1 | // This file is part of Eigen, a lightweight C++ template library |
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2 | // for linear algebra. Eigen itself is part of the KDE project. |
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3 | // |
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4 | // Copyright (C) 2008 Gael Guennebaud <g.gael@free.fr> |
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5 | // Copyright (C) 2008 Benoit Jacob <jacob.benoit.1@gmail.com> |
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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 | // no include guard, we'll include this twice from All.h from Eigen2Support, and it's internal anyway |
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12 | |
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13 | namespace Eigen { |
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14 | |
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15 | /** \geometry_module \ingroup Geometry_Module |
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16 | * |
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17 | * \class Hyperplane |
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18 | * |
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19 | * \brief A hyperplane |
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20 | * |
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21 | * A hyperplane is an affine subspace of dimension n-1 in a space of dimension n. |
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22 | * For example, a hyperplane in a plane is a line; a hyperplane in 3-space is a plane. |
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23 | * |
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24 | * \param _Scalar the scalar type, i.e., the type of the coefficients |
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25 | * \param _AmbientDim the dimension of the ambient space, can be a compile time value or Dynamic. |
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26 | * Notice that the dimension of the hyperplane is _AmbientDim-1. |
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27 | * |
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28 | * This class represents an hyperplane as the zero set of the implicit equation |
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29 | * \f$ n \cdot x + d = 0 \f$ where \f$ n \f$ is a unit normal vector of the plane (linear part) |
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30 | * and \f$ d \f$ is the distance (offset) to the origin. |
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31 | */ |
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32 | template <typename _Scalar, int _AmbientDim> |
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33 | class Hyperplane |
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34 | { |
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35 | public: |
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36 | EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(_Scalar,_AmbientDim==Dynamic ? Dynamic : _AmbientDim+1) |
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37 | enum { AmbientDimAtCompileTime = _AmbientDim }; |
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38 | typedef _Scalar Scalar; |
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39 | typedef typename NumTraits<Scalar>::Real RealScalar; |
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40 | typedef Matrix<Scalar,AmbientDimAtCompileTime,1> VectorType; |
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41 | typedef Matrix<Scalar,int(AmbientDimAtCompileTime)==Dynamic |
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42 | ? Dynamic |
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43 | : int(AmbientDimAtCompileTime)+1,1> Coefficients; |
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44 | typedef Block<Coefficients,AmbientDimAtCompileTime,1> NormalReturnType; |
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45 | |
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46 | /** Default constructor without initialization */ |
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47 | inline explicit Hyperplane() {} |
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48 | |
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49 | /** Constructs a dynamic-size hyperplane with \a _dim the dimension |
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50 | * of the ambient space */ |
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51 | inline explicit Hyperplane(int _dim) : m_coeffs(_dim+1) {} |
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52 | |
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53 | /** Construct a plane from its normal \a n and a point \a e onto the plane. |
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54 | * \warning the vector normal is assumed to be normalized. |
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55 | */ |
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56 | inline Hyperplane(const VectorType& n, const VectorType& e) |
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57 | : m_coeffs(n.size()+1) |
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58 | { |
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59 | normal() = n; |
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60 | offset() = -e.eigen2_dot(n); |
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61 | } |
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62 | |
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63 | /** Constructs a plane from its normal \a n and distance to the origin \a d |
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64 | * such that the algebraic equation of the plane is \f$ n \cdot x + d = 0 \f$. |
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65 | * \warning the vector normal is assumed to be normalized. |
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66 | */ |
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67 | inline Hyperplane(const VectorType& n, Scalar d) |
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68 | : m_coeffs(n.size()+1) |
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69 | { |
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70 | normal() = n; |
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71 | offset() = d; |
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72 | } |
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73 | |
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74 | /** Constructs a hyperplane passing through the two points. If the dimension of the ambient space |
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75 | * is greater than 2, then there isn't uniqueness, so an arbitrary choice is made. |
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76 | */ |
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77 | static inline Hyperplane Through(const VectorType& p0, const VectorType& p1) |
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78 | { |
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79 | Hyperplane result(p0.size()); |
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80 | result.normal() = (p1 - p0).unitOrthogonal(); |
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81 | result.offset() = -result.normal().eigen2_dot(p0); |
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82 | return result; |
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83 | } |
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84 | |
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85 | /** Constructs a hyperplane passing through the three points. The dimension of the ambient space |
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86 | * is required to be exactly 3. |
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87 | */ |
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88 | static inline Hyperplane Through(const VectorType& p0, const VectorType& p1, const VectorType& p2) |
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89 | { |
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90 | EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(VectorType, 3) |
