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/* |
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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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#ifndef MATH_QUATERNION_HPP |
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#define MATH_QUATERNION_HPP |
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#include "math/Vector.hpp" |
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#include "math/Vector3.hpp" |
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#include "math/SquareMatrix.hpp" |
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#define ISZERO(a,eps) ( (a)>-(eps) && (a)<(eps) ) |
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const RealType tiny=1.0e-6; |
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namespace oopse{ |
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|
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/** |
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* @class Quaternion Quaternion.hpp "math/Quaternion.hpp" |
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* Quaternion is a sort of a higher-level complex number. |
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* It is defined as Q = w + x*i + y*j + z*k, |
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* where w, x, y, and z are numbers of type T (e.g. double), and |
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* i*i = -1; j*j = -1; k*k = -1; |
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* i*j = k; j*k = i; k*i = j; |
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*/ |
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template<typename Real> |
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class Quaternion : public Vector<Real, 4> { |
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public: |
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Quaternion() : Vector<Real, 4>() {} |
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/** |
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* @class Quaternion Quaternion.hpp "math/Quaternion.hpp" |
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* Quaternion is a sort of a higher-level complex number. |
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* It is defined as Q = w + x*i + y*j + z*k, |
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* where w, x, y, and z are numbers of type T (e.g. RealType), and |
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* i*i = -1; j*j = -1; k*k = -1; |
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* i*j = k; j*k = i; k*i = j; |
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*/ |
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template<typename Real> |
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class Quaternion : public Vector<Real, 4> { |
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|
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/** Constructs and initializes a Quaternion from w, x, y, z values */ |
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Quaternion(Real w, Real x, Real y, Real z) { |
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data_[0] = w; |
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data_[1] = x; |
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data_[2] = y; |
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data_[3] = z; |
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} |
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public: |
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Quaternion() : Vector<Real, 4>() {} |
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|
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/** Constructs and initializes a Quaternion from w, x, y, z values */ |
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Quaternion(Real w, Real x, Real y, Real z) { |
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this->data_[0] = w; |
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this->data_[1] = x; |
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this->data_[2] = y; |
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this->data_[3] = z; |
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} |
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/** Constructs and initializes a Quaternion from a Vector<Real,4> */ |
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Quaternion(const Vector<Real,4>& v) |
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: Vector<Real, 4>(v){ |
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} |
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/** Constructs and initializes a Quaternion from a Vector<Real,4> */ |
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Quaternion(const Vector<Real,4>& v) |
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: Vector<Real, 4>(v){ |
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} |
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/** copy assignment */ |
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Quaternion& operator =(const Vector<Real, 4>& v){ |
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if (this == & v) |
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return *this; |
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/** copy assignment */ |
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Quaternion& operator =(const Vector<Real, 4>& v){ |
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if (this == & v) |
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return *this; |
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|
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Vector<Real, 4>::operator=(v); |
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return *this; |
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} |
