1 | // This file is part of Eigen, a lightweight C++ template library
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2 | // for linear algebra.
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3 | //
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4 | // Copyright (C) 2008-2010 Gael Guennebaud <gael.guennebaud@inria.fr>
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5 | // Copyright (C) 2009 Mathieu Gautier <mathieu.gautier@cea.fr>
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6 | //
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7 | // This Source Code Form is subject to the terms of the Mozilla
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8 | // Public License v. 2.0. If a copy of the MPL was not distributed
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9 | // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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10 |
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11 | #ifndef EIGEN_QUATERNION_H
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12 | #define EIGEN_QUATERNION_H
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13 | namespace Eigen {
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14 |
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15 |
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16 | /***************************************************************************
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17 | * Definition of QuaternionBase<Derived>
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18 | * The implementation is at the end of the file
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19 | ***************************************************************************/
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20 |
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21 | namespace internal {
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22 | template<typename Other,
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23 | int OtherRows=Other::RowsAtCompileTime,
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24 | int OtherCols=Other::ColsAtCompileTime>
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25 | struct quaternionbase_assign_impl;
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26 | }
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27 |
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28 | /** \geometry_module \ingroup Geometry_Module
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29 | * \class QuaternionBase
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30 | * \brief Base class for quaternion expressions
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31 | * \tparam Derived derived type (CRTP)
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32 | * \sa class Quaternion
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33 | */
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34 | template<class Derived>
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35 | class QuaternionBase : public RotationBase<Derived, 3>
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36 | {
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37 | typedef RotationBase<Derived, 3> Base;
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38 | public:
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39 | using Base::operator*;
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40 | using Base::derived;
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41 |
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42 | typedef typename internal::traits<Derived>::Scalar Scalar;
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43 | typedef typename NumTraits<Scalar>::Real RealScalar;
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44 | typedef typename internal::traits<Derived>::Coefficients Coefficients;
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45 | enum {
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46 | Flags = Eigen::internal::traits<Derived>::Flags
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47 | };
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48 |
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49 | // typedef typename Matrix<Scalar,4,1> Coefficients;
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50 | /** the type of a 3D vector */
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51 | typedef Matrix<Scalar,3,1> Vector3;
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52 | /** the equivalent rotation matrix type */
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53 | typedef Matrix<Scalar,3,3> Matrix3;
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54 | /** the equivalent angle-axis type */
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55 | typedef AngleAxis<Scalar> AngleAxisType;
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56 |
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57 |
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58 |
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59 | /** \returns the \c x coefficient */
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60 | inline Scalar x() const { return this->derived().coeffs().coeff(0); }
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61 | /** \returns the \c y coefficient */
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62 | inline Scalar y() const { return this->derived().coeffs().coeff(1); }
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63 | /** \returns the \c z coefficient */
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64 | inline Scalar z() const { return this->derived().coeffs().coeff(2); }
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65 | /** \returns the \c w coefficient */
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66 | inline Scalar w() const { return this->derived().coeffs().coeff(3); }
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67 |
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68 | /** \returns a reference to the \c x coefficient */
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69 | inline Scalar& x() { return this->derived().coeffs().coeffRef(0); }
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70 | /** \returns a reference to the \c y coefficient */
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71 | inline Scalar& y() { return this->derived().coeffs().coeffRef(1); }
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72 | /** \returns a reference to the \c z coefficient */
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73 | inline Scalar& z() { return this->derived().coeffs().coeffRef(2); }
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74 | /** \returns a reference to the \c w coefficient */
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75 | inline Scalar& w() { return this->derived().coeffs().coeffRef(3); }
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76 |
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77 | /** \returns a read-only vector expression of the imaginary part (x,y,z) */
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78 | inline const VectorBlock<const Coefficients,3> vec() const { return coeffs().template head<3>(); }
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79 |
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80 | /** \returns a vector expression of the imaginary part (x,y,z) */
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81 | inline VectorBlock<Coefficients,3> vec() { return coeffs().template head<3>(); }
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82 |
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83 | /** \returns a read-only vector expression of the coefficients (x,y,z,w) */
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84 | inline const typename internal::traits<Derived>::Coefficients& coeffs() const { return derived().coeffs(); }
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85 |
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86 | /** \returns a vector expression of the coefficients (x,y,z,w) */
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87 | inline typename internal::traits<Derived>::Coefficients& coeffs() { return derived().coeffs(); }
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88 |
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89 | EIGEN_STRONG_INLINE QuaternionBase<Derived>& operator=(const QuaternionBase<Derived>& other);
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90 | template<class OtherDerived> EIGEN_STRONG_INLINE Derived& operator=(const QuaternionBase<OtherDerived>& other);
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91 |
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92 | // disabled this copy operator as it is giving very strange compilation errors when compiling
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93 | // test_stdvector with GCC 4.4.2. This looks like a GCC bug though, so feel free to re-enable it if it's
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94 | // useful; however notice that we already have the templated operator= above and e.g. in MatrixBase
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95 | // we didn't have to add, in addition to templated operator=, such a non-templated copy operator.