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91 | Hyperplane result(p0.size()); |
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92 | result.normal() = (p2 - p0).cross(p1 - p0).normalized(); |
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93 | result.offset() = -result.normal().eigen2_dot(p0); |
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94 | return result; |
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95 | } |
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96 | |
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97 | /** Constructs a hyperplane passing through the parametrized line \a parametrized. |
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98 | * If the dimension of the ambient space is greater than 2, then there isn't uniqueness, |
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99 | * so an arbitrary choice is made. |
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100 | */ |
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101 | // FIXME to be consitent with the rest this could be implemented as a static Through function ?? |
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102 | explicit Hyperplane(const ParametrizedLine<Scalar, AmbientDimAtCompileTime>& parametrized) |
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103 | { |
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104 | normal() = parametrized.direction().unitOrthogonal(); |
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105 | offset() = -normal().eigen2_dot(parametrized.origin()); |
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106 | } |
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107 | |
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108 | ~Hyperplane() {} |
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109 | |
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110 | /** \returns the dimension in which the plane holds */ |
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111 | inline int dim() const { return int(AmbientDimAtCompileTime)==Dynamic ? m_coeffs.size()-1 : int(AmbientDimAtCompileTime); } |
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112 | |
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113 | /** normalizes \c *this */ |
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114 | void normalize(void) |
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115 | { |
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116 | m_coeffs /= normal().norm(); |
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117 | } |
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118 | |
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119 | /** \returns the signed distance between the plane \c *this and a point \a p. |
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120 | * \sa absDistance() |
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121 | */ |
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122 | inline Scalar signedDistance(const VectorType& p) const { return p.eigen2_dot(normal()) + offset(); } |
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123 | |
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124 | /** \returns the absolute distance between the plane \c *this and a point \a p. |
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125 | * \sa signedDistance() |
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126 | */ |
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127 | inline Scalar absDistance(const VectorType& p) const { return ei_abs(signedDistance(p)); } |
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128 | |
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129 | /** \returns the projection of a point \a p onto the plane \c *this. |
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130 | */ |
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131 | inline VectorType projection(const VectorType& p) const { return p - signedDistance(p) * normal(); } |
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132 | |
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133 | /** \returns a constant reference to the unit normal vector of the plane, which corresponds |
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134 | * to the linear part of the implicit equation. |
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135 | */ |
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136 | inline const NormalReturnType normal() const { return NormalReturnType(*const_cast<Coefficients*>(&m_coeffs),0,0,dim(),1); } |
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137 | |
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138 | /** \returns a non-constant reference to the unit normal vector of the plane, which corresponds |
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139 | * to the linear part of the implicit equation. |
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140 | */ |
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141 | inline NormalReturnType normal() { return NormalReturnType(m_coeffs,0,0,dim(),1); } |
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142 | |
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143 | /** \returns the distance to the origin, which is also the "constant term" of the implicit equation |
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144 | * \warning the vector normal is assumed to be normalized. |
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145 | */ |
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146 | inline const Scalar& offset() const { return m_coeffs.coeff(dim()); } |
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147 | |
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148 | /** \returns a non-constant reference to the distance to the origin, which is also the constant part |
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149 | * of the implicit equation */ |
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150 | inline Scalar& offset() { return m_coeffs(dim()); } |
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151 | |
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152 | /** \returns a constant reference to the coefficients c_i of the plane equation: |
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153 | * \f$ c_0*x_0 + ... + c_{d-1}*x_{d-1} + c_d = 0 \f$ |
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154 | */ |
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155 | inline const Coefficients& coeffs() const { return m_coeffs; } |
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156 | |
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157 | /** \returns a non-constant reference to the coefficients c_i of the plane equation: |
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158 | * \f$ c_0*x_0 + ... + c_{d-1}*x_{d-1} + c_d = 0 \f$ |
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159 | */ |
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160 | inline Coefficients& coeffs() { return m_coeffs; } |
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161 | |
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162 | /** \returns the intersection of *this with \a other. |
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163 | * |
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164 | * \warning The ambient space must be a plane, i.e. have dimension 2, so that \c *this and \a other are lines. |
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165 | * |
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166 | * \note If \a other is approximately parallel to *this, this method will return any point on *this. |
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167 | */ |
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168 | VectorType intersection(const Hyperplane& other) |
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169 | { |
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170 | EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(VectorType, 2) |
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171 | Scalar det = coeffs().coeff(0) * other.coeffs().coeff(1) - coeffs().coeff(1) * other.coeffs().coeff(0); |
