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/* |
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* Copyright (c) 2005 The University of Notre Dame. All Rights Reserved. |
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* |
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* The University of Notre Dame grants you ("Licensee") a |
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* non-exclusive, royalty free, license to use, modify and |
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* redistribute this software in source and binary code form, provided |
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* that the following conditions are met: |
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* |
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* 1. Redistributions of source code must retain the above copyright |
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* notice, this list of conditions and the following disclaimer. |
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* |
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* 2. Redistributions in binary form must reproduce the above copyright |
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* notice, this list of conditions and the following disclaimer in the |
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* documentation and/or other materials provided with the |
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* distribution. |
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* |
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* This software is provided "AS IS," without a warranty of any |
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* kind. All express or implied conditions, representations and |
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* warranties, including any implied warranty of merchantability, |
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* fitness for a particular purpose or non-infringement, are hereby |
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* excluded. The University of Notre Dame and its licensors shall not |
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* be liable for any damages suffered by licensee as a result of |
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* using, modifying or distributing the software or its |
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* derivatives. In no event will the University of Notre Dame or its |
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* licensors be liable for any lost revenue, profit or data, or for |
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* direct, indirect, special, consequential, incidental or punitive |
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* damages, however caused and regardless of the theory of liability, |
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* arising out of the use of or inability to use software, even if the |
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* University of Notre Dame has been advised of the possibility of |
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* such damages. |
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* |
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* SUPPORT OPEN SCIENCE! If you use OpenMD or its source code in your |
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* research, please cite the appropriate papers when you publish your |
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* work. Good starting points are: |
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* |
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* [1] Meineke, et al., J. Comp. Chem. 26, 252-271 (2005). |
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* [2] Fennell & Gezelter, J. Chem. Phys. 124, 234104 (2006). |
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* [3] Sun, Lin & Gezelter, J. Chem. Phys. 128, 24107 (2008). |
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* [4] Kuang & Gezelter, J. Chem. Phys. 133, 164101 (2010). |
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* [5] Vardeman, Stocker & Gezelter, J. Chem. Theory Comput. 7, 834 (2011). |
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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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#include "config.h" |
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#include <cmath> |
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#include <vector> |