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96 | // Derived& operator=(const QuaternionBase& other)
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97 | // { return operator=<Derived>(other); }
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98 |
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99 | Derived& operator=(const AngleAxisType& aa);
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100 | template<class OtherDerived> Derived& operator=(const MatrixBase<OtherDerived>& m);
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101 |
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102 | /** \returns a quaternion representing an identity rotation
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103 | * \sa MatrixBase::Identity()
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104 | */
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105 | static inline Quaternion<Scalar> Identity() { return Quaternion<Scalar>(Scalar(1), Scalar(0), Scalar(0), Scalar(0)); }
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106 |
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107 | /** \sa QuaternionBase::Identity(), MatrixBase::setIdentity()
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108 | */
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109 | inline QuaternionBase& setIdentity() { coeffs() << Scalar(0), Scalar(0), Scalar(0), Scalar(1); return *this; }
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110 |
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111 | /** \returns the squared norm of the quaternion's coefficients
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112 | * \sa QuaternionBase::norm(), MatrixBase::squaredNorm()
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113 | */
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114 | inline Scalar squaredNorm() const { return coeffs().squaredNorm(); }
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115 |
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116 | /** \returns the norm of the quaternion's coefficients
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117 | * \sa QuaternionBase::squaredNorm(), MatrixBase::norm()
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118 | */
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119 | inline Scalar norm() const { return coeffs().norm(); }
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120 |
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121 | /** Normalizes the quaternion \c *this
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122 | * \sa normalized(), MatrixBase::normalize() */
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123 | inline void normalize() { coeffs().normalize(); }
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124 | /** \returns a normalized copy of \c *this
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125 | * \sa normalize(), MatrixBase::normalized() */
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126 | inline Quaternion<Scalar> normalized() const { return Quaternion<Scalar>(coeffs().normalized()); }
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127 |
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128 | /** \returns the dot product of \c *this and \a other
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129 | * Geometrically speaking, the dot product of two unit quaternions
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130 | * corresponds to the cosine of half the angle between the two rotations.
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131 | * \sa angularDistance()
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132 | */
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133 | template<class OtherDerived> inline Scalar dot(const QuaternionBase<OtherDerived>& other) const { return coeffs().dot(other.coeffs()); }
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134 |
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135 | template<class OtherDerived> Scalar angularDistance(const QuaternionBase<OtherDerived>& other) const;
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136 |
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137 | /** \returns an equivalent 3x3 rotation matrix */
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138 | Matrix3 toRotationMatrix() const;
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139 |
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140 | /** \returns the quaternion which transform \a a into \a b through a rotation */
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141 | template<typename Derived1, typename Derived2>
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142 | Derived& setFromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b);
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143 |
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144 | template<class OtherDerived> EIGEN_STRONG_INLINE Quaternion<Scalar> operator* (const QuaternionBase<OtherDerived>& q) const;
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145 | template<class OtherDerived> EIGEN_STRONG_INLINE Derived& operator*= (const QuaternionBase<OtherDerived>& q);
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146 |
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147 | /** \returns the quaternion describing the inverse rotation */
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148 | Quaternion<Scalar> inverse() const;
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149 |
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150 | /** \returns the conjugated quaternion */
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151 | Quaternion<Scalar> conjugate() const;
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152 |
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153 | template<class OtherDerived> Quaternion<Scalar> slerp(const Scalar& t, const QuaternionBase<OtherDerived>& other) const;
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154 |
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155 | /** \returns \c true if \c *this is approximately equal to \a other, within the precision
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156 | * determined by \a prec.
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157 | *
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158 | * \sa MatrixBase::isApprox() */
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159 | template<class OtherDerived>
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160 | bool isApprox(const QuaternionBase<OtherDerived>& other, const RealScalar& prec = NumTraits<Scalar>::dummy_precision()) const
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161 | { return coeffs().isApprox(other.coeffs(), prec); }
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162 |
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163 | /** return the result vector of \a v through the rotation*/
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164 | EIGEN_STRONG_INLINE Vector3 _transformVector(const Vector3& v) const;
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165 |
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166 | /** \returns \c *this with scalar type casted to \a NewScalarType
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167 | *
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168 | * Note that if \a NewScalarType is equal to the current scalar type of \c *this
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169 | * then this function smartly returns a const reference to \c *this.
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170 | */
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171 | template<typename NewScalarType>
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172 | inline typename internal::cast_return_type<Derived,Quaternion<NewScalarType> >::type cast() const
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173 | {
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174 | return typename internal::cast_return_type<Derived,Quaternion<NewScalarType> >::type(derived());
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175 | }
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176 |
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177 | #ifdef EIGEN_QUATERNIONBASE_PLUGIN
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178 | # include EIGEN_QUATERNIONBASE_PLUGIN
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179 | #endif
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180 | };
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181 |
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182 | /***************************************************************************
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183 | * Definition/implementation of Quaternion<Scalar>
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184 | ***************************************************************************/
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185 |
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186 | /** \geometry_module \ingroup Geometry_Module
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187 | *
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188 | * \class Quaternion
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189 | *
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190 | * \brief The quaternion class used to represent 3D orientations and rotations
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191 | *
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192 | * \tparam _Scalar the scalar type, i.e., the type of the coefficients
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193 | * \tparam _Options controls the memory alignment of the coefficients. Can be \# AutoAlign or \# DontAlign. Default is AutoAlign.
