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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_SQUAREMATRIX#_HPP |
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#define MATH_SQUAREMATRIX#_HPP |
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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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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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*this = q.toRotationMatrix3(); |
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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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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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/** 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 Vector3d& euler); |
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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 theta |
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* @psi theta |
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*/ |
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void setupRotMat(double phi, double theta, double psi); |
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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 Vector4d& quat); |
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void setupRotMat(const Quaternion<Real>& quat) { |
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*this = quat.toRotationMatrix3(); |
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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 q0 |
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* @param q1 |
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* @param q2 |
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* @parma q3 |
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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(double q0, double q1, double q2, double q4); |
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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 rotMatToQuat(); |
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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 quaternion 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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Vector3d rotMatToEuler(); |
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Vector3<Real> toEulerAngles() { |
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Vector<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(min(1.0,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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/** |
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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 |
255 |
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* implementation of inverse in SquareMatrix class |
256 |
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*/ |
257 |
< |
void inverse(); |
257 |
> |
SquareMatrix3<Real> inverse() { |
258 |
> |
SquareMatrix3<Real> m; |
259 |
> |
double det = determinant(); |
260 |
> |
if (fabs(det) <= oopse::epsilon) { |
261 |
> |
//"The method was called on a matrix with |determinant| <= 1e-6.", |
262 |
> |
//"This is a runtime or a programming error in your application."); |
263 |
> |
} |
264 |
|
|
265 |
< |
void diagonalize(); |
265 |
> |
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]; |
267 |
> |
m(2, 0) = data_[1][0]*data_[2][1] - data_[1][1]*data_[2][0]; |
268 |
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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]; |
270 |
> |
m(2, 1) = data_[2][0]*data_[0][1] - data_[2][1]*data_[0][0]; |
271 |
> |
m(0, 2) = data_[0][1]*data_[1][2] - data_[0][2]*data_[1][1]; |
272 |
> |
m(1, 2) = data_[0][2]*data_[1][0] - data_[0][0]*data_[1][2]; |
273 |
> |
m(2, 2) = data_[0][0]*data_[1][1] - data_[0][1]*data_[1][0]; |
274 |
|
|
275 |
< |
} |
275 |
> |
m /= det; |
276 |
> |
return m; |
277 |
> |
} |
278 |
|
|
279 |
+ |
void diagonalize(SquareMatrix3<Real>& a, Vector3<Real>& w, SquareMatrix3<Real>& v) { |