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|
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/** |
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* Returns the value of the first element of this quaternion. |
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* @return the value of the first element of this quaternion |
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*/ |
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Real w() const { |
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return this->data_[0]; |
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} |
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Vector<Real, 4>::operator=(v); |
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return *this; |
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} |
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/** |
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* Returns the reference of the first element of this quaternion. |
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* @return the reference of the first element of this quaternion |
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*/ |
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Real& w() { |
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return this->data_[0]; |
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} |
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/** |
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* Returns the value of the first element of this quaternion. |
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* @return the value of the first element of this quaternion |
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*/ |
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Real w() const { |
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return data_[0]; |
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} |
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/** |
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* Returns the value of the first element of this quaternion. |
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* @return the value of the first element of this quaternion |
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*/ |
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Real x() const { |
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return this->data_[1]; |
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} |
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/** |
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* Returns the reference of the first element of this quaternion. |
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* @return the reference of the first element of this quaternion |
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*/ |
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Real& w() { |
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return data_[0]; |
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} |
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/** |
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* Returns the reference of the second element of this quaternion. |
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* @return the reference of the second element of this quaternion |
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*/ |
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Real& x() { |
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return this->data_[1]; |
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} |
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/** |
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* Returns the value of the first element of this quaternion. |
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* @return the value of the first element of this quaternion |
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*/ |
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Real x() const { |
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return data_[1]; |
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} |
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/** |
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* Returns the value of the thirf element of this quaternion. |
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* @return the value of the third element of this quaternion |
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*/ |
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Real y() const { |
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return this->data_[2]; |
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} |
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/** |
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* Returns the reference of the second element of this quaternion. |
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* @return the reference of the second element of this quaternion |
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*/ |
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Real& x() { |
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return data_[1]; |
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} |
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/** |
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* Returns the reference of the third element of this quaternion. |
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* @return the reference of the third element of this quaternion |
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*/ |
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Real& y() { |
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return this->data_[2]; |
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} |
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/** |
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* Returns the value of the thirf element of this quaternion. |