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194 | *
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195 | * This class represents a quaternion \f$ w+xi+yj+zk \f$ that is a convenient representation of
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196 | * orientations and rotations of objects in three dimensions. Compared to other representations
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197 | * like Euler angles or 3x3 matrices, quaternions offer the following advantages:
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198 | * \li \b compact storage (4 scalars)
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199 | * \li \b efficient to compose (28 flops),
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200 | * \li \b stable spherical interpolation
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201 | *
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202 | * The following two typedefs are provided for convenience:
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203 | * \li \c Quaternionf for \c float
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204 | * \li \c Quaterniond for \c double
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205 | *
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206 | * \warning Operations interpreting the quaternion as rotation have undefined behavior if the quaternion is not normalized.
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207 | *
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208 | * \sa class AngleAxis, class Transform
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209 | */
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210 |
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211 | namespace internal {
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212 | template<typename _Scalar,int _Options>
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213 | struct traits<Quaternion<_Scalar,_Options> >
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214 | {
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215 | typedef Quaternion<_Scalar,_Options> PlainObject;
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216 | typedef _Scalar Scalar;
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217 | typedef Matrix<_Scalar,4,1,_Options> Coefficients;
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218 | enum{
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219 | IsAligned = internal::traits<Coefficients>::Flags & AlignedBit,
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220 | Flags = IsAligned ? (AlignedBit | LvalueBit) : LvalueBit
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221 | };
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222 | };
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223 | }
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224 |
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225 | template<typename _Scalar, int _Options>
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226 | class Quaternion : public QuaternionBase<Quaternion<_Scalar,_Options> >
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227 | {
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228 | typedef QuaternionBase<Quaternion<_Scalar,_Options> > Base;
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229 | enum { IsAligned = internal::traits<Quaternion>::IsAligned };
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230 |
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231 | public:
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232 | typedef _Scalar Scalar;
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233 |
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234 | EIGEN_INHERIT_ASSIGNMENT_OPERATORS(Quaternion)
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235 | using Base::operator*=;
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236 |
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237 | typedef typename internal::traits<Quaternion>::Coefficients Coefficients;
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238 | typedef typename Base::AngleAxisType AngleAxisType;
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239 |
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240 | /** Default constructor leaving the quaternion uninitialized. */
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241 | inline Quaternion() {}
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242 |
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243 | /** Constructs and initializes the quaternion \f$ w+xi+yj+zk \f$ from
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244 | * its four coefficients \a w, \a x, \a y and \a z.
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245 | *
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246 | * \warning Note the order of the arguments: the real \a w coefficient first,
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247 | * while internally the coefficients are stored in the following order:
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248 | * [\c x, \c y, \c z, \c w]
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249 | */
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250 | inline Quaternion(const Scalar& w, const Scalar& x, const Scalar& y, const Scalar& z) : m_coeffs(x, y, z, w){}
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251 |
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252 | /** Constructs and initialize a quaternion from the array data */
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253 | inline Quaternion(const Scalar* data) : m_coeffs(data) {}
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254 |
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255 | /** Copy constructor */
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256 | template<class Derived> EIGEN_STRONG_INLINE Quaternion(const QuaternionBase<Derived>& other) { this->Base::operator=(other); }
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257 |
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258 | /** Constructs and initializes a quaternion from the angle-axis \a aa */
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259 | explicit inline Quaternion(const AngleAxisType& aa) { *this = aa; }
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260 |
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261 | /** Constructs and initializes a quaternion from either:
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262 | * - a rotation matrix expression,
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263 | * - a 4D vector expression representing quaternion coefficients.
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264 | */
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265 | template<typename Derived>
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266 | explicit inline Quaternion(const MatrixBase<Derived>& other) { *this = other; }
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267 |
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268 | /** Explicit copy constructor with scalar conversion */
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269 | template<typename OtherScalar, int OtherOptions>
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270 | explicit inline Quaternion(const Quaternion<OtherScalar, OtherOptions>& other)
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271 | { m_coeffs = other.coeffs().template cast<Scalar>(); }
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272 |
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273 | template<typename Derived1, typename Derived2>
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274 | static Quaternion FromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b);
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275 |
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276 | inline Coefficients& coeffs() { return m_coeffs;}
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277 | inline const Coefficients& coeffs() const { return m_coeffs;}
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278 |
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279 | EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF(bool(IsAligned))
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280 |
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281 | protected:
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282 | Coefficients m_coeffs;
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283 |
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284 | #ifndef EIGEN_PARSED_BY_DOXYGEN
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285 | static EIGEN_STRONG_INLINE void _check_template_params()
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286 | {
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287 | EIGEN_STATIC_ASSERT( (_Options & DontAlign) == _Options,
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288 | INVALID_MATRIX_TEMPLATE_PARAMETERS)
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289 | }
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290 | #endif
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291 | };
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292 |
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293 | /** \ingroup Geometry_Module
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294 | * single precision quaternion type */
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295 | typedef Quaternion<float> Quaternionf;
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296 | /** \ingroup Geometry_Module
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297 | * double precision quaternion type */
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298 | typedef Quaternion<double> Quaterniond;
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299 |
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300 | /***************************************************************************
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301 | * Specialization of Map<Quaternion<Scalar>>
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302 | ***************************************************************************/
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303 |
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304 | namespace internal {
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305 | template<typename _Scalar, int _Options>
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306 | struct traits<Map<Quaternion<_Scalar>, _Options> > : traits<Quaternion<_Scalar, (int(_Options)&Aligned)==Aligned ? AutoAlign : DontAlign> >
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307 | {
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308 | typedef Map<Matrix<_Scalar,4,1>, _Options> Coefficients;
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309 | };
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310 | }
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311 |
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312 | namespace internal {
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313 | template<typename _Scalar, int _Options>
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314 | struct traits<Map<const Quaternion<_Scalar>, _Options> > : traits<Quaternion<_Scalar, (int(_Options)&Aligned)==Aligned ? AutoAlign : DontAlign> >
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315 | {
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316 | typedef Map<const Matrix<_Scalar,4,1>, _Options> Coefficients;
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317 | typedef traits<Quaternion<_Scalar, (int(_Options)&Aligned)==Aligned ? AutoAlign : DontAlign> > TraitsBase;
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318 | enum {
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319 | Flags = TraitsBase::Flags & ~LvalueBit
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320 | };
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321 | };
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322 | }
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323 |
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324 | /** \ingroup Geometry_Module
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325 | * \brief Quaternion expression mapping a constant memory buffer
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326 | *
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327 | * \tparam _Scalar the type of the Quaternion coefficients
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328 | * \tparam _Options see class Map
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329 | *
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330 | * This is a specialization of class Map for Quaternion. This class allows to view
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331 | * a 4 scalar memory buffer as an Eigen's Quaternion object.