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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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#include "utils/NumericConstant.hpp" |
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namespace OpenMD { |
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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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SquareMatrix3<Real>& operator =(const Quaternion<Real>& q) { |
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this->setupRotMat(q); |
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return *this; |
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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 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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this->data_[0][0] = cpsi * cphi - ctheta * sphi * spsi; |
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this->data_[0][1] = cpsi * sphi + ctheta * cphi * spsi; |
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this->data_[0][2] = spsi * stheta; |
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|
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this->data_[1][0] = -spsi * ctheta - ctheta * sphi * cpsi; |
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this->data_[1][1] = -spsi * stheta + ctheta * cphi * cpsi; |
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this->data_[1][2] = cpsi * stheta; |
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|
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this->data_[2][0] = stheta * sphi; |
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this->data_[2][1] = -stheta * cphi; |
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this->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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void setupSkewMat(Vector3<Real> v) { |
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setupSkewMat(v[0], v[1], v[2]); |
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} |
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|
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void setupSkewMat(Real v1, Real v2, Real v3) { |
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this->data_[0][0] = 0; |
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this->data_[0][1] = -v3; |
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this->data_[0][2] = v2; |
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this->data_[1][0] = v3; |
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this->data_[1][1] = 0; |
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this->data_[1][2] = -v1; |
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this->data_[2][0] = -v2; |
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this->data_[2][1] = v1; |
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this->data_[2][2] = 0; |
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|
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|
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} |
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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 = this->data_[0][0] + this->data_[1][1] + this->data_[2][2] + 1.0; |
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|
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if( t > NumericConstant::epsilon ){ |
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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] = (this->data_[1][2] - this->data_[2][1]) * s; |
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q[2] = (this->data_[2][0] - this->data_[0][2]) * s; |
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q[3] = (this->data_[0][1] - this->data_[1][0]) * s; |
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} else { |
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|
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ad1 = this->data_[0][0]; |
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ad2 = this->data_[1][1]; |
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ad3 = this->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 = 0.5 / sqrt( 1.0 + this->data_[0][0] - this->data_[1][1] - this->data_[2][2] ); |
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q[0] = (this->data_[1][2] - this->data_[2][1]) * s; |
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q[1] = 0.25 / s; |
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q[2] = (this->data_[0][1] + this->data_[1][0]) * s; |
