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 Gael Guennebaud <g.gael@free.fr>
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5 | //
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6 | // This Source Code Form is subject to the terms of the Mozilla
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7 | // Public License v. 2.0. If a copy of the MPL was not distributed
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8 | // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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9 |
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10 | // no include guard, we'll include this twice from All.h from Eigen2Support, and it's internal anyway
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11 |
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12 | namespace Eigen {
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13 |
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14 | template<typename Other,
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15 | int OtherRows=Other::RowsAtCompileTime,
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16 | int OtherCols=Other::ColsAtCompileTime>
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17 | struct ei_quaternion_assign_impl;
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18 |
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19 | /** \geometry_module \ingroup Geometry_Module
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20 | *
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21 | * \class Quaternion
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22 | *
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23 | * \brief The quaternion class used to represent 3D orientations and rotations
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24 | *
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25 | * \param _Scalar the scalar type, i.e., the type of the coefficients
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26 | *
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27 | * This class represents a quaternion \f$ w+xi+yj+zk \f$ that is a convenient representation of
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28 | * orientations and rotations of objects in three dimensions. Compared to other representations
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29 | * like Euler angles or 3x3 matrices, quatertions offer the following advantages:
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30 | * \li \b compact storage (4 scalars)
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31 | * \li \b efficient to compose (28 flops),
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32 | * \li \b stable spherical interpolation
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33 | *
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34 | * The following two typedefs are provided for convenience:
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35 | * \li \c Quaternionf for \c float
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36 | * \li \c Quaterniond for \c double
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37 | *
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38 | * \sa class AngleAxis, class Transform
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39 | */
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40 |
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41 | template<typename _Scalar> struct ei_traits<Quaternion<_Scalar> >
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42 | {
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43 | typedef _Scalar Scalar;
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44 | };
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45 |
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46 | template<typename _Scalar>
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47 | class Quaternion : public RotationBase<Quaternion<_Scalar>,3>
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48 | {
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49 | typedef RotationBase<Quaternion<_Scalar>,3> Base;
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50 |
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51 | public:
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52 | EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(_Scalar,4)
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53 |
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54 | using Base::operator*;
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55 |
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56 | /** the scalar type of the coefficients */
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57 | typedef _Scalar Scalar;
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58 |
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59 | /** the type of the Coefficients 4-vector */
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60 | typedef Matrix<Scalar, 4, 1> Coefficients;
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61 | /** the type of a 3D vector */
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62 | typedef Matrix<Scalar,3,1> Vector3;
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63 | /** the equivalent rotation matrix type */
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64 | typedef Matrix<Scalar,3,3> Matrix3;
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65 | /** the equivalent angle-axis type */
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66 | typedef AngleAxis<Scalar> AngleAxisType;
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67 |
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68 | /** \returns the \c x coefficient */
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69 | inline Scalar x() const { return m_coeffs.coeff(0); }
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70 | /** \returns the \c y coefficient */
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71 | inline Scalar y() const { return m_coeffs.coeff(1); }
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72 | /** \returns the \c z coefficient */
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73 | inline Scalar z() const { return m_coeffs.coeff(2); }
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74 | /** \returns the \c w coefficient */
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75 | inline Scalar w() const { return m_coeffs.coeff(3); }
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76 |
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77 | /** \returns a reference to the \c x coefficient */
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78 | inline Scalar& x() { return m_coeffs.coeffRef(0); }
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79 | /** \returns a reference to the \c y coefficient */
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80 | inline Scalar& y() { return m_coeffs.coeffRef(1); }
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81 | /** \returns a reference to the \c z coefficient */
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82 | inline Scalar& z() { return m_coeffs.coeffRef(2); }
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83 | /** \returns a reference to the \c w coefficient */
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84 | inline Scalar& w() { return m_coeffs.coeffRef(3); }
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85 |
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86 | /** \returns a read-only vector expression of the imaginary part (x,y,z) */
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87 | inline const Block<const Coefficients,3,1> vec() const { return m_coeffs.template start<3>(); }
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88 |
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89 | /** \returns a vector expression of the imaginary part (x,y,z) */
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90 | inline Block<Coefficients,3,1> vec() { return m_coeffs.template start<3>(); }
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91 |
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92 | /** \returns a read-only vector expression of the coefficients (x,y,z,w) */
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93 | inline const Coefficients& coeffs() const { return m_coeffs; }
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94 |
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95 | /** \returns a vector expression of the coefficients (x,y,z,w) */
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96 | inline Coefficients& coeffs() { return m_coeffs; }
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97 |
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98 | /** Default constructor leaving the quaternion uninitialized. */
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99 | inline Quaternion() {}
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100 |
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101 | /** Constructs and initializes the quaternion \f$ w+xi+yj+zk \f$ from
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102 | * its four coefficients \a w, \a x, \a y and \a z.