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* @return the value of the third element of this quaternion |
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*/ |
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Real y() const { |
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return data_[2]; |
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} |
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/** |
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* Returns the value of the fourth element of this quaternion. |
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* @return the value of the fourth element of this quaternion |
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*/ |
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Real z() const { |
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return this->data_[3]; |
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} |
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/** |
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* Returns the reference of the fourth element of this quaternion. |
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* @return the reference of the fourth element of this quaternion |
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*/ |
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Real& z() { |
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return this->data_[3]; |
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} |
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/** |
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* Returns the reference of the third element of this quaternion. |
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* @return the reference of the third element of this quaternion |
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*/ |
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Real& y() { |
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return data_[2]; |
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} |
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/** |
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* Tests if this quaternion is equal to other quaternion |
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* @return true if equal, otherwise return false |
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* @param q quaternion to be compared |
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*/ |
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inline bool operator ==(const Quaternion<Real>& q) { |
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/** |
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* Returns the value of the fourth element of this quaternion. |
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* @return the value of the fourth element of this quaternion |
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*/ |
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Real z() const { |
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return data_[3]; |
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} |
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/** |
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* Returns the reference of the fourth element of this quaternion. |
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* @return the reference of the fourth element of this quaternion |
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*/ |
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Real& z() { |
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return data_[3]; |
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} |
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|
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/** |
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* Tests if this quaternion is equal to other quaternion |
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* @return true if equal, otherwise return false |
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* @param q quaternion to be compared |
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*/ |
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inline bool operator ==(const Quaternion<Real>& q) { |
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|
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for (unsigned int i = 0; i < 4; i ++) { |
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if (!equal(data_[i], q[i])) { |
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return false; |
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} |
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} |
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for (unsigned int i = 0; i < 4; i ++) { |
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if (!equal(this->data_[i], q[i])) { |
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return false; |
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} |
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} |
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return true; |
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} |
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return true; |
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} |
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|
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/** |
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* Returns the inverse of this quaternion |
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* @return inverse |
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* @note since quaternion is a complex number, the inverse of quaternion |
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* q = w + xi + yj+ zk is inv_q = (w -xi - yj - zk)/(|q|^2) |
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*/ |