280 |
+ |
int i,j,k,maxI; |
281 |
+ |
Real tmp, maxVal; |
282 |
+ |
Vector3<Real> v_maxI, v_k, v_j; |
283 |
+ |
|
284 |
+ |
// diagonalize using Jacobi |
285 |
+ |
jacobi(a, w, v); |
286 |
+ |
|
287 |
+ |
// if all the eigenvalues are the same, return identity matrix |
288 |
+ |
if (w[0] == w[1] && w[0] == w[2] ){ |
289 |
+ |
v = SquareMatrix3<Real>::identity(); |
290 |
+ |
return |
291 |
+ |
} |
292 |
+ |
|
293 |
+ |
// transpose temporarily, it makes it easier to sort the eigenvectors |
294 |
+ |
v = v.tanspose(); |
295 |
+ |
|
296 |
+ |
// if two eigenvalues are the same, re-orthogonalize to optimally line |
297 |
+ |
// up the eigenvectors with the x, y, and z axes |
298 |
+ |
for (i = 0; i < 3; i++) { |
299 |
+ |
if (w((i+1)%3) == w((i+2)%3)) {// two eigenvalues are the same |
300 |
+ |
// find maximum element of the independant eigenvector |
301 |
+ |
maxVal = fabs(v(i, 0)); |
302 |
+ |
maxI = 0; |
303 |
+ |
for (j = 1; j < 3; j++) { |
304 |
+ |
if (maxVal < (tmp = fabs(v(i, j)))){ |
305 |
+ |
maxVal = tmp; |
306 |
+ |
maxI = j; |
307 |
+ |
} |
308 |
+ |
} |
309 |
+ |
|
310 |
+ |
// swap the eigenvector into its proper position |
311 |
+ |
if (maxI != i) { |
312 |
+ |
tmp = w(maxI); |
313 |
+ |
w(maxI) = w(i); |
314 |
+ |
w(i) = tmp; |
315 |
+ |
|
316 |
+ |
v.swapRow(i, maxI); |
317 |
+ |
} |
318 |
+ |
// maximum element of eigenvector should be positive |
319 |
+ |
if (v(maxI, maxI) < 0) { |
320 |
+ |
v(maxI, 0) = -v(maxI, 0); |
321 |
+ |
v(maxI, 1) = -v(maxI, 1); |
322 |
+ |
v(maxI, 2) = -v(maxI, 2); |
323 |
+ |
} |
324 |
+ |
|
325 |
+ |
// re-orthogonalize the other two eigenvectors |
326 |
+ |
j = (maxI+1)%3; |
327 |
+ |
k = (maxI+2)%3; |
328 |
+ |
|
329 |
+ |
v(j, 0) = 0.0; |
330 |
+ |
v(j, 1) = 0.0; |
331 |
+ |
v(j, 2) = 0.0; |
332 |
+ |
v(j, j) = 1.0; |
333 |
+ |
|
334 |
+ |
/** @todo */ |
335 |
+ |
v_maxI = v.getRow(maxI); |
336 |
+ |
v_j = v.getRow(j); |
337 |
+ |
v_k = cross(v_maxI, v_j); |
338 |
+ |
v_k.normailze(); |
339 |
+ |
v_j = cross(v_k, v_maxI); |
340 |
+ |
v.setRow(j, v_j); |
341 |
+ |
v.setRow(k, v_k); |
342 |
+ |
|
343 |
+ |
|
344 |
+ |
// transpose vectors back to columns |
345 |
+ |
v = v.transpose(); |
346 |
+ |
return; |
347 |
+ |
} |
348 |
+ |
} |
349 |
+ |
|
350 |
+ |
// the three eigenvalues are different, just sort the eigenvectors |
351 |
+ |
// to align them with the x, y, and z axes |
352 |
+ |
|
353 |
+ |
// find the vector with the largest x element, make that vector |
354 |
+ |
// the first vector |
355 |
+ |
maxVal = fabs(v(0, 0)); |
356 |
+ |
maxI = 0; |
357 |
+ |
for (i = 1; i < 3; i++) { |
358 |
+ |
if (maxVal < (tmp = fabs(v(i, 0)))) { |
359 |
+ |
maxVal = tmp; |
360 |
+ |
maxI = i; |
361 |
+ |
} |
362 |
+ |
} |
363 |
+ |
|
364 |
+ |
// swap eigenvalue and eigenvector |
365 |
+ |
if (maxI != 0) { |
366 |
+ |
tmp = w(maxI); |
367 |
+ |
w(maxI) = w(0); |
368 |
+ |
w(0) = tmp; |
369 |
+ |
v.swapRow(maxI, 0); |
370 |
+ |
} |
371 |
+ |
// do the same for the y element |
372 |
+ |
if (fabs(v(1, 1)) < fabs(v(2, 1))) { |
373 |
+ |
tmp = w(2); |
374 |
+ |
w(2) = w(1); |
375 |
+ |
w(1) = tmp; |
376 |
+ |
v.swapRow(2, 1); |
377 |
+ |
} |
378 |
+ |
|
379 |
+ |
// ensure that the sign of the eigenvectors is correct |
380 |
+ |
for (i = 0; i < 2; i++) { |
381 |
+ |
if (v(i, i) < 0) { |
382 |
+ |
v(i, 0) = -v(i, 0); |
383 |
+ |
v(i, 1) = -v(i, 1); |
384 |
+ |
v(i, 2) = -v(i, 2); |
385 |
+ |
} |
386 |
+ |
} |
387 |
+ |
|
388 |
+ |
// set sign of final eigenvector to ensure that determinant is positive |
389 |
+ |
if (determinant(v) < 0) { |
390 |
+ |
v(2, 0) = -v(2, 0); |
391 |
+ |
v(2, 1) = -v(2, 1); |
392 |
+ |
v(2, 2) = -v(2, 2); |
393 |
+ |
} |
394 |
+ |
|
395 |
+ |
// transpose the eigenvectors back again |
396 |
+ |
v = v.transpose(); |
397 |
+ |
return ; |
398 |
+ |
} |
399 |
|
}; |
400 |
|
|
401 |
< |
} |
402 |
< |
#endif // MATH_SQUAREMATRIX#_HPP |
401 |
> |
typedef SquareMatrix3<double> Mat3x3d; |
402 |
> |
typedef SquareMatrix3<double> RotMat3x3d; |
403 |
> |
|
404 |
> |
} //namespace oopse |
405 |
> |
#endif // MATH_SQUAREMATRIX_HPP |