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103 | *
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104 | * \warning Note the order of the arguments: the real \a w coefficient first,
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105 | * while internally the coefficients are stored in the following order:
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106 | * [\c x, \c y, \c z, \c w]
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107 | */
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108 | inline Quaternion(Scalar w, Scalar x, Scalar y, Scalar z)
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109 | { m_coeffs << x, y, z, w; }
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110 |
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111 | /** Copy constructor */
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112 | inline Quaternion(const Quaternion& other) { m_coeffs = other.m_coeffs; }
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113 |
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114 | /** Constructs and initializes a quaternion from the angle-axis \a aa */
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115 | explicit inline Quaternion(const AngleAxisType& aa) { *this = aa; }
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116 |
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117 | /** Constructs and initializes a quaternion from either:
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118 | * - a rotation matrix expression,
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119 | * - a 4D vector expression representing quaternion coefficients.
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120 | * \sa operator=(MatrixBase<Derived>)
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121 | */
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122 | template<typename Derived>
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123 | explicit inline Quaternion(const MatrixBase<Derived>& other) { *this = other; }
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124 |
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125 | Quaternion& operator=(const Quaternion& other);
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126 | Quaternion& operator=(const AngleAxisType& aa);
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127 | template<typename Derived>
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128 | Quaternion& operator=(const MatrixBase<Derived>& m);
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129 |
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130 | /** \returns a quaternion representing an identity rotation
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131 | * \sa MatrixBase::Identity()
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132 | */
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133 | static inline Quaternion Identity() { return Quaternion(1, 0, 0, 0); }
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134 |
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135 | /** \sa Quaternion::Identity(), MatrixBase::setIdentity()
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136 | */
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137 | inline Quaternion& setIdentity() { m_coeffs << 0, 0, 0, 1; return *this; }
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138 |
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139 | /** \returns the squared norm of the quaternion's coefficients
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140 | * \sa Quaternion::norm(), MatrixBase::squaredNorm()
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141 | */
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142 | inline Scalar squaredNorm() const { return m_coeffs.squaredNorm(); }
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143 |
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144 | /** \returns the norm of the quaternion's coefficients
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145 | * \sa Quaternion::squaredNorm(), MatrixBase::norm()
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146 | */
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147 | inline Scalar norm() const { return m_coeffs.norm(); }
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148 |
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149 | /** Normalizes the quaternion \c *this
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150 | * \sa normalized(), MatrixBase::normalize() */
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151 | inline void normalize() { m_coeffs.normalize(); }
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152 | /** \returns a normalized version of \c *this
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153 | * \sa normalize(), MatrixBase::normalized() */
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154 | inline Quaternion normalized() const { return Quaternion(m_coeffs.normalized()); }
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155 |
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156 | /** \returns the dot product of \c *this and \a other
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157 | * Geometrically speaking, the dot product of two unit quaternions
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158 | * corresponds to the cosine of half the angle between the two rotations.