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332 | *
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333 | * \sa class Map, class Quaternion, class QuaternionBase
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334 | */
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335 | template<typename _Scalar, int _Options>
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336 | class Map<const Quaternion<_Scalar>, _Options >
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337 | : public QuaternionBase<Map<const Quaternion<_Scalar>, _Options> >
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338 | {
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339 | typedef QuaternionBase<Map<const Quaternion<_Scalar>, _Options> > Base;
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340 |
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341 | public:
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342 | typedef _Scalar Scalar;
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343 | typedef typename internal::traits<Map>::Coefficients Coefficients;
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344 | EIGEN_INHERIT_ASSIGNMENT_OPERATORS(Map)
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345 | using Base::operator*=;
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346 |
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347 | /** Constructs a Mapped Quaternion object from the pointer \a coeffs
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348 | *
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349 | * The pointer \a coeffs must reference the four coefficients of Quaternion in the following order:
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350 | * \code *coeffs == {x, y, z, w} \endcode
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351 | *
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352 | * If the template parameter _Options is set to #Aligned, then the pointer coeffs must be aligned. */
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353 | EIGEN_STRONG_INLINE Map(const Scalar* coeffs) : m_coeffs(coeffs) {}
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354 |
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355 | inline const Coefficients& coeffs() const { return m_coeffs;}
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356 |
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357 | protected:
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358 | const Coefficients m_coeffs;
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359 | };
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360 |
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361 | /** \ingroup Geometry_Module
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362 | * \brief Expression of a quaternion from a memory buffer
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363 | *
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364 | * \tparam _Scalar the type of the Quaternion coefficients
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365 | * \tparam _Options see class Map
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366 | *
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367 | * This is a specialization of class Map for Quaternion. This class allows to view
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368 | * a 4 scalar memory buffer as an Eigen's Quaternion object.
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369 | *
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370 | * \sa class Map, class Quaternion, class QuaternionBase
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371 | */
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372 | template<typename _Scalar, int _Options>
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373 | class Map<Quaternion<_Scalar>, _Options >
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374 | : public QuaternionBase<Map<Quaternion<_Scalar>, _Options> >
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375 | {
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376 | typedef QuaternionBase<Map<Quaternion<_Scalar>, _Options> > Base;
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377 |
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378 | public:
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379 | typedef _Scalar Scalar;
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380 | typedef typename internal::traits<Map>::Coefficients Coefficients;
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381 | EIGEN_INHERIT_ASSIGNMENT_OPERATORS(Map)
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382 | using Base::operator*=;
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383 |
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384 | /** Constructs a Mapped Quaternion object from the pointer \a coeffs
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385 | *
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386 | * The pointer \a coeffs must reference the four coefficients of Quaternion in the following order:
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387 | * \code *coeffs == {x, y, z, w} \endcode
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388 | *
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389 | * If the template parameter _Options is set to #Aligned, then the pointer coeffs must be aligned. */
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390 | EIGEN_STRONG_INLINE Map(Scalar* coeffs) : m_coeffs(coeffs) {}
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391 |
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392 | inline Coefficients& coeffs() { return m_coeffs; }
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393 | inline const Coefficients& coeffs() const { return m_coeffs; }
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394 |
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395 | protected:
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396 | Coefficients m_coeffs;
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397 | };
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398 |
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399 | /** \ingroup Geometry_Module
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400 | * Map an unaligned array of single precision scalars as a quaternion */
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401 | typedef Map<Quaternion<float>, 0> QuaternionMapf;
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402 | /** \ingroup Geometry_Module
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403 | * Map an unaligned array of double precision scalars as a quaternion */
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404 | typedef Map<Quaternion<double>, 0> QuaternionMapd;
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405 | /** \ingroup Geometry_Module
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406 | * Map a 16-byte aligned array of single precision scalars as a quaternion */
|
---|
407 | typedef Map<Quaternion<float>, Aligned> QuaternionMapAlignedf;
|
---|
408 | /** \ingroup Geometry_Module
|
---|
409 | * Map a 16-byte aligned array of double precision scalars as a quaternion */
|
---|
410 | typedef Map<Quaternion<double>, Aligned> QuaternionMapAlignedd;
|
---|
411 |
|
---|
412 | /***************************************************************************
|
---|
413 | * Implementation of QuaternionBase methods
|
---|
414 | ***************************************************************************/
|
---|
415 |
|
---|
416 | // Generic Quaternion * Quaternion product
|
---|
417 | // This product can be specialized for a given architecture via the Arch template argument.