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q[3] = (this->data_[0][2] + this->data_[2][0]) * s; |
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} else if ( ad2 >= ad1 && ad2 >= ad3 ) { |
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s = 0.5 / sqrt( 1.0 + this->data_[1][1] - this->data_[0][0] - this->data_[2][2] ); |
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q[0] = (this->data_[2][0] - this->data_[0][2] ) * s; |
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q[1] = (this->data_[0][1] + this->data_[1][0]) * s; |
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q[2] = 0.25 / s; |
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q[3] = (this->data_[1][2] + this->data_[2][1]) * s; |
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} else { |
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|
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s = 0.5 / sqrt( 1.0 + this->data_[2][2] - this->data_[0][0] - this->data_[1][1] ); |
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q[0] = (this->data_[0][1] - this->data_[1][0]) * s; |
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q[1] = (this->data_[0][2] + this->data_[2][0]) * s; |
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q[2] = (this->data_[1][2] + this->data_[2][1]) * s; |
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q[3] = 0.25 / 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, |
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* psi), where the first rotation is by an angle phi about the z-axis, |
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* the second is by an angle theta (0 <= theta <= 180) about the x-axis, |
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* and the third is by an angle psi about the 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; |
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Real theta; |
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Real psi; |
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Real ctheta; |
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Real 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((RealType)1.0, std::max((RealType)-1.0,this->data_[2][2]))); |
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ctheta = this->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 |
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// singularity In this case, we can assign an arbitary value to |
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// phi (or psi), and then determine the psi (or phi) or |
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// vice-versa. We'll assume that phi always gets the rotation, |
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// and psi is 0 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 |
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// non-negative. Therefore, it will never change the sign of both of |
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// the parameters passed to atan2. |
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|
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if (fabs(stheta) < 1e-6){ |
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psi = 0.0; |
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phi = atan2(-this->data_[1][0], this->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(this->data_[2][0], -this->data_[2][1]); |
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psi = atan2(this->data_[0][2], this->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 += 2.0 * M_PI; |
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|
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if (psi < 0) |
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psi += 2.0 * 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 = this->data_[0][0] * (this->data_[1][1] * this->data_[2][2] - this->data_[1][2] * this->data_[2][1]); |
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y = this->data_[0][1] * (this->data_[1][2] * this->data_[2][0] - this->data_[1][0] * this->data_[2][2]); |
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z = this->data_[0][2] * (this->data_[1][0] * this->data_[2][1] - this->data_[1][1] * this->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 this->data_[0][0] + this->data_[1][1] + this->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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RealType det = determinant(); |
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if (fabs(det) <= OpenMD::epsilon) { |
| 328 |
//"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."); |