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159 | * \sa angularDistance()
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160 | */
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161 | inline Scalar eigen2_dot(const Quaternion& other) const { return m_coeffs.eigen2_dot(other.m_coeffs); }
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162 |
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163 | inline Scalar angularDistance(const Quaternion& other) const;
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164 |
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165 | Matrix3 toRotationMatrix(void) const;
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166 |
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167 | template<typename Derived1, typename Derived2>
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168 | Quaternion& setFromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b);
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169 |
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170 | inline Quaternion operator* (const Quaternion& q) const;
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171 | inline Quaternion& operator*= (const Quaternion& q);
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172 |
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173 | Quaternion inverse(void) const;
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174 | Quaternion conjugate(void) const;
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175 |
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176 | Quaternion slerp(Scalar t, const Quaternion& other) const;
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177 |
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178 | template<typename Derived>
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179 | Vector3 operator* (const MatrixBase<Derived>& vec) const;
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180 |
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181 | /** \returns \c *this with scalar type casted to \a NewScalarType
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182 | *
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183 | * Note that if \a NewScalarType is equal to the current scalar type of \c *this
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184 | * then this function smartly returns a const reference to \c *this.
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185 | */
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186 | template<typename NewScalarType>
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187 | inline typename internal::cast_return_type<Quaternion,Quaternion<NewScalarType> >::type cast() const
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188 | { return typename internal::cast_return_type<Quaternion,Quaternion<NewScalarType> >::type(*this); }
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189 |
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190 | /** Copy constructor with scalar type conversion */
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191 | template<typename OtherScalarType>
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192 | inline explicit Quaternion(const Quaternion<OtherScalarType>& other)
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193 | { m_coeffs = other.coeffs().template cast<Scalar>(); }
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194 |
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195 | /** \returns \c true if \c *this is approximately equal to \a other, within the precision
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196 | * determined by \a prec.
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197 | *
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198 | * \sa MatrixBase::isApprox() */
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199 | bool isApprox(const Quaternion& other, typename NumTraits<Scalar>::Real prec = precision<Scalar>()) const
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200 | { return m_coeffs.isApprox(other.m_coeffs, prec); }
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201 |
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202 | protected:
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203 | Coefficients m_coeffs;
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204 | };
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205 |
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206 | /** \ingroup Geometry_Module
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207 | * single precision quaternion type */
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208 | typedef Quaternion<float> Quaternionf;
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209 | /** \ingroup Geometry_Module
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210 | * double precision quaternion type */
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211 | typedef Quaternion<double> Quaterniond;
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212 |
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213 | // Generic Quaternion * Quaternion product
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214 | template<typename Scalar> inline Quaternion<Scalar>
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215 | ei_quaternion_product(const Quaternion<Scalar>& a, const Quaternion<Scalar>& b)
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216 | {
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217 | return Quaternion<Scalar>
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218 | (
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219 | a.w() * b.w() - a.x() * b.x() - a.y() * b.y() - a.z() * b.z(),
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220 | a.w() * b.x() + a.x() * b.w() + a.y() * b.z() - a.z() * b.y(),
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221 | a.w() * b.y() + a.y() * b.w() + a.z() * b.x() - a.x() * b.z(),
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222 | a.w() * b.z() + a.z() * b.w() + a.x() * b.y() - a.y() * b.x()
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223 | );
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224 | }
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225 |
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226 | /** \returns the concatenation of two rotations as a quaternion-quaternion product */
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227 | template <typename Scalar>
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228 | inline Quaternion<Scalar> Quaternion<Scalar>::operator* (const Quaternion& other) const
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229 | {
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230 | return ei_quaternion_product(*this,other);
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231 | }
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232 |
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233 | /** \sa operator*(Quaternion) */
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234 | template <typename Scalar>
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235 | inline Quaternion<Scalar>& Quaternion<Scalar>::operator*= (const Quaternion& other)
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236 | {
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237 | return (*this = *this * other);
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238 | }
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239 |
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240 | /** Rotation of a vector by a quaternion.
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241 | * \remarks If the quaternion is used to rotate several points (>1)
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242 | * then it is much more efficient to first convert it to a 3x3 Matrix.