|
---|
418 | namespace internal {
|
---|
419 | template<int Arch, class Derived1, class Derived2, typename Scalar, int _Options> struct quat_product
|
---|
420 | {
|
---|
421 | static EIGEN_STRONG_INLINE Quaternion<Scalar> run(const QuaternionBase<Derived1>& a, const QuaternionBase<Derived2>& b){
|
---|
422 | return Quaternion<Scalar>
|
---|
423 | (
|
---|
424 | a.w() * b.w() - a.x() * b.x() - a.y() * b.y() - a.z() * b.z(),
|
---|
425 | a.w() * b.x() + a.x() * b.w() + a.y() * b.z() - a.z() * b.y(),
|
---|
426 | a.w() * b.y() + a.y() * b.w() + a.z() * b.x() - a.x() * b.z(),
|
---|
427 | a.w() * b.z() + a.z() * b.w() + a.x() * b.y() - a.y() * b.x()
|
---|
428 | );
|
---|
429 | }
|
---|
430 | };
|
---|
431 | }
|
---|
432 |
|
---|
433 | /** \returns the concatenation of two rotations as a quaternion-quaternion product */
|
---|
434 | template <class Derived>
|
---|
435 | template <class OtherDerived>
|
---|
436 | EIGEN_STRONG_INLINE Quaternion<typename internal::traits<Derived>::Scalar>
|
---|
437 | QuaternionBase<Derived>::operator* (const QuaternionBase<OtherDerived>& other) const
|
---|
438 | {
|
---|
439 | EIGEN_STATIC_ASSERT((internal::is_same<typename Derived::Scalar, typename OtherDerived::Scalar>::value),
|
---|
440 | YOU_MIXED_DIFFERENT_NUMERIC_TYPES__YOU_NEED_TO_USE_THE_CAST_METHOD_OF_MATRIXBASE_TO_CAST_NUMERIC_TYPES_EXPLICITLY)
|
---|
441 | return internal::quat_product<Architecture::Target, Derived, OtherDerived,
|
---|
442 | typename internal::traits<Derived>::Scalar,
|
---|
443 | internal::traits<Derived>::IsAligned && internal::traits<OtherDerived>::IsAligned>::run(*this, other);
|
---|
444 | }
|
---|
445 |
|
---|
446 | /** \sa operator*(Quaternion) */
|
---|
447 | template <class Derived>
|
---|
448 | template <class OtherDerived>
|
---|
449 | EIGEN_STRONG_INLINE Derived& QuaternionBase<Derived>::operator*= (const QuaternionBase<OtherDerived>& other)
|
---|
450 | {
|
---|
451 | derived() = derived() * other.derived();
|
---|
452 | return derived();
|
---|
453 | }
|
---|
454 |
|
---|
455 | /** Rotation of a vector by a quaternion.
|
---|
456 | * \remarks If the quaternion is used to rotate several points (>1)
|
---|
457 | * then it is much more efficient to first convert it to a 3x3 Matrix.
|
---|
458 | * Comparison of the operation cost for n transformations:
|
---|
459 | * - Quaternion2: 30n
|
---|
460 | * - Via a Matrix3: 24 + 15n
|
---|
461 | */
|
---|
462 | template <class Derived>
|
---|
463 | EIGEN_STRONG_INLINE typename QuaternionBase<Derived>::Vector3
|
---|
464 | QuaternionBase<Derived>::_transformVector(const Vector3& v) const
|
---|
465 | {
|
---|
466 | // Note that this algorithm comes from the optimization by hand
|
---|
467 | // of the conversion to a Matrix followed by a Matrix/Vector product.
|
---|
468 | // It appears to be much faster than the common algorithm found
|
---|
469 | // in the literature (30 versus 39 flops). It also requires two
|
---|
470 | // Vector3 as temporaries.