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Quaternion<Real> inverse() { |
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Quaternion<Real> q; |
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Real d = this->lengthSquare(); |
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/** |
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* Returns the inverse of this quaternion |
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* @return inverse |
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* @note since quaternion is a complex number, the inverse of quaternion |
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* q = w + xi + yj+ zk is inv_q = (w -xi - yj - zk)/(|q|^2) |
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*/ |
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Quaternion<Real> inverse() { |
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Quaternion<Real> q; |
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Real d = this->lengthSquare(); |
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q.w() = w() / d; |
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q.x() = -x() / d; |
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q.y() = -y() / d; |
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q.z() = -z() / d; |
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q.w() = w() / d; |
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q.x() = -x() / d; |
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q.y() = -y() / d; |
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q.z() = -z() / d; |
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return q; |
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} |
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return q; |
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} |
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/** |
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* Sets the value to the multiplication of itself and another quaternion |
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* @param q the other quaternion |
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*/ |
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void mul(const Quaternion<Real>& q) { |
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Quaternion<Real> tmp(*this); |
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/** |
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* Sets the value to the multiplication of itself and another quaternion |
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* @param q the other quaternion |
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*/ |
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void mul(const Quaternion<Real>& q) { |
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Quaternion<Real> tmp(*this); |
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data_[0] = (tmp[0]*q[0]) -(tmp[1]*q[1]) - (tmp[2]*q[2]) - (tmp[3]*q[3]); |
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data_[1] = (tmp[0]*q[1]) + (tmp[1]*q[0]) + (tmp[2]*q[3]) - (tmp[3]*q[2]); |
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data_[2] = (tmp[0]*q[2]) + (tmp[2]*q[0]) + (tmp[3]*q[1]) - (tmp[1]*q[3]); |
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data_[3] = (tmp[0]*q[3]) + (tmp[3]*q[0]) + (tmp[1]*q[2]) - (tmp[2]*q[1]); |
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} |
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this->data_[0] = (tmp[0]*q[0]) -(tmp[1]*q[1]) - (tmp[2]*q[2]) - (tmp[3]*q[3]); |
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this->data_[1] = (tmp[0]*q[1]) + (tmp[1]*q[0]) + (tmp[2]*q[3]) - (tmp[3]*q[2]); |
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this->data_[2] = (tmp[0]*q[2]) + (tmp[2]*q[0]) + (tmp[3]*q[1]) - (tmp[1]*q[3]); |
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this->data_[3] = (tmp[0]*q[3]) + (tmp[3]*q[0]) + (tmp[1]*q[2]) - (tmp[2]*q[1]); |
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} |
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|
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void mul(const Real& s) { |
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data_[0] *= s; |
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data_[1] *= s; |
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data_[2] *= s; |
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data_[3] *= s; |
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} |
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> |
void mul(const Real& s) { |
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this->data_[0] *= s; |
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this->data_[1] *= s; |
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this->data_[2] *= s; |
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this->data_[3] *= s; |
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} |
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|
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/** Set the value of this quaternion to the division of itself by another quaternion */ |
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void div(Quaternion<Real>& q) { |
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mul(q.inverse()); |
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} |
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/** Set the value of this quaternion to the division of itself by another quaternion */ |
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> |
void div(Quaternion<Real>& q) { |
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mul(q.inverse()); |
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} |
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|
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< |
void div(const Real& s) { |