| 330 |
std::vector<int> zeroDiagElementIndex; |
| 331 |
for (int i =0; i < 3; ++i) { |
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if (fabs(this->data_[i][i]) <= OpenMD::epsilon) { |
| 333 |
zeroDiagElementIndex.push_back(i); |
| 334 |
} |
| 335 |
} |
| 336 |
|
| 337 |
if (zeroDiagElementIndex.size() == 2) { |
| 338 |
int index = zeroDiagElementIndex[0]; |
| 339 |
m(index, index) = 1.0 / this->data_[index][index]; |
| 340 |
}else if (zeroDiagElementIndex.size() == 1) { |
| 341 |
|
| 342 |
int a = (zeroDiagElementIndex[0] + 1) % 3; |
| 343 |
int b = (zeroDiagElementIndex[0] + 2) %3; |
| 344 |
RealType denom = this->data_[a][a] * this->data_[b][b] - this->data_[b][a]*this->data_[a][b]; |
| 345 |
m(a, a) = this->data_[b][b] /denom; |
| 346 |
m(b, a) = -this->data_[b][a]/denom; |
| 347 |
|
| 348 |
m(a,b) = -this->data_[a][b]/denom; |
| 349 |
m(b, b) = this->data_[a][a]/denom; |
| 350 |
|
| 351 |
} |
| 352 |
|
| 353 |
/* |
| 354 |
for(std::vector<int>::iterator iter = zeroDiagElementIndex.begin(); iter != zeroDiagElementIndex.end() ++iter) { |
| 355 |
if (this->data_[*iter][0] > OpenMD::epsilon || this->data_[*iter][1] ||this->data_[*iter][2] || |
| 356 |
this->data_[0][*iter] > OpenMD::epsilon || this->data_[1][*iter] ||this->data_[2][*iter] ) { |
| 357 |
std::cout << "can not inverse matrix" << std::endl; |
| 358 |
} |
| 359 |
} |
| 360 |
*/ |
| 361 |
} else { |
| 362 |
|
| 363 |
m(0, 0) = this->data_[1][1]*this->data_[2][2] - this->data_[1][2]*this->data_[2][1]; |
| 364 |
m(1, 0) = this->data_[1][2]*this->data_[2][0] - this->data_[1][0]*this->data_[2][2]; |
| 365 |
m(2, 0) = this->data_[1][0]*this->data_[2][1] - this->data_[1][1]*this->data_[2][0]; |
| 366 |
m(0, 1) = this->data_[2][1]*this->data_[0][2] - this->data_[2][2]*this->data_[0][1]; |
| 367 |
m(1, 1) = this->data_[2][2]*this->data_[0][0] - this->data_[2][0]*this->data_[0][2]; |
| 368 |
m(2, 1) = this->data_[2][0]*this->data_[0][1] - this->data_[2][1]*this->data_[0][0]; |
| 369 |
m(0, 2) = this->data_[0][1]*this->data_[1][2] - this->data_[0][2]*this->data_[1][1]; |
| 370 |
m(1, 2) = this->data_[0][2]*this->data_[1][0] - this->data_[0][0]*this->data_[1][2]; |
| 371 |
m(2, 2) = this->data_[0][0]*this->data_[1][1] - this->data_[0][1]*this->data_[1][0]; |
| 372 |
|
| 373 |
m /= det; |
| 374 |
} |
| 375 |
return m; |
| 376 |
} |
| 377 |
|
| 378 |
SquareMatrix3<Real> transpose() const{ |
| 379 |
SquareMatrix3<Real> result; |
| 380 |
|
| 381 |
for (unsigned int i = 0; i < 3; i++) |
| 382 |
for (unsigned int j = 0; j < 3; j++) |
| 383 |
result(j, i) = this->data_[i][j]; |
| 384 |
|
| 385 |
return result; |
| 386 |
} |
| 387 |
/** |
| 388 |
* Extract the eigenvalues and eigenvectors from a 3x3 matrix. |
| 389 |
* The eigenvectors (the columns of V) will be normalized. |
| 390 |
* The eigenvectors are aligned optimally with the x, y, and z |
| 391 |
* axes respectively. |
| 392 |
* @param a symmetric matrix whose eigenvectors are to be computed. On return, the matrix is |
| 393 |
* overwritten |
| 394 |
* @param w will contain the eigenvalues of the matrix On return of this function |
| 395 |
* @param v the columns of this matrix will contain the eigenvectors. The eigenvectors are |
| 396 |
* normalized and mutually orthogonal. |
| 397 |
* @warning a will be overwritten |
| 398 |
*/ |
| 399 |
static void diagonalize(SquareMatrix3<Real>& a, Vector3<Real>& w, SquareMatrix3<Real>& v); |
| 400 |
}; |
| 401 |
/*========================================================================= |
| 402 |
|
| 403 |
Program: Visualization Toolkit |
| 404 |
Module: $RCSfile: SquareMatrix3.hpp,v $ |
| 405 |
|
| 406 |
Copyright (c) Ken Martin, Will Schroeder, Bill Lorensen |
| 407 |
All rights reserved. |
| 408 |
See Copyright.txt or http://www.kitware.com/Copyright.htm for details. |
| 409 |
|
| 410 |
This software is distributed WITHOUT ANY WARRANTY; without even |
| 411 |
the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR |
| 412 |
PURPOSE. See the above copyright notice for more information. |
| 413 |
|
| 414 |
=========================================================================*/ |
| 415 |
template<typename Real> |
| 416 |
void SquareMatrix3<Real>::diagonalize(SquareMatrix3<Real>& a, Vector3<Real>& w, |
| 417 |
SquareMatrix3<Real>& v) { |
| 418 |
int i,j,k,maxI; |
| 419 |
Real tmp, maxVal; |
| 420 |
Vector3<Real> v_maxI, v_k, v_j; |
| 421 |
|
| 422 |
// diagonalize using Jacobi |
| 423 |
SquareMatrix3<Real>::jacobi(a, w, v); |
| 424 |
// if all the eigenvalues are the same, return identity matrix |
| 425 |