|
---|
471 | Vector3 uv = this->vec().cross(v);
|
---|
472 | uv += uv;
|
---|
473 | return v + this->w() * uv + this->vec().cross(uv);
|
---|
474 | }
|
---|
475 |
|
---|
476 | template<class Derived>
|
---|
477 | EIGEN_STRONG_INLINE QuaternionBase<Derived>& QuaternionBase<Derived>::operator=(const QuaternionBase<Derived>& other)
|
---|
478 | {
|
---|
479 | coeffs() = other.coeffs();
|
---|
480 | return derived();
|
---|
481 | }
|
---|
482 |
|
---|
483 | template<class Derived>
|
---|
484 | template<class OtherDerived>
|
---|
485 | EIGEN_STRONG_INLINE Derived& QuaternionBase<Derived>::operator=(const QuaternionBase<OtherDerived>& other)
|
---|
486 | {
|
---|
487 | coeffs() = other.coeffs();
|
---|
488 | return derived();
|
---|
489 | }
|
---|
490 |
|
---|
491 | /** Set \c *this from an angle-axis \a aa and returns a reference to \c *this
|
---|
492 | */
|
---|
493 | template<class Derived>
|
---|
494 | EIGEN_STRONG_INLINE Derived& QuaternionBase<Derived>::operator=(const AngleAxisType& aa)
|
---|
495 | {
|
---|
496 | using std::cos;
|
---|
497 | using std::sin;
|
---|
498 | Scalar ha = Scalar(0.5)*aa.angle(); // Scalar(0.5) to suppress precision loss warnings
|
---|
499 | this->w() = cos(ha);
|
---|
500 | this->vec() = sin(ha) * aa.axis();
|
---|
501 | return derived();
|
---|
502 | }
|
---|
503 |
|
---|
504 | /** Set \c *this from the expression \a xpr:
|
---|
505 | * - if \a xpr is a 4x1 vector, then \a xpr is assumed to be a quaternion
|
---|
506 | * - if \a xpr is a 3x3 matrix, then \a xpr is assumed to be rotation matrix
|
---|
507 | * and \a xpr is converted to a quaternion
|
---|
508 | */
|
---|
509 |
|
---|
510 | template<class Derived>
|
---|
511 | template<class MatrixDerived>
|
---|
512 | inline Derived& QuaternionBase<Derived>::operator=(const MatrixBase<MatrixDerived>& xpr)
|
---|
513 | {
|
---|
514 | EIGEN_STATIC_ASSERT((internal::is_same<typename Derived::Scalar, typename MatrixDerived::Scalar>::value),
|
---|
515 | YOU_MIXED_DIFFERENT_NUMERIC_TYPES__YOU_NEED_TO_USE_THE_CAST_METHOD_OF_MATRIXBASE_TO_CAST_NUMERIC_TYPES_EXPLICITLY)
|
---|
516 | internal::quaternionbase_assign_impl<MatrixDerived>::run(*this, xpr.derived());
|
---|
517 | return derived();
|
---|
518 | }
|
---|
519 |
|
---|
520 | /** Convert the quaternion to a 3x3 rotation matrix. The quaternion is required to
|
---|
521 | * be normalized, otherwise the result is undefined.
|
---|
522 | */
|
---|
523 | template<class Derived>
|
---|
524 | inline typename QuaternionBase<Derived>::Matrix3
|
---|
525 | QuaternionBase<Derived>::toRotationMatrix(void) const
|
---|
526 | {
|
---|
527 | // NOTE if inlined, then gcc 4.2 and 4.4 get rid of the temporary (not gcc 4.3 !!)
|
---|
528 | // if not inlined then the cost of the return by value is huge ~ +35%,
|
---|
529 | // however, not inlining this function is an order of magnitude slower, so
|
---|
530 | // it has to be inlined, and so the return by value is not an issue
|
---|
531 | Matrix3 res;
|
---|
532 |
|
---|
533 | const Scalar tx = Scalar(2)*this->x();
|
---|
534 | const Scalar ty = Scalar(2)*this->y();
|
---|
535 | const Scalar tz = Scalar(2)*this->z();
|
---|
536 | const Scalar twx = tx*this->w();
|
---|
537 | const Scalar twy = ty*this->w();
|
---|
538 | const Scalar twz = tz*this->w();
|
---|
539 | const Scalar txx = tx*this->x();
|
---|
540 | const Scalar txy = ty*this->x();
|
---|
541 | const Scalar txz = tz*this->x();
|
---|
542 | const Scalar tyy = ty*this->y();
|
---|
543 | const Scalar tyz = tz*this->y();
|
---|
544 | const Scalar tzz = tz*this->z();
|
---|
545 |
|
---|
546 | res.coeffRef(0,0) = Scalar(1)-(tyy+tzz);
|
---|
547 | res.coeffRef(0,1) = txy-twz;
|
---|
548 | res.coeffRef(0,2) = txz+twy;
|
---|
549 | res.coeffRef(1,0) = txy+twz;
|
---|
550 | res.coeffRef(1,1) = Scalar(1)-(txx+tzz);
|
---|
551 | res.coeffRef(1,2) = tyz-twx;
|
---|
552 | res.coeffRef(2,0) = txz-twy;
|
---|
553 | res.coeffRef(2,1) = tyz+twx;
|
---|
554 | res.coeffRef(2,2) = Scalar(1)-(txx+tyy);
|
---|
555 |
|
---|
556 | return res;
|
---|
557 | }
|
---|
558 |
|
---|
559 | /** Sets \c *this to be a quaternion representing a rotation between
|
---|
560 | * the two arbitrary vectors \a a and \a b. In other words, the built
|
---|
561 | * rotation represent a rotation sending the line of direction \a a
|
---|
562 | * to the line of direction \a b, both lines passing through the origin.
|
---|
563 | *
|
---|
564 | * \returns a reference to \c *this.
|
---|
565 | *
|
---|
566 | * Note that the two input vectors do \b not have to be normalized, and
|
---|
567 | * do not need to have the same norm.