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172 | // since the line equations ax+by=c are normalized with a^2+b^2=1, the following tests |
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173 | // whether the two lines are approximately parallel. |
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174 | if(ei_isMuchSmallerThan(det, Scalar(1))) |
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175 | { // special case where the two lines are approximately parallel. Pick any point on the first line. |
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176 | if(ei_abs(coeffs().coeff(1))>ei_abs(coeffs().coeff(0))) |
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177 | return VectorType(coeffs().coeff(1), -coeffs().coeff(2)/coeffs().coeff(1)-coeffs().coeff(0)); |
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178 | else |
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179 | return VectorType(-coeffs().coeff(2)/coeffs().coeff(0)-coeffs().coeff(1), coeffs().coeff(0)); |
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180 | } |
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181 | else |
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182 | { // general case |
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183 | Scalar invdet = Scalar(1) / det; |
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184 | return VectorType(invdet*(coeffs().coeff(1)*other.coeffs().coeff(2)-other.coeffs().coeff(1)*coeffs().coeff(2)), |
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185 | invdet*(other.coeffs().coeff(0)*coeffs().coeff(2)-coeffs().coeff(0)*other.coeffs().coeff(2))); |
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186 | } |
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187 | } |
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188 | |
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189 | /** Applies the transformation matrix \a mat to \c *this and returns a reference to \c *this. |
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190 | * |
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191 | * \param mat the Dim x Dim transformation matrix |
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192 | * \param traits specifies whether the matrix \a mat represents an Isometry |
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193 | * or a more generic Affine transformation. The default is Affine. |
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194 | */ |
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195 | template<typename XprType> |
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196 | inline Hyperplane& transform(const MatrixBase<XprType>& mat, TransformTraits traits = Affine) |
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197 | { |
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198 | if (traits==Affine) |
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199 | normal() = mat.inverse().transpose() * normal(); |
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200 | else if (traits==Isometry) |
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201 | normal() = mat * normal(); |
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202 | else |
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203 | { |
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204 | ei_assert("invalid traits value in Hyperplane::transform()"); |
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205 | } |
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206 | return *this; |
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207 | } |
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208 | |
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209 | /** Applies the transformation \a t to \c *this and returns a reference to \c *this. |
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210 | * |
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211 | * \param t the transformation of dimension Dim |
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212 | * \param traits specifies whether the transformation \a t represents an Isometry |
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213 | * or a more generic Affine transformation. The default is Affine. |
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214 | * Other kind of transformations are not supported. |
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215 | */ |
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216 | inline Hyperplane& transform(const Transform<Scalar,AmbientDimAtCompileTime>& t, |
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217 | TransformTraits traits = Affine) |
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218 | { |
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219 | transform(t.linear(), traits); |
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220 | offset() -= t.translation().eigen2_dot(normal()); |
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221 | return *this; |
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222 | } |
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223 | |
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224 | /** \returns \c *this with scalar type casted to \a NewScalarType |
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225 | * |
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226 | * Note that if \a NewScalarType is equal to the current scalar type of \c *this |
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227 | * then this function smartly returns a const reference to \c *this. |
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228 | */ |
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229 | template<typename NewScalarType> |
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230 | inline typename internal::cast_return_type<Hyperplane, |
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231 | Hyperplane<NewScalarType,AmbientDimAtCompileTime> >::type cast() const |
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232 | { |
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233 | return typename internal::cast_return_type<Hyperplane, |
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234 | Hyperplane<NewScalarType,AmbientDimAtCompileTime> >::type(*this); |
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235 | } |
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236 | |
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237 | /** Copy constructor with scalar type conversion */ |
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238 | template<typename OtherScalarType> |
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239 | inline explicit Hyperplane(const Hyperplane<OtherScalarType,AmbientDimAtCompileTime>& other) |
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240 | { m_coeffs = other.coeffs().template cast<Scalar>(); } |
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241 | |
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242 | /** \returns \c true if \c *this is approximately equal to \a other, within the precision |
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243 | * determined by \a prec. |
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244 | * |
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245 | * \sa MatrixBase::isApprox() */ |
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246 | bool isApprox(const Hyperplane& other, typename NumTraits<Scalar>::Real prec = precision<Scalar>()) const |
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247 | { return m_coeffs.isApprox(other.m_coeffs, prec); } |
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248 | |
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249 | protected: |
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250 | |
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251 | Coefficients m_coeffs; |
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252 | }; |
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253 | |
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254 | } // end namespace Eigen |
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