; |
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} |
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} |
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|
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return q; |
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|
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} |
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|
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/** |
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* Returns the euler angles from this rotation matrix |
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* @return the euler angles in a vector |
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* @exception invalid rotation matrix |
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* We use so-called "x-convention", which is the most common definition. |
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* In this convention, the rotation given by Euler angles (phi, theta, psi), where the first |
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* rotation is by an angle phi about the z-axis, the second is by an angle |
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* theta (0 <= theta <= 180)about the x-axis, and thethird is by an angle psi about the |
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* z-axis (again). |
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*/ |
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Vector3<Real> toEulerAngles() { |
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Vector3<Real> myEuler; |
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Real phi,theta,psi,eps; |
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Real ctheta,stheta; |
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|
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// set the tolerance for Euler angles and rotation elements |
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|
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theta = acos(std::min(1.0, std::max(-1.0,data_[2][2]))); |
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ctheta = data_[2][2]; |
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stheta = sqrt(1.0 - ctheta * ctheta); |
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|
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// when sin(theta) is close to 0, we need to consider singularity |
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// In this case, we can assign an arbitary value to phi (or psi), and then determine |
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// the psi (or phi) or vice-versa. We'll assume that phi always gets the rotation, and psi is 0 |
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// in cases of singularity. |
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// we use atan2 instead of atan, since atan2 will give us -Pi to Pi. |
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// Since 0 <= theta <= 180, sin(theta) will be always non-negative. Therefore, it never |
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// change the sign of both of the parameters passed to atan2. |
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|
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if (fabs(stheta) <= oopse::epsilon){ |
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psi = 0.0; |
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phi = atan2(-data_[1][0], data_[0][0]); |
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} |
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// we only have one unique solution |
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else{ |
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phi = atan2(data_[2][0], -data_[2][1]); |
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psi = atan2(data_[0][2], data_[1][2]); |
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} |
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|
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//wrap phi and psi, make sure they are in the range from 0 to 2*Pi |
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if (phi < 0) |
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phi += M_PI; |
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|
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if (psi < 0) |
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psi += M_PI; |
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|
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myEuler[0] = phi; |
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myEuler[1] = theta; |
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myEuler[2] = psi; |
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|
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return myEuler; |
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} |
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|
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/** Returns the determinant of this matrix. */ |
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Real determinant() const { |
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Real x,y,z; |
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|
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x = data_[0][0] * (data_[1][1] * data_[2][2] - data_[1][2] * data_[2][1]); |
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y = data_[0][1] * (data_[1][2] * data_[2][0] - data_[1][0] * data_[2][2]); |
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z = data_[0][2] * (data_[1][0] * data_[2][1] - data_[1][1] * data_[2][0]); |
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|
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return(x + y + z); |
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} |
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|
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/** Returns the trace of this matrix. */ |
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Real trace() const { |
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return data_[0][0] + data_[1][1] + data_[2][2]; |
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} |
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|
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/** |
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* Sets the value of this matrix to the inversion of itself. |
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* @note since simple algorithm can be applied to inverse the 3 by 3 matrix, we hide the |
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* implementation of inverse in SquareMatrix class |
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*/ |
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SquareMatrix3<Real> inverse() const { |
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SquareMatrix3<Real> m; |
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double det = determinant(); |