|
---|
568 | */
|
---|
569 | template<class Derived>
|
---|
570 | template<typename Derived1, typename Derived2>
|
---|
571 | inline Derived& QuaternionBase<Derived>::setFromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b)
|
---|
572 | {
|
---|
573 | using std::max;
|
---|
574 | using std::sqrt;
|
---|
575 | Vector3 v0 = a.normalized();
|
---|
576 | Vector3 v1 = b.normalized();
|
---|
577 | Scalar c = v1.dot(v0);
|
---|
578 |
|
---|
579 | // if dot == -1, vectors are nearly opposites
|
---|
580 | // => accurately compute the rotation axis by computing the
|
---|
581 | // intersection of the two planes. This is done by solving:
|
---|
582 | // x^T v0 = 0
|
---|
583 | // x^T v1 = 0
|
---|
584 | // under the constraint:
|
---|
585 | // ||x|| = 1
|
---|
586 | // which yields a singular value problem
|
---|
587 | if (c < Scalar(-1)+NumTraits<Scalar>::dummy_precision())
|
---|
588 | {
|
---|
589 | c = (max)(c,Scalar(-1));
|
---|
590 | Matrix<Scalar,2,3> m; m << v0.transpose(), v1.transpose();
|
---|
591 | JacobiSVD<Matrix<Scalar,2,3> > svd(m, ComputeFullV);
|
---|
592 | Vector3 axis = svd.matrixV().col(2);
|
---|
593 |
|
---|
594 | Scalar w2 = (Scalar(1)+c)*Scalar(0.5);
|
---|
595 | this->w() = sqrt(w2);
|
---|
596 | this->vec() = axis * sqrt(Scalar(1) - w2);
|
---|
597 | return derived();
|
---|
598 | }
|
---|
599 | Vector3 axis = v0.cross(v1);
|
---|
600 | Scalar s = sqrt((Scalar(1)+c)*Scalar(2));
|
---|
601 | Scalar invs = Scalar(1)/s;
|
---|
602 | this->vec() = axis * invs;
|
---|
603 | this->w() = s * Scalar(0.5);
|
---|
604 |
|
---|
605 | return derived();
|
---|
606 | }
|
---|
607 |
|
---|
608 |
|
---|
609 | /** Returns a quaternion representing a rotation between
|
---|
610 | * the two arbitrary vectors \a a and \a b. In other words, the built
|
---|
611 | * rotation represent a rotation sending the line of direction \a a
|
---|
612 | * to the line of direction \a b, both lines passing through the origin.
|
---|
613 | *
|
---|
614 | * \returns resulting quaternion
|
---|
615 | *
|
---|
616 | * Note that the two input vectors do \b not have to be normalized, and
|
---|
617 | * do not need to have the same norm.
|
---|
618 | */
|
---|
619 | template<typename Scalar, int Options>
|
---|
620 | template<typename Derived1, typename Derived2>
|
---|
621 | Quaternion<Scalar,Options> Quaternion<Scalar,Options>::FromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b)
|
---|
622 | {
|
---|
623 | Quaternion quat;
|
---|
624 | quat.setFromTwoVectors(a, b);
|
---|
625 | return quat;
|
---|
626 | }
|
---|
627 |
|
---|
628 |
|
---|
629 | /** \returns the multiplicative inverse of \c *this
|
---|
630 | * Note that in most cases, i.e., if you simply want the opposite rotation,
|
---|
631 | * and/or the quaternion is normalized, then it is enough to use the conjugate.
|
---|
632 | *
|
---|
633 | * \sa QuaternionBase::conjugate()
|
---|
634 | */
|
---|
635 | template <class Derived>
|
---|
636 | inline Quaternion<typename internal::traits<Derived>::Scalar> QuaternionBase<Derived>::inverse() const
|
---|
637 | {
|
---|
638 | // FIXME should this function be called multiplicativeInverse and conjugate() be called inverse() or opposite() ??
|
---|
639 | Scalar n2 = this->squaredNorm();
|
---|
640 | if (n2 > Scalar(0))
|
---|
641 | return Quaternion<Scalar>(conjugate().coeffs() / n2);
|
---|
642 | else
|
---|
643 | {
|
---|
644 | // return an invalid result to flag the error
|
---|
645 | return Quaternion<Scalar>(Coefficients::Zero());
|
---|
646 | }
|
---|
647 | }
|
---|
648 |
|
---|
649 | /** \returns the conjugate of the \c *this which is equal to the multiplicative inverse
|
---|
650 | * if the quaternion is normalized.
|
---|
651 | * The conjugate of a quaternion represents the opposite rotation.
|
---|
652 | *
|
---|
653 | * \sa Quaternion2::inverse()
|
---|
654 | */
|
---|
655 | template <class Derived>
|
---|
656 | inline Quaternion<typename internal::traits<Derived>::Scalar>
|
---|
657 | QuaternionBase<Derived>::conjugate() const
|
---|
658 | {
|
---|
659 | return Quaternion<Scalar>(this->w(),-this->x(),-this->y(),-this->z());
|
---|
660 | }
|
---|
661 |
|
---|
662 | /** \returns the angle (in radian) between two rotations
|
---|
663 | * \sa dot()
|
---|
664 | */
|
---|
665 | template <class Derived>
|
---|
666 | template <class OtherDerived>
|
---|
667 | inline typename internal::traits<Derived>::Scalar
|
---|
668 | QuaternionBase<Derived>::angularDistance(const QuaternionBase<OtherDerived>& other) const
|
---|
669 | {
|
---|
670 | using std::atan2;
|
---|
671 | using std::abs;
|
---|
672 | Quaternion<Scalar> d = (*this) * other.conjugate();
|
---|
673 | return Scalar(2) * atan2( d.vec().norm(), abs(d.w()) );
|
---|
674 | }
|
---|
675 |
|
---|
676 |
|
---|
677 |
|
---|
678 | /** \returns the spherical linear interpolation between the two quaternions
|
---|
679 | * \c *this and \a other at the parameter \a t in [0;1].