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243 | * Comparison of the operation cost for n transformations:
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244 | * - Quaternion: 30n
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245 | * - Via a Matrix3: 24 + 15n
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246 | */
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247 | template <typename Scalar>
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248 | template<typename Derived>
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249 | inline typename Quaternion<Scalar>::Vector3
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250 | Quaternion<Scalar>::operator* (const MatrixBase<Derived>& v) const
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251 | {
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252 | // Note that this algorithm comes from the optimization by hand
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253 | // of the conversion to a Matrix followed by a Matrix/Vector product.
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254 | // It appears to be much faster than the common algorithm found
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255 | // in the litterature (30 versus 39 flops). It also requires two
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256 | // Vector3 as temporaries.
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257 | Vector3 uv;
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258 | uv = 2 * this->vec().cross(v);
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259 | return v + this->w() * uv + this->vec().cross(uv);
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260 | }
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261 |
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262 | template<typename Scalar>
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263 | inline Quaternion<Scalar>& Quaternion<Scalar>::operator=(const Quaternion& other)
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264 | {
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265 | m_coeffs = other.m_coeffs;
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266 | return *this;
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267 | }
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268 |
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269 | /** Set \c *this from an angle-axis \a aa and returns a reference to \c *this
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270 | */
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271 | template<typename Scalar>
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272 | inline Quaternion<Scalar>& Quaternion<Scalar>::operator=(const AngleAxisType& aa)
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273 | {
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274 | Scalar ha = Scalar(0.5)*aa.angle(); // Scalar(0.5) to suppress precision loss warnings
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275 | this->w() = ei_cos(ha);
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276 | this->vec() = ei_sin(ha) * aa.axis();
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277 | return *this;
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278 | }
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279 |
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280 | /** Set \c *this from the expression \a xpr:
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281 | * - if \a xpr is a 4x1 vector, then \a xpr is assumed to be a quaternion
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282 | * - if \a xpr is a 3x3 matrix, then \a xpr is assumed to be rotation matrix
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283 | * and \a xpr is converted to a quaternion
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284 | */
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285 | template<typename Scalar>
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286 | template<typename Derived>
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287 | inline Quaternion<Scalar>& Quaternion<Scalar>::operator=(const MatrixBase<Derived>& xpr)
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288 | {
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289 | ei_quaternion_assign_impl<Derived>::run(*this, xpr.derived());
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290 | return *this;
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291 | }
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292 |
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293 | /** Convert the quaternion to a 3x3 rotation matrix */
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294 | template<typename Scalar>
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295 | inline typename Quaternion<Scalar>::Matrix3
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296 | Quaternion<Scalar>::toRotationMatrix(void) const
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297 | {
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298 | // NOTE if inlined, then gcc 4.2 and 4.4 get rid of the temporary (not gcc 4.3 !!)
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299 | // if not inlined then the cost of the return by value is huge ~ +35%,
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300 | // however, not inlining this function is an order of magnitude slower, so
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301 | // it has to be inlined, and so the return by value is not an issue
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302 | Matrix3 res;
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303 |
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304 | const Scalar tx = Scalar(2)*this->x();
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305 | const Scalar ty = Scalar(2)*this->y();
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306 | const Scalar tz = Scalar(2)*this->z();
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307 | const Scalar twx = tx*this->w();
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308 | const Scalar twy = ty*this->w();
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309 | const Scalar twz = tz*this->w();
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310 | const Scalar txx = tx*this->x();
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311 | const Scalar txy = ty*this->x();
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312 | const Scalar txz = tz*this->x();
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313 | const Scalar tyy = ty*this->y();
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314 | const Scalar tyz = tz*this->y();
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315 | const Scalar tzz = tz*this->z();
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316 |
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317 | res.coeffRef(0,0) = Scalar(1)-(tyy+tzz);
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318 | res.coeffRef(0,1) = txy-twz;
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319 | res.coeffRef(0,2) = txz+twy;
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320 | res.coeffRef(1,0) = txy+twz;
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321 | res.coeffRef(1,1) = Scalar(1)-(txx+tzz);
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322 | res.coeffRef(1,2) = tyz-twx;
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323 | res.coeffRef(2,0) = txz-twy;
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324 | res.coeffRef(2,1) = tyz+twx;
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325 | res.coeffRef(2,2) = Scalar(1)-(txx+tyy);
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326 |
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327 | return res;
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328 | }
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329 |
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330 | /** Sets *this to be a quaternion representing a rotation sending the vector \a a to the vector \a b.