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data_[0] /= s; |
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data_[1] /= s; |
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data_[2] /= s; |
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data_[3] /= s; |
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} |
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> |
void div(const Real& s) { |
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> |
this->data_[0] /= s; |
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> |
this->data_[1] /= s; |
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> |
this->data_[2] /= s; |
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this->data_[3] /= s; |
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> |
} |
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|
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< |
Quaternion<Real>& operator *=(const Quaternion<Real>& q) { |
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< |
mul(q); |
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< |
return *this; |
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< |
} |
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> |
Quaternion<Real>& operator *=(const Quaternion<Real>& q) { |
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> |
mul(q); |
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> |
return *this; |
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> |
} |
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|
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< |
Quaternion<Real>& operator *=(const Real& s) { |
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< |
mul(s); |
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< |
return *this; |
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< |
} |
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> |
Quaternion<Real>& operator *=(const Real& s) { |
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> |
mul(s); |
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> |
return *this; |
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> |
} |
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|
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< |
Quaternion<Real>& operator /=(Quaternion<Real>& q) { |
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< |
*this *= q.inverse(); |
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< |
return *this; |
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< |
} |
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> |
Quaternion<Real>& operator /=(Quaternion<Real>& q) { |
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> |
*this *= q.inverse(); |
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> |
return *this; |
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> |
} |
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|
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< |
Quaternion<Real>& operator /=(const Real& s) { |
| 241 |
< |
div(s); |
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< |
return *this; |
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< |
} |
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< |
/** |
| 245 |
< |
* Returns the conjugate quaternion of this quaternion |
| 246 |
< |
* @return the conjugate quaternion of this quaternion |
| 247 |
< |
*/ |
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< |
Quaternion<Real> conjugate() { |
| 249 |
< |
return Quaternion<Real>(w(), -x(), -y(), -z()); |
| 250 |
< |
} |
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> |
Quaternion<Real>& operator /=(const Real& s) { |
| 241 |
> |
div(s); |
| 242 |
> |
return *this; |
| 243 |
> |
} |
| 244 |
> |
/** |
| 245 |
> |
* Returns the conjugate quaternion of this quaternion |
| 246 |
> |
* @return the conjugate quaternion of this quaternion |
| 247 |
> |
*/ |
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> |
Quaternion<Real> conjugate() const { |
| 249 |
> |
return Quaternion<Real>(w(), -x(), -y(), -z()); |
| 250 |
> |
} |
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|
| 249 |
– |
/** |
| 250 |
– |
* Returns the corresponding rotation matrix (3x3) |
| 251 |
– |
* @return a 3x3 rotation matrix |
| 252 |
– |
*/ |
| 253 |
– |
SquareMatrix<Real, 3> toRotationMatrix3() { |
| 254 |
– |
SquareMatrix<Real, 3> rotMat3; |
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|
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< |
Real w2; |
| 254 |
< |
Real x2; |
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< |
Real y2; |
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< |
Real z2; |
| 253 |
> |
/** |
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> |
return rotation angle from -PI to PI |
| 255 |
> |
*/ |
| 256 |
> |
inline Real get_rotation_angle() const{ |
| 257 |
> |
if( w < (Real)0.0 ) |
| 258 |
> |
return 2.0*atan2(-sqrt( x()*x() + y()*y() + z()*z() ), -w() ); |
| 259 |
> |
else |
| 260 |
> |
return 2.0*atan2( sqrt( x()*x() + y()*y() + z()*z() ), w() ); |
| 261 |
> |
} |
| 262 |
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|
| 263 |
< |
if (!isNormalized()) |
| 264 |
< |
normalize(); |
| 265 |
< |
|
| 266 |
< |
w2 = w() * w(); |
| 267 |
< |
x2 = x() * x(); |
| 268 |
< |
y2 = y() * y(); |
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< |
z2 = z() * z(); |
| 263 |
> |
/** |
| 264 |
> |
create a unit quaternion from axis angle representation |
| 265 |
> |
*/ |
| 266 |
> |
Quaternion<Real> fromAxisAngle(const Vector3<Real>& axis, |
| 267 |
> |
const Real& angle){ |
| 268 |
> |
Vector3<Real> v(axis); |
| 269 |
> |
v.normalize(); |
| 270 |
> |
Real half_angle = angle*0.5; |
| 271 |
> |
Real sin_a = sin(half_angle); |
| 272 |
> |
*this = Quaternion<Real>(cos(half_angle), |
| 273 |
> |
v.x()*sin_a, |
| 274 |
> |
v.y()*sin_a, |
| 275 |
> |
v.z()*sin_a); |
| 276 |
> |
} |
| 277 |
> |
|
| 278 |
> |
/** |
| 279 |