|
---|
680 | *
|
---|
681 | * This represents an interpolation for a constant motion between \c *this and \a other,
|
---|
682 | * see also http://en.wikipedia.org/wiki/Slerp.
|
---|
683 | */
|
---|
684 | template <class Derived>
|
---|
685 | template <class OtherDerived>
|
---|
686 | Quaternion<typename internal::traits<Derived>::Scalar>
|
---|
687 | QuaternionBase<Derived>::slerp(const Scalar& t, const QuaternionBase<OtherDerived>& other) const
|
---|
688 | {
|
---|
689 | using std::acos;
|
---|
690 | using std::sin;
|
---|
691 | using std::abs;
|
---|
692 | static const Scalar one = Scalar(1) - NumTraits<Scalar>::epsilon();
|
---|
693 | Scalar d = this->dot(other);
|
---|
694 | Scalar absD = abs(d);
|
---|
695 |
|
---|
696 | Scalar scale0;
|
---|
697 | Scalar scale1;
|
---|
698 |
|
---|
699 | if(absD>=one)
|
---|
700 | {
|
---|
701 | scale0 = Scalar(1) - t;
|
---|
702 | scale1 = t;
|
---|
703 | }
|
---|
704 | else
|
---|
705 | {
|
---|
706 | // theta is the angle between the 2 quaternions
|
---|
707 | Scalar theta = acos(absD);
|
---|
708 | Scalar sinTheta = sin(theta);
|
---|
709 |
|
---|
710 | scale0 = sin( ( Scalar(1) - t ) * theta) / sinTheta;
|
---|
711 | scale1 = sin( ( t * theta) ) / sinTheta;
|
---|
712 | }
|
---|
713 | if(d<Scalar(0)) scale1 = -scale1;
|
---|
714 |
|
---|
715 | return Quaternion<Scalar>(scale0 * coeffs() + scale1 * other.coeffs());
|
---|
716 | }
|
---|
717 |
|
---|
718 | namespace internal {
|
---|
719 |
|
---|
720 | // set from a rotation matrix
|
---|
721 | template<typename Other>
|
---|
722 | struct quaternionbase_assign_impl<Other,3,3>
|
---|
723 | {
|
---|
724 | typedef typename Other::Scalar Scalar;
|
---|
725 | typedef DenseIndex Index;
|
---|
726 | template<class Derived> static inline void run(QuaternionBase<Derived>& q, const Other& mat)
|
---|
727 | {
|
---|
728 | using std::sqrt;
|
---|
729 | // This algorithm comes from "Quaternion Calculus and Fast Animation",
|
---|
730 | // Ken Shoemake, 1987 SIGGRAPH course notes
|
---|
731 | Scalar t = mat.trace();
|
---|
732 | if (t > Scalar(0))
|
---|
733 | {
|
---|
734 | t = sqrt(t + Scalar(1.0));
|
---|
735 | q.w() = Scalar(0.5)*t;
|
---|
736 | t = Scalar(0.5)/t;
|
---|
737 | q.x() = (mat.coeff(2,1) - mat.coeff(1,2)) * t;
|
---|
738 | q.y() = (mat.coeff(0,2) - mat.coeff(2,0)) * t;
|
---|
739 | q.z() = (mat.coeff(1,0) - mat.coeff(0,1)) * t;
|
---|
740 | }
|
---|
741 | else
|
---|
742 | {
|
---|
743 | DenseIndex i = 0;
|
---|
744 | if (mat.coeff(1,1) > mat.coeff(0,0))
|
---|
745 | i = 1;
|
---|
746 | if (mat.coeff(2,2) > mat.coeff(i,i))
|
---|
747 | i = 2;
|
---|
748 | DenseIndex j = (i+1)%3;
|
---|
749 | DenseIndex k = (j+1)%3;
|
---|
750 |
|
---|
751 | t = sqrt(mat.coeff(i,i)-mat.coeff(j,j)-mat.coeff(k,k) + Scalar(1.0));
|
---|
752 | q.coeffs().coeffRef(i) = Scalar(0.5) * t;
|
---|
753 | t = Scalar(0.5)/t;
|
---|
754 | q.w() = (mat.coeff(k,j)-mat.coeff(j,k))*t;
|
---|
755 | q.coeffs().coeffRef(j) = (mat.coeff(j,i)+mat.coeff(i,j))*t;
|
---|
756 | q.coeffs().coeffRef(k) = (mat.coeff(k,i)+mat.coeff(i,k))*t;
|
---|
757 | }
|
---|
758 | }
|
---|
759 | };
|
---|
760 |
|
---|
761 | // set from a vector of coefficients assumed to be a quaternion
|
---|
762 | template<typename Other>
|
---|
763 | struct quaternionbase_assign_impl<Other,4,1>
|
---|
764 | {
|
---|
765 | typedef typename Other::Scalar Scalar;
|
---|
766 | template<class Derived> static inline void run(QuaternionBase<Derived>& q, const Other& vec)
|
---|
767 | {
|
---|
768 | q.coeffs() = vec;
|
---|
769 | }
|
---|
770 | };
|
---|
771 |
|
---|
772 | } // end namespace internal
|
---|
773 |
|
---|
774 | } // end namespace Eigen
|
---|
775 |
|
---|
776 | #endif // EIGEN_QUATERNION_H
|
---|