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331 | *
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332 | * \returns a reference to *this.
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333 | *
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334 | * Note that the two input vectors do \b not have to be normalized.
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335 | */
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336 | template<typename Scalar>
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337 | template<typename Derived1, typename Derived2>
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338 | inline Quaternion<Scalar>& Quaternion<Scalar>::setFromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b)
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339 | {
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340 | Vector3 v0 = a.normalized();
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341 | Vector3 v1 = b.normalized();
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342 | Scalar c = v0.eigen2_dot(v1);
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343 |
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344 | // if dot == 1, vectors are the same
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345 | if (ei_isApprox(c,Scalar(1)))
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346 | {
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347 | // set to identity
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348 | this->w() = 1; this->vec().setZero();
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349 | return *this;
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350 | }
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351 | // if dot == -1, vectors are opposites
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352 | if (ei_isApprox(c,Scalar(-1)))
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353 | {
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354 | this->vec() = v0.unitOrthogonal();
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355 | this->w() = 0;
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356 | return *this;
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357 | }
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358 |
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359 | Vector3 axis = v0.cross(v1);
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360 | Scalar s = ei_sqrt((Scalar(1)+c)*Scalar(2));
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361 | Scalar invs = Scalar(1)/s;
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362 | this->vec() = axis * invs;
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363 | this->w() = s * Scalar(0.5);
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364 |
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365 | return *this;
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366 | }
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367 |
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368 | /** \returns the multiplicative inverse of \c *this
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369 | * Note that in most cases, i.e., if you simply want the opposite rotation,
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370 | * and/or the quaternion is normalized, then it is enough to use the conjugate.
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371 | *
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372 | * \sa Quaternion::conjugate()
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373 | */
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374 | template <typename Scalar>
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375 | inline Quaternion<Scalar> Quaternion<Scalar>::inverse() const
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376 | {
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377 | // FIXME should this function be called multiplicativeInverse and conjugate() be called inverse() or opposite() ??
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378 | Scalar n2 = this->squaredNorm();
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379 | if (n2 > 0)
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380 | return Quaternion(conjugate().coeffs() / n2);
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381 | else
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382 | {
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383 | // return an invalid result to flag the error
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384 | return Quaternion(Coefficients::Zero());
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385 | }
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386 | }
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387 |
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388 | /** \returns the conjugate of the \c *this which is equal to the multiplicative inverse
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389 | * if the quaternion is normalized.
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390 | * The conjugate of a quaternion represents the opposite rotation.