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if (fabs(det) <= oopse::epsilon) { |
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//"The method was called on a matrix with |determinant| <= 1e-6.", |
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//"This is a runtime or a programming error in your application."); |
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} |
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|
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m(0, 0) = data_[1][1]*data_[2][2] - data_[1][2]*data_[2][1]; |
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m(1, 0) = data_[1][2]*data_[2][0] - data_[1][0]*data_[2][2]; |
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m(2, 0) = data_[1][0]*data_[2][1] - data_[1][1]*data_[2][0]; |
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m(0, 1) = data_[2][1]*data_[0][2] - data_[2][2]*data_[0][1]; |
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m(1, 1) = data_[2][2]*data_[0][0] - data_[2][0]*data_[0][2]; |
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m(2, 1) = data_[2][0]*data_[0][1] - data_[2][1]*data_[0][0]; |
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m(0, 2) = data_[0][1]*data_[1][2] - data_[0][2]*data_[1][1]; |
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m(1, 2) = data_[0][2]*data_[1][0] - data_[0][0]*data_[1][2]; |
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m(2, 2) = data_[0][0]*data_[1][1] - data_[0][1]*data_[1][0]; |
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|
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m /= det; |
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return m; |
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} |
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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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|
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Program: Visualization Toolkit |
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Module: $RCSfile: SquareMatrix3.hpp,v $ |
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|
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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. |
| 317 |
|
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This software is distributed WITHOUT ANY WARRANTY; without even |
| 319 |
the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR |
| 320 |
PURPOSE. See the above copyright notice for more information. |
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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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|
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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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|
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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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|
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v.swapRow(i, maxI); |
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} |
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// maximum element of eigenvector should be positive |
| 364 |
if (v(maxI, maxI) < 0) { |
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v(maxI, 0) = -v(maxI, 0); |
| 366 |
v(maxI, 1) = -v(maxI, 1); |
| 367 |
v(maxI, 2) = -v(maxI, 2); |
| 368 |
} |
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|
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// re-orthogonalize the other two eigenvectors |
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j = (maxI+1)%3; |
| 372 |
k = (maxI+2)%3; |
| 373 |
|
| 374 |
v(j, 0) = 0.0; |
| 375 |
v(j, 1) = 0.0; |
| 376 |
v(j, 2) = 0.0; |
| 377 |
v(j, j) = 1.0; |
| 378 |
|
| 379 |
/** @todo */ |
| 380 |
v_maxI = v.getRow(maxI); |
| 381 |
v_j = v.getRow(j); |
| 382 |
v_k = cross(v_maxI, v_j); |
| 383 |
v_k.normalize(); |
| 384 |
v_j = cross(v_k, v_maxI); |
| 385 |
v.setRow(j, v_j); |
| 386 |
v.setRow(k, v_k); |
| 387 |
|
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|
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// transpose vectors back to columns |
| 390 |
v = v.transpose(); |
| 391 |
return; |
| 392 |
} |
| 393 |
} |
| 394 |
|
| 395 |
// the three eigenvalues are different, just sort the eigenvectors |
| 396 |
// to align them with the x, y, and z axes |
| 397 |
|
| 398 |
// find the vector with the largest x element, make that vector |
| 399 |
// the first vector |
| 400 |
maxVal = fabs(v(0, 0)); |
| 401 |
maxI = 0; |
| 402 |
for (i = 1; i < 3; i++) { |
| 403 |
if (maxVal < (tmp = fabs(v(i, 0)))) { |
| 404 |
maxVal = tmp; |
| 405 |
maxI = i; |
| 406 |
} |
| 407 |
} |
| 408 |
|
| 409 |
// swap eigenvalue and eigenvector |
| 410 |
if (maxI != 0) { |
| 411 |
tmp = w(maxI); |
| 412 |
w(maxI) = w(0); |
| 413 |
w(0) = tmp; |
| 414 |
v.swapRow(maxI, 0); |
| 415 |
} |
| 416 |
// do the same for the y element |
| 417 |
if (fabs(v(1, 1)) < fabs(v(2, 1))) { |
| 418 |
tmp = w(2); |
| 419 |
w(2) = w(1); |
| 420 |
w(1) = tmp; |
| 421 |
v.swapRow(2, 1); |
| 422 |
} |
| 423 |
|
| 424 |
// ensure that the sign of the eigenvectors is correct |
| 425 |
for (i = 0; i < 2; i++) { |
| 426 |
if (v(i, i) < 0) { |
| 427 |
v(i, 0) = -v(i, 0); |
| 428 |
v(i, 1) = -v(i, 1); |
| 429 |
v(i, 2) = -v(i, 2); |
| 430 |
} |
| 431 |
} |
| 432 |
|
| 433 |
// set sign of final eigenvector to ensure that determinant is positive |
| 434 |
if (v.determinant() < 0) { |
| 435 |
v(2, 0) = -v(2, 0); |
| 436 |
v(2, 1) = -v(2, 1); |
| 437 |
v(2, 2) = -v(2, 2); |
| 438 |
} |
| 439 |
|
| 440 |
// transpose the eigenvectors back again |
| 441 |
v = v.transpose(); |
| 442 |
return ; |
| 443 |
} |
| 444 |
|
| 445 |
/** |
| 446 |
* Return the multiplication of two matrixes (m1 * m2). |
| 447 |
* @return the multiplication of two matrixes |
| 448 |
* @param m1 the first matrix |
| 449 |
* @param m2 the second matrix |
| 450 |
*/ |
| 451 |
template<typename Real> |
| 452 |
inline SquareMatrix3<Real> operator *(const SquareMatrix3<Real>& m1, const SquareMatrix3<Real>& m2) { |
| 453 |
SquareMatrix3<Real> result; |
| 454 |
|
| 455 |
for (unsigned int i = 0; i < 3; i++) |
| 456 |
for (unsigned int j = 0; j < 3; j++) |
| 457 |
for (unsigned int k = 0; k < 3; k++) |
| 458 |
result(i, j) += m1(i, k) * m2(k, j); |
| 459 |
|
| 460 |
return result; |
| 461 |
} |
| 462 |
|
| 463 |
template<typename Real> |
| 464 |
inline SquareMatrix3<Real> outProduct(const Vector3<Real>& v1, const Vector3<Real>& v2) { |
| 465 |
SquareMatrix3<Real> result; |
| 466 |
|
| 467 |
for (unsigned int i = 0; i < 3; i++) { |
| 468 |
for (unsigned int j = 0; j < 3; j++) { |
| 469 |
result(i, j) = v1[i] * v2[j]; |
| 470 |
} |
| 471 |
} |
| 472 |
|
| 473 |
return result; |
| 474 |
} |
| 475 |
|
| 476 |
|
| 477 |
typedef SquareMatrix3<double> Mat3x3d; |
| 478 |
typedef SquareMatrix3<double> RotMat3x3d; |
| 479 |
|
| 480 |
} //namespace oopse |
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#endif // MATH_SQUAREMATRIX_HPP |
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