> |
convert a quaternion to axis angle representation, |
| 280 |
> |
preserve the axis direction and angle from -PI to +PI |
| 281 |
> |
*/ |
| 282 |
> |
void toAxisAngle(Vector3<Real>& axis, Real& angle)const { |
| 283 |
> |
Real vl = sqrt( x()*x() + y()*y() + z()*z() ); |
| 284 |
> |
if( vl > tiny ) { |
| 285 |
> |
Real ivl = 1.0/vl; |
| 286 |
> |
axis.x() = x() * ivl; |
| 287 |
> |
axis.y() = y() * ivl; |
| 288 |
> |
axis.z() = z() * ivl; |
| 289 |
|
|
| 290 |
< |
rotMat3(0, 0) = w2 + x2 - y2 - z2; |
| 291 |
< |
rotMat3(0, 1) = 2.0 * ( x() * y() + w() * z() ); |
| 292 |
< |
rotMat3(0, 2) = 2.0 * ( x() * z() - w() * y() ); |
| 290 |
> |
if( w() < 0 ) |
| 291 |
> |
angle = 2.0*atan2(-vl, -w()); //-PI,0 |
| 292 |
> |
else |
| 293 |
> |
angle = 2.0*atan2( vl, w()); //0,PI |
| 294 |
> |
} else { |
| 295 |
> |
axis = Vector3<Real>(0.0,0.0,0.0); |
| 296 |
> |
angle = 0.0; |
| 297 |
> |
} |
| 298 |
> |
} |
| 299 |
|
|
| 300 |
< |
rotMat3(1, 0) = 2.0 * ( x() * y() - w() * z() ); |
| 301 |
< |
rotMat3(1, 1) = w2 - x2 + y2 - z2; |
| 302 |
< |
rotMat3(1, 2) = 2.0 * ( y() * z() + w() * x() ); |
| 300 |
> |
/** |
| 301 |
> |
shortest arc quaternion rotate one vector to another by shortest path. |
| 302 |
> |
create rotation from -> to, for any length vectors. |
| 303 |
> |
*/ |
| 304 |
> |
Quaternion<Real> fromShortestArc(const Vector3d& from, |
| 305 |
> |
const Vector3d& to ) { |
| 306 |
> |
|
| 307 |
> |
Vector3d c( cross(from,to) ); |
| 308 |
> |
*this = Quaternion<Real>(dot(from,to), |
| 309 |
> |
c.x(), |
| 310 |
> |
c.y(), |
| 311 |
> |
c.z()); |
| 312 |
|
|
| 313 |
< |
rotMat3(2, 0) = 2.0 * ( x() * z() + w() * y() ); |
| 314 |
< |
rotMat3(2, 1) = 2.0 * ( y() * z() - w() * x() ); |
| 315 |
< |
rotMat3(2, 2) = w2 - x2 -y2 +z2; |
| 313 |
> |
this->normalize(); // if "from" or "to" not unit, normalize quat |
| 314 |
> |
w += 1.0f; // reducing angle to halfangle |
| 315 |
> |
if( w <= 1e-6 ) { // angle close to PI |
| 316 |
> |
if( ( from.z()*from.z() ) > ( from.x()*from.x() ) ) { |
| 317 |
> |
this->data_[0] = w; |
| 318 |
> |
this->data_[1] = 0.0; //cross(from , Vector3d(1,0,0)) |
| 319 |
> |
this->data_[2] = from.z(); |
| 320 |
> |
this->data_[3] = -from.y(); |
| 321 |
> |
} else { |
| 322 |
> |
this->data_[0] = w; |
| 323 |
> |
this->data_[1] = from.y(); //cross(from, Vector3d(0,0,1)) |
| 324 |
> |
this->data_[2] = -from.x(); |
| 325 |
> |
this->data_[3] = 0.0; |
| 326 |
> |
} |
| 327 |
> |
} |
| 328 |
> |
this->normalize(); |
| 329 |
> |
} |
| 330 |
|
|
| 331 |
< |
return rotMat3; |
| 332 |
< |
} |
| 331 |
> |
Real ComputeTwist(const Quaternion& q) { |
| 332 |
> |
return (Real)2.0 * atan2(q.z(), q.w()); |
| 333 |
> |
} |
| 334 |
|
|
| 335 |
< |
};//end Quaternion |
| 335 |
> |
void RemoveTwist(Quaternion& q) { |
| 336 |
> |
Real t = ComputeTwist(q); |
| 337 |
> |
Quaternion rt = fromAxisAngle(V3Z, t); |
| 338 |
> |
|
| 339 |
> |
q *= rt.inverse(); |
| 340 |
> |
} |
| 341 |
|
|
| 342 |
+ |
void getTwistSwingAxisAngle(Real& twistAngle, Real& swingAngle, |
| 343 |
+ |
Vector3<Real>& swingAxis) { |
| 344 |
+ |
|
| 345 |
+ |
twistAngle = (Real)2.0 * atan2(z(), w()); |
| 346 |
+ |
Quaternion rt, rs; |
| 347 |
+ |
rt.fromAxisAngle(V3Z, twistAngle); |
| 348 |
+ |
rs = *this * rt.inverse(); |
| 349 |
+ |
|
| 350 |
+ |
Real vl = sqrt( rs.x()*rs.x() + rs.y()*rs.y() + rs.z()*rs.z() ); |
| 351 |
+ |
if( vl > tiny ) { |
| 352 |
+ |
Real ivl = 1.0 / vl; |
| 353 |
+ |
swingAxis.x() = rs.x() * ivl; |
| 354 |
+ |
swingAxis.y() = rs.y() * ivl; |
| 355 |
+ |
swingAxis.z() = rs.z() * ivl; |
| 356 |
|
|
| 357 |
< |
/** |
| 358 |
< |
* Returns the vaule of scalar multiplication of this quaterion q (q * s). |
| 359 |
< |
* @return the vaule of scalar multiplication of this vector |
| 360 |
< |
* @param q the source quaternion |
| 361 |
< |
* @param s the scalar value |
| 362 |
< |
*/ |
| 363 |
< |
template<typename Real, unsigned int Dim> |
| 364 |
< |
Quaternion<Real> operator * ( const Quaternion<Real>& q, Real s) { |
| 295 |
< |
Quaternion<Real> result(q); |
| 296 |
< |
result.mul(s); |
| 297 |
< |
return result; |
| 357 |
> |
if( rs.w() < 0.0 ) |
| 358 |
> |
swingAngle = 2.0*atan2(-vl, -rs.w()); //-PI,0 |
| 359 |
> |
else |
| 360 |
> |
swingAngle = 2.0*atan2( vl, rs.w()); //0,PI |
| 361 |
> |
} else { |
| 362 |
> |
swingAxis = Vector3<Real>(1.0,0.0,0.0); |
| 363 |
> |
swingAngle = 0.0; |
| 364 |
> |
} |
| 365 |
|
} |
| 299 |
– |
|
| 300 |
– |
/** |
| 301 |
– |
* Returns the vaule of scalar multiplication of this quaterion q (q * s). |
| 302 |
– |
* @return the vaule of scalar multiplication of this vector |
| 303 |
– |
* @param s the scalar value |
| 304 |
– |
* @param q the source quaternion |
| 305 |
– |
*/ |
| 306 |
– |
template<typename Real, unsigned int Dim> |
| 307 |
– |
Quaternion<Real> operator * ( const Real& s, const Quaternion<Real>& q ) { |
| 308 |
– |
Quaternion<Real> result(q); |
| 309 |
– |
result.mul(s); |
| 310 |
– |
return result; |
| 311 |
– |
} |
| 366 |
|
|
| 367 |
< |
/** |
| 368 |
< |
* Returns the multiplication of two quaternion |
| 369 |
< |
* @return the multiplication of two quaternion |
| 370 |
< |
* @param q1 the first quaternion |
| 371 |
< |
* @param q2 the second quaternion |
| 372 |
< |
*/ |
| 373 |
< |
template<typename Real> |
| 374 |
< |
inline Quaternion<Real> operator *(const Quaternion<Real>& q1, const Quaternion<Real>& q2) { |
| 375 |
< |
Quaternion<Real> result(q1); |
| 376 |
< |
result *= q2; |
| 377 |
< |
return result; |
| 367 |
> |
|
| 368 |
> |
Vector3<Real> rotate(const Vector3<Real>& v) { |
| 369 |
> |
|
| 370 |
> |
Quaternion<Real> q(v.x() * w() + v.z() * y() - v.y() * z(), |
| 371 |
> |
v.y() * w() + v.x() * z() - v.z() * x(), |
| 372 |
> |
v.z() * w() + v.y() * x() - v.x() * y(), |
| 373 |
> |
v.x() * x() + v.y() * y() + v.z() * z()); |
| 374 |
> |
|
| 375 |
> |
return Vector3<Real>(w()*q.x() + x()*q.w() + y()*q.z() - z()*q.y(), |
| 376 |
> |
w()*q.y() + y()*q.w() + z()*q.x() - x()*q.z(), |