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391 | *
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392 | * \sa Quaternion::inverse()
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393 | */
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394 | template <typename Scalar>
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395 | inline Quaternion<Scalar> Quaternion<Scalar>::conjugate() const
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396 | {
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397 | return Quaternion(this->w(),-this->x(),-this->y(),-this->z());
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398 | }
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399 |
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400 | /** \returns the angle (in radian) between two rotations
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401 | * \sa eigen2_dot()
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402 | */
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403 | template <typename Scalar>
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404 | inline Scalar Quaternion<Scalar>::angularDistance(const Quaternion& other) const
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405 | {
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406 | double d = ei_abs(this->eigen2_dot(other));
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407 | if (d>=1.0)
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408 | return 0;
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409 | return Scalar(2) * std::acos(d);
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410 | }
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411 |
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412 | /** \returns the spherical linear interpolation between the two quaternions
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413 | * \c *this and \a other at the parameter \a t
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414 | */
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415 | template <typename Scalar>
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416 | Quaternion<Scalar> Quaternion<Scalar>::slerp(Scalar t, const Quaternion& other) const
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417 | {
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418 | static const Scalar one = Scalar(1) - machine_epsilon<Scalar>();
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419 | Scalar d = this->eigen2_dot(other);
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420 | Scalar absD = ei_abs(d);
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421 |
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422 | Scalar scale0;
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423 | Scalar scale1;
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424 |
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425 | if (absD>=one)
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426 | {
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427 | scale0 = Scalar(1) - t;
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428 | scale1 = t;
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429 | }
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430 | else
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431 | {
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432 | // theta is the angle between the 2 quaternions
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433 | Scalar theta = std::acos(absD);
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434 | Scalar sinTheta = ei_sin(theta);
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435 |
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436 | scale0 = ei_sin( ( Scalar(1) - t ) * theta) / sinTheta;
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437 | scale1 = ei_sin( ( t * theta) ) / sinTheta;
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438 | if (d<0)
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439 | scale1 = -scale1;
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440 | }
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441 |
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442 | return Quaternion<Scalar>(scale0 * coeffs() + scale1 * other.coeffs());
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443 | }
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444 |
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445 | // set from a rotation matrix
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446 | template<typename Other>
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447 | struct ei_quaternion_assign_impl<Other,3,3>
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448 | {
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449 | typedef typename Other::Scalar Scalar;
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450 | static inline void run(Quaternion<Scalar>& q, const Other& mat)
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451 | {
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452 | // This algorithm comes from "Quaternion Calculus and Fast Animation",
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453 | // Ken Shoemake, 1987 SIGGRAPH course notes
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454 | Scalar t = mat.trace();
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455 | if (t > 0)
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456 | {
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457 | t = ei_sqrt(t + Scalar(1.0));
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458 | q.w() = Scalar(0.5)*t;
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459 | t = Scalar(0.5)/t;
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460 | q.x() = (mat.coeff(2,1) - mat.coeff(1,2)) * t;
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461 | q.y() = (mat.coeff(0,2) - mat.coeff(2,0)) * t;
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462 | q.z() = (mat.coeff(1,0) - mat.coeff(0,1)) * t;
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463 | }
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464 | else
|
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465 | {
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466 | int i = 0;
|
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467 | if (mat.coeff(1,1) > mat.coeff(0,0))
|
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468 | i = 1;
|
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469 | if (mat.coeff(2,2) > mat.coeff(i,i))
|
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470 | i = 2;
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471 | int j = (i+1)%3;
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472 | int k = (j+1)%3;
|
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473 |
|
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474 | t = ei_sqrt(mat.coeff(i,i)-mat.coeff(j,j)-mat.coeff(k,k) + Scalar(1.0));
|
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475 | q.coeffs().coeffRef(i) = Scalar(0.5) * t;
|
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476 | t = Scalar(0.5)/t;
|
---|
477 | q.w() = (mat.coeff(k,j)-mat.coeff(j,k))*t;
|
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478 | q.coeffs().coeffRef(j) = (mat.coeff(j,i)+mat.coeff(i,j))*t;
|
---|
479 | q.coeffs().coeffRef(k) = (mat.coeff(k,i)+mat.coeff(i,k))*t;
|
---|
480 | }
|
---|
481 | }
|
---|
482 | };
|
---|
483 |
|
---|
484 | // set from a vector of coefficients assumed to be a quaternion
|
---|
485 | template<typename Other>
|
---|
486 | struct ei_quaternion_assign_impl<Other,4,1>
|
---|
487 | {
|
---|
488 | typedef typename Other::Scalar Scalar;
|
---|
489 | static inline void run(Quaternion<Scalar>& q, const Other& vec)
|
---|
490 | {
|
---|
491 | q.coeffs() = vec;
|
---|
492 | }
|
---|
493 | };
|
---|
494 |
|
---|
495 | } // end namespace Eigen
|
---|