if (w[0] == w[1] && w[0] == w[2] ) { |
| 426 |
v = SquareMatrix3<Real>::identity(); |
| 427 |
return; |
| 428 |
} |
| 429 |
|
| 430 |
// transpose temporarily, it makes it easier to sort the eigenvectors |
| 431 |
v = v.transpose(); |
| 432 |
|
| 433 |
// if two eigenvalues are the same, re-orthogonalize to optimally line |
| 434 |
// up the eigenvectors with the x, y, and z axes |
| 435 |
for (i = 0; i < 3; i++) { |
| 436 |
if (w((i+1)%3) == w((i+2)%3)) {// two eigenvalues are the same |
| 437 |
// find maximum element of the independant eigenvector |
| 438 |
maxVal = fabs(v(i, 0)); |
| 439 |
maxI = 0; |
| 440 |
for (j = 1; j < 3; j++) { |
| 441 |
if (maxVal < (tmp = fabs(v(i, j)))){ |
| 442 |
maxVal = tmp; |
| 443 |
maxI = j; |
| 444 |
} |
| 445 |
} |
| 446 |
|
| 447 |
// swap the eigenvector into its proper position |
| 448 |
if (maxI != i) { |
| 449 |
tmp = w(maxI); |
| 450 |
w(maxI) = w(i); |
| 451 |
w(i) = tmp; |
| 452 |
|
| 453 |
v.swapRow(i, maxI); |
| 454 |
} |
| 455 |
// maximum element of eigenvector should be positive |
| 456 |
if (v(maxI, maxI) < 0) { |
| 457 |
v(maxI, 0) = -v(maxI, 0); |
| 458 |
v(maxI, 1) = -v(maxI, 1); |
| 459 |
v(maxI, 2) = -v(maxI, 2); |
| 460 |
} |
| 461 |
|
| 462 |
// re-orthogonalize the other two eigenvectors |
| 463 |
j = (maxI+1)%3; |
| 464 |
k = (maxI+2)%3; |
| 465 |
|
| 466 |
v(j, 0) = 0.0; |
| 467 |
v(j, 1) = 0.0; |
| 468 |
v(j, 2) = 0.0; |
| 469 |
v(j, j) = 1.0; |
| 470 |
|
| 471 |
/** @todo */ |
| 472 |
v_maxI = v.getRow(maxI); |
| 473 |
v_j = v.getRow(j); |
| 474 |
v_k = cross(v_maxI, v_j); |
| 475 |
v_k.normalize(); |
| 476 |
v_j = cross(v_k, v_maxI); |
| 477 |
v.setRow(j, v_j); |
| 478 |
v.setRow(k, v_k); |
| 479 |
|
| 480 |
|
| 481 |
// transpose vectors back to columns |
| 482 |
v = v.transpose(); |
| 483 |
return; |
| 484 |
} |
| 485 |
} |
| 486 |
|
| 487 |
// the three eigenvalues are different, just sort the eigenvectors |
| 488 |
// to align them with the x, y, and z axes |
| 489 |
|
| 490 |
// find the vector with the largest x element, make that vector |
| 491 |
// the first vector |
| 492 |
maxVal = fabs(v(0, 0)); |
| 493 |
maxI = 0; |
| 494 |
for (i = 1; i < 3; i++) { |
| 495 |
if (maxVal < (tmp = fabs(v(i, 0)))) { |
| 496 |
maxVal = tmp; |
| 497 |
maxI = i; |
| 498 |
} |
| 499 |
} |
| 500 |
|
| 501 |
// swap eigenvalue and eigenvector |
| 502 |
if (maxI != 0) { |
| 503 |
tmp = w(maxI); |
| 504 |
w(maxI) = w(0); |
| 505 |
w(0) = tmp; |
| 506 |
v.swapRow(maxI, 0); |
| 507 |
} |
| 508 |
// do the same for the y element |
| 509 |
if (fabs(v(1, 1)) < fabs(v(2, 1))) { |
| 510 |
tmp = w(2); |
| 511 |
w(2) = w(1); |
| 512 |
w(1) = tmp; |
| 513 |
v.swapRow(2, 1); |
| 514 |
} |
| 515 |
|
| 516 |
// ensure that the sign of the eigenvectors is correct |
| 517 |
for (i = 0; i < 2; i++) { |
| 518 |
if (v(i, i) < 0) { |
| 519 |
v(i, 0) = -v(i, 0); |
| 520 |
v(i, 1) = -v(i, 1); |
| 521 |
v(i, 2) = -v(i, 2); |
| 522 |
} |
| 523 |
} |
| 524 |
|
| 525 |
// set sign of final eigenvector to ensure that determinant is positive |
| 526 |
if (v.determinant() < 0) { |
| 527 |
v(2, 0) = -v(2, 0); |
| 528 |
v(2, 1) = -v(2, 1); |
| 529 |
v(2, 2) = -v(2, 2); |
| 530 |
} |
| 531 |
|
| 532 |
// transpose the eigenvectors back again |
| 533 |
v = v.transpose(); |
| 534 |
return ; |
| 535 |
} |
| 536 |
|
| 537 |
/** |
| 538 |
* Return the multiplication of two matrixes (m1 * m2). |
| 539 |
* @return the multiplication of two matrixes |
| 540 |
* @param m1 the first matrix |
| 541 |
* @param m2 the second matrix |
| 542 |
*/ |
| 543 |
template<typename Real> |
| 544 |
inline SquareMatrix3<Real> operator *(const SquareMatrix3<Real>& m1, const SquareMatrix3<Real>& m2) { |
| 545 |
SquareMatrix3<Real> result; |
| 546 |
|
| 547 |
for (unsigned int i = 0; i < 3; i++) |
| 548 |
for (unsigned int j = 0; j < 3; j++) |
| 549 |
for (unsigned int k = 0; k < 3; k++) |
| 550 |
result(i, j) += m1(i, k) * m2(k, j); |
| 551 |
|
| 552 |
return result; |
| 553 |
} |
| 554 |
|
| 555 |
template<typename Real> |
| 556 |
inline SquareMatrix3<Real> outProduct(const Vector3<Real>& v1, const Vector3<Real>& v2) { |
| 557 |
SquareMatrix3<Real> result; |
| 558 |
|
| 559 |
for (unsigned int i = 0; i < 3; i++) { |
| 560 |
for (unsigned int j = 0; j < 3; j++) { |
| 561 |
result(i, j) = v1[i] * v2[j]; |
| 562 |
} |
| 563 |
} |
| 564 |
|
| 565 |
return result; |
| 566 |
} |
| 567 |
|
| 568 |
|
| 569 |
typedef SquareMatrix3<RealType> Mat3x3d; |
| 570 |
typedef SquareMatrix3<RealType> RotMat3x3d; |
| 571 |
|
| 572 |
} //namespace OpenMD |
| 573 |
#endif // MATH_SQUAREMATRIX_HPP |
| 574 |
|