| 377 |
> |
w()*q.z() + z()*q.w() + x()*q.y() - y()*q.x())* |
| 378 |
> |
( 1.0/this->lengthSquare() ); |
| 379 |
> |
} |
| 380 |
> |
|
| 381 |
> |
Quaternion<Real>& align (const Vector3<Real>& V1, |
| 382 |
> |
const Vector3<Real>& V2) { |
| 383 |
> |
|
| 384 |
> |
// If V1 and V2 are not parallel, the axis of rotation is the unit-length |
| 385 |
> |
// vector U = Cross(V1,V2)/Length(Cross(V1,V2)). The angle of rotation, |
| 386 |
> |
// A, is the angle between V1 and V2. The quaternion for the rotation is |
| 387 |
> |
// q = cos(A/2) + sin(A/2)*(ux*i+uy*j+uz*k) where U = (ux,uy,uz). |
| 388 |
> |
// |
| 389 |
> |
// (1) Rather than extract A = acos(Dot(V1,V2)), multiply by 1/2, then |
| 390 |
> |
// compute sin(A/2) and cos(A/2), we reduce the computational costs |
| 391 |
> |
// by computing the bisector B = (V1+V2)/Length(V1+V2), so cos(A/2) = |
| 392 |
> |
// Dot(V1,B). |
| 393 |
> |
// |
| 394 |
> |
// (2) The rotation axis is U = Cross(V1,B)/Length(Cross(V1,B)), but |
| 395 |
> |
// Length(Cross(V1,B)) = Length(V1)*Length(B)*sin(A/2) = sin(A/2), in |
| 396 |
> |
// which case sin(A/2)*(ux*i+uy*j+uz*k) = (cx*i+cy*j+cz*k) where |
| 397 |
> |
// C = Cross(V1,B). |
| 398 |
> |
// |
| 399 |
> |
// If V1 = V2, then B = V1, cos(A/2) = 1, and U = (0,0,0). If V1 = -V2, |
| 400 |
> |
// then B = 0. This can happen even if V1 is approximately -V2 using |
| 401 |
> |
// floating point arithmetic, since Vector3::Normalize checks for |
| 402 |
> |
// closeness to zero and returns the zero vector accordingly. The test |
| 403 |
> |
// for exactly zero is usually not recommend for floating point |
| 404 |
> |
// arithmetic, but the implementation of Vector3::Normalize guarantees |
| 405 |
> |
// the comparison is robust. In this case, the A = pi and any axis |
| 406 |
> |
// perpendicular to V1 may be used as the rotation axis. |
| 407 |
> |
|
| 408 |
> |
Vector3<Real> Bisector = V1 + V2; |
| 409 |
> |
Bisector.normalize(); |
| 410 |
> |
|
| 411 |
> |
Real CosHalfAngle = dot(V1,Bisector); |
| 412 |
> |
|
| 413 |
> |
this->data_[0] = CosHalfAngle; |
| 414 |
> |
|
| 415 |
> |
if (CosHalfAngle != (Real)0.0) { |
| 416 |
> |
Vector3<Real> Cross = cross(V1, Bisector); |
| 417 |
> |
this->data_[1] = Cross.x(); |
| 418 |
> |
this->data_[2] = Cross.y(); |
| 419 |
> |
this->data_[3] = Cross.z(); |
| 420 |
> |
} else { |
| 421 |
> |
Real InvLength; |
| 422 |
> |
if (fabs(V1[0]) >= fabs(V1[1])) { |
| 423 |
> |
// V1.x or V1.z is the largest magnitude component |
| 424 |
> |
InvLength = (Real)1.0/sqrt(V1[0]*V1[0] + V1[2]*V1[2]); |
| 425 |
> |
|
| 426 |
> |
this->data_[1] = -V1[2]*InvLength; |
| 427 |
> |
this->data_[2] = (Real)0.0; |
| 428 |
> |
this->data_[3] = +V1[0]*InvLength; |
| 429 |
> |
} else { |
| 430 |
> |
// V1.y or V1.z is the largest magnitude component |
| 431 |
> |
InvLength = (Real)1.0 / sqrt(V1[1]*V1[1] + V1[2]*V1[2]); |
| 432 |
> |
|
| 433 |
> |
this->data_[1] = (Real)0.0; |
| 434 |
> |
this->data_[2] = +V1[2]*InvLength; |
| 435 |
> |
this->data_[3] = -V1[1]*InvLength; |
| 436 |
> |
} |
| 437 |
> |
} |
| 438 |
> |
return *this; |
| 439 |
|
} |
| 440 |
|
|
| 441 |
+ |
void toTwistSwing ( Real& tw, Real& sx, Real& sy ) { |
| 442 |
+ |
|
| 443 |
+ |
// First test if the swing is in the singularity: |
| 444 |
+ |
|
| 445 |
+ |
if ( ISZERO(z(),tiny) && ISZERO(w(),tiny) ) { sx=sy=M_PI; tw=0; return; } |
| 446 |
+ |
|
| 447 |
+ |
// Decompose into twist-swing by solving the equation: |
| 448 |
+ |
// |
| 449 |
+ |
// Qtwist(t*2) * Qswing(s*2) = q |
| 450 |
+ |
// |
| 451 |
+ |
// note: (x,y) is the normalized swing axis (x*x+y*y=1) |
| 452 |
+ |
// |
| 453 |
+ |
// ( Ct 0 0 St ) * ( Cs xSs ySs 0 ) = ( qw qx qy qz ) |
| 454 |
+ |
// ( CtCs xSsCt-yStSs xStSs+ySsCt StCs ) = ( qw qx qy qz ) (1) |
| 455 |
+ |
// From (1): CtCs / StCs = qw/qz => Ct/St = qw/qz => tan(t) = qz/qw (2) |
| 456 |
+ |
// |
| 457 |
+ |
// The swing rotation/2 s comes from: |
| 458 |
+ |
// |
| 459 |
+ |
// From (1): (CtCs)^2 + (StCs)^2 = qw^2 + qz^2 => |
| 460 |
+ |
// Cs = sqrt ( qw^2 + qz^2 ) (3) |
| 461 |
+ |
// |
| 462 |
+ |
// From (1): (xSsCt-yStSs)^2 + (xStSs+ySsCt)^2 = qx^2 + qy^2 => |
| 463 |
+ |
// Ss = sqrt ( qx^2 + qy^2 ) (4) |
| 464 |
+ |
// From (1): |SsCt -StSs| |x| = |qx| |
| 465 |
+ |
// |StSs +SsCt| |y| |qy| (5) |
| 466 |
+ |
|
| 467 |
+ |
Real qw, qx, qy, qz; |
| 468 |
+ |
|
| 469 |
+ |
if ( w()<0 ) { |
| 470 |
+ |
qw=-w(); |
| 471 |
+ |
qx=-x(); |
| 472 |
+ |
qy=-y(); |
| 473 |
+ |
qz=-z(); |
| 474 |
+ |
} else { |
| 475 |
+ |
qw=w(); |
| 476 |
+ |
qx=x(); |
| 477 |
+ |
qy=y(); |
| 478 |
+ |
qz=z(); |
| 479 |
+ |
} |
| 480 |
+ |
|
| 481 |
+ |
Real t = atan2 ( qz, qw ); // from (2) |
| 482 |
+ |
Real s = atan2( sqrt(qx*qx+qy*qy), sqrt(qz*qz+qw*qw) ); // from (3) |
| 483 |
+ |
// and (4) |
| 484 |
+ |
|
| 485 |
+ |
Real x=0.0, y=0.0, sins=sin(s); |
| 486 |
+ |
|
| 487 |
+ |
if ( !ISZERO(sins,tiny) ) { |
| 488 |
+ |
Real sint = sin(t); |
| 489 |
+ |
Real cost = cos(t); |
| 490 |
+ |
|
| 491 |
+ |
// by solving the linear system in (5): |
| 492 |
+ |
y = (-qx*sint + qy*cost)/sins; |
| 493 |
+ |
x = ( qx*cost + qy*sint)/sins; |
| 494 |
+ |
} |
| 495 |
+ |
|
| 496 |
+ |
tw = (Real)2.0*t; |
| 497 |
+ |
sx = (Real)2.0*x*s; |
| 498 |
+ |
sy = (Real)2.0*y*s; |
| 499 |
+ |
} |
| 500 |
+ |
|
| 501 |
+ |
void toSwingTwist(Real& sx, Real& sy, Real& tw ) { |
| 502 |
+ |
|
| 503 |
+ |
// Decompose q into swing-twist using a similar development as |
| 504 |
+ |
// in function toTwistSwing |
| 505 |
+ |
|
| 506 |
+ |
if ( ISZERO(z(),tiny) && ISZERO(w(),tiny) ) { sx=sy=M_PI; tw=0; } |
| 507 |
+ |
|
| 508 |
+ |
Real qw, qx, qy, qz; |
| 509 |
+ |
if ( w() < 0 ){ |
| 510 |
+ |
qw=-w(); |
| 511 |
+ |
qx=-x(); |
| 512 |
+ |
qy=-y(); |
| 513 |
+ |
qz=-z(); |
| 514 |
+ |
} else { |
| 515 |
+ |
qw=w(); |
| 516 |
+ |
qx=x(); |
| 517 |
+ |
qy=y(); |
| 518 |
+ |
qz=z(); |
| 519 |
+ |
} |
| 520 |
+ |
|
| 521 |
+ |
// Get the twist t: |
| 522 |
+ |
Real t = 2.0 * atan2(qz,qw); |
| 523 |
+ |
|
| 524 |
+ |
Real bet = atan2( sqrt(qx*qx+qy*qy), sqrt(qz*qz+qw*qw) ); |
| 525 |
+ |
Real gam = t/2.0; |
| 526 |
+ |
Real sinc = ISZERO(bet,tiny)? 1.0 : sin(bet)/bet; |
| 527 |
+ |
Real singam = sin(gam); |
| 528 |
+ |
Real cosgam = cos(gam); |
| 529 |
+ |
|
| 530 |
+ |
sx = Real( (2.0/sinc) * (cosgam*qx - singam*qy) ); |
| 531 |
+ |
sy = Real( (2.0/sinc) * (singam*qx + cosgam*qy) ); |
| 532 |
+ |
tw = Real( t ); |
| 533 |
+ |
} |
| 534 |
+ |
|
| 535 |
+ |
|
| 536 |
+ |
|
| 537 |
|
/** |
| 538 |
< |
* Returns the division of two quaternion |
| 539 |
< |
* @param q1 divisor |
| 329 |
< |
* @param q2 dividen |
| 538 |
> |
* Returns the corresponding rotation matrix (3x3) |
| 539 |
> |
* @return a 3x3 rotation matrix |
| 540 |
|
*/ |
| 541 |
+ |
SquareMatrix<Real, 3> toRotationMatrix3() { |
| 542 |
+ |
SquareMatrix<Real, 3> rotMat3; |
| 543 |
+ |
|
| 544 |
+ |
Real w2; |
| 545 |
+ |
Real x2; |
| 546 |
+ |
Real y2; |
| 547 |
+ |
Real z2; |
| 548 |
|
|
| 549 |
< |
template<typename Real> |
| 550 |
< |
inline Quaternion<Real> operator /( Quaternion<Real>& q1, Quaternion<Real>& q2) { |
| 551 |
< |
return q1 * q2.inverse(); |
| 549 |
> |
if (!this->isNormalized()) |
| 550 |
> |
this->normalize(); |
| 551 |
> |
|
| 552 |
> |
w2 = w() * w(); |
| 553 |
> |
x2 = x() * x(); |
| 554 |
> |
y2 = y() * y(); |
| 555 |
> |
z2 = z() * z(); |
| 556 |
> |
|
| 557 |
> |
rotMat3(0, 0) = w2 + x2 - y2 - z2; |
| 558 |
> |
rotMat3(0, 1) = 2.0 * ( x() * y() + w() * z() ); |
| 559 |
> |
rotMat3(0, 2) = 2.0 * ( x() * z() - w() * y() ); |
| 560 |
> |
|
| 561 |
> |
rotMat3(1, 0) = 2.0 * ( x() * y() - w() * z() ); |
| 562 |
> |
rotMat3(1, 1) = w2 - x2 + y2 - z2; |
| 563 |
> |
rotMat3(1, 2) = 2.0 * ( y() * z() + w() * x() ); |
| 564 |
> |
|
| 565 |
> |
rotMat3(2, 0) = 2.0 * ( x() * z() + w() * y() ); |
| 566 |
> |
rotMat3(2, 1) = 2.0 * ( y() * z() - w() * x() ); |
| 567 |
> |
rotMat3(2, 2) = w2 - x2 -y2 +z2; |
| 568 |
> |
|
| 569 |
> |
return rotMat3; |
| 570 |
|
} |
| 571 |
|
|
| 572 |
+ |
};//end Quaternion |
| 573 |
+ |
|
| 574 |
+ |
|
| 575 |
|
/** |
| 576 |
< |
* Returns the value of the division of a scalar by a quaternion |
| 577 |
< |
* @return the value of the division of a scalar by a quaternion |
| 578 |
< |
* @param s scalar |
| 579 |
< |
* @param q quaternion |
| 342 |
< |
* @note for a quaternion q, 1/q = q.inverse() |
| 576 |
> |
* Returns the vaule of scalar multiplication of this quaterion q (q * s). |
| 577 |
> |
* @return the vaule of scalar multiplication of this vector |
| 578 |
> |
* @param q the source quaternion |
| 579 |
> |
* @param s the scalar value |
| 580 |
|
*/ |
| 581 |
< |
template<typename Real> |
| 582 |
< |
Quaternion<Real> operator /(const Real& s, Quaternion<Real>& q) { |
| 581 |
> |
template<typename Real, unsigned int Dim> |
| 582 |
> |
Quaternion<Real> operator * ( const Quaternion<Real>& q, Real s) { |
| 583 |
> |
Quaternion<Real> result(q); |
| 584 |
> |
result.mul(s); |
| 585 |
> |
return result; |
| 586 |
> |
} |
| 587 |
> |
|
| 588 |
> |
/** |
| 589 |
> |
* Returns the vaule of scalar multiplication of this quaterion q (q * s). |
| 590 |
> |
* @return the vaule of scalar multiplication of this vector |
| 591 |
> |
* @param s the scalar value |
| 592 |
> |
* @param q the source quaternion |
| 593 |
> |
*/ |
| 594 |
> |
template<typename Real, unsigned int Dim> |
| 595 |
> |
Quaternion<Real> operator * ( const Real& s, const Quaternion<Real>& q ) { |
| 596 |
> |
Quaternion<Real> result(q); |
| 597 |
> |
result.mul(s); |
| 598 |
> |
return result; |
| 599 |
> |
} |
| 600 |
|
|
| 601 |
< |
Quaternion<Real> x; |
| 602 |
< |
x = q.inverse(); |
| 603 |
< |
x *= s; |
| 604 |
< |
return x; |
| 605 |
< |
} |
| 601 |
> |
/** |
| 602 |
> |
* Returns the multiplication of two quaternion |
| 603 |
> |
* @return the multiplication of two quaternion |
| 604 |
> |
* @param q1 the first quaternion |
| 605 |
> |
* @param q2 the second quaternion |
| 606 |
> |
*/ |
| 607 |
> |
template<typename Real> |
| 608 |
> |
inline Quaternion<Real> operator *(const Quaternion<Real>& q1, const Quaternion<Real>& q2) { |
| 609 |
> |
Quaternion<Real> result(q1); |
| 610 |
> |
result *= q2; |
| 611 |
> |
return result; |
| 612 |
> |
} |
| 613 |
> |
|
| 614 |
> |
/** |
| 615 |
> |
* Returns the division of two quaternion |
| 616 |
> |
* @param q1 divisor |
| 617 |
> |
* @param q2 dividen |
| 618 |
> |
*/ |
| 619 |
> |
|
| 620 |
> |
template<typename Real> |
| 621 |
> |
inline Quaternion<Real> operator /( Quaternion<Real>& q1, Quaternion<Real>& q2) { |
| 622 |
> |
return q1 * q2.inverse(); |
| 623 |
> |
} |
| 624 |
> |
|
| 625 |
> |
/** |
| 626 |
> |
* Returns the value of the division of a scalar by a quaternion |
| 627 |
> |
* @return the value of the division of a scalar by a quaternion |
| 628 |
> |
* @param s scalar |
| 629 |
> |
* @param q quaternion |
| 630 |
> |
* @note for a quaternion q, 1/q = q.inverse() |
| 631 |
> |
*/ |
| 632 |
> |
template<typename Real> |
| 633 |
> |
Quaternion<Real> operator /(const Real& s, Quaternion<Real>& q) { |
| 634 |
> |
|
| 635 |
> |
Quaternion<Real> x; |
| 636 |
> |
x = q.inverse(); |
| 637 |
> |
x *= s; |
| 638 |
> |
return x; |
| 639 |
> |
} |
| 640 |
|
|
| 641 |
< |
template <class T> |
| 642 |
< |
inline bool operator==(const Quaternion<T>& lhs, const Quaternion<T>& rhs) { |
| 643 |
< |
return equal(lhs[0] ,rhs[0]) && equal(lhs[1] , rhs[1]) && equal(lhs[2], rhs[2]) && equal(lhs[3], rhs[3]); |
| 644 |
< |
} |
| 641 |
> |
template <class T> |
| 642 |
> |
inline bool operator==(const Quaternion<T>& lhs, const Quaternion<T>& rhs) { |
| 643 |
> |
return equal(lhs[0] ,rhs[0]) && equal(lhs[1] , rhs[1]) && equal(lhs[2], rhs[2]) && equal(lhs[3], rhs[3]); |
| 644 |
> |
} |
| 645 |
|
|
| 646 |
< |
typedef Quaternion<double> Quat4d; |
| 646 |
> |
typedef Quaternion<RealType> Quat4d; |
| 647 |
|
} |
| 648 |
|
#endif //MATH_QUATERNION_HPP |
