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 <gael.guennebaud@inria.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 | #ifndef EIGEN_HYPERPLANE_H
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12 | #define EIGEN_HYPERPLANE_H
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13 |
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14 | namespace Eigen {
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15 |
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16 | /** \geometry_module \ingroup Geometry_Module
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17 | *
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18 | * \class Hyperplane
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19 | *
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20 | * \brief A hyperplane
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21 | *
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22 | * A hyperplane is an affine subspace of dimension n-1 in a space of dimension n.
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23 | * For example, a hyperplane in a plane is a line; a hyperplane in 3-space is a plane.
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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 | * \param _AmbientDim the dimension of the ambient space, can be a compile time value or Dynamic.
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27 | * Notice that the dimension of the hyperplane is _AmbientDim-1.
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28 | *
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29 | * This class represents an hyperplane as the zero set of the implicit equation
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30 | * \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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31 | * and \f$ d \f$ is the distance (offset) to the origin.
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32 | */
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33 | template <typename _Scalar, int _AmbientDim, int _Options>
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34 | class Hyperplane
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35 | {
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36 | public:
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37 | EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(_Scalar,_AmbientDim==Dynamic ? Dynamic : _AmbientDim+1)
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38 | enum {
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39 | AmbientDimAtCompileTime = _AmbientDim,
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40 | Options = _Options
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41 | };
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42 | typedef _Scalar Scalar;
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43 | typedef typename NumTraits<Scalar>::Real RealScalar;
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44 | typedef DenseIndex Index;
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45 | typedef Matrix<Scalar,AmbientDimAtCompileTime,1> VectorType;
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46 | typedef Matrix<Scalar,Index(AmbientDimAtCompileTime)==Dynamic
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47 | ? Dynamic
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48 | : Index(AmbientDimAtCompileTime)+1,1,Options> Coefficients;
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49 | typedef Block<Coefficients,AmbientDimAtCompileTime,1> NormalReturnType;
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50 | typedef const Block<const Coefficients,AmbientDimAtCompileTime,1> ConstNormalReturnType;
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51 |
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52 | /** Default constructor without initialization */
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53 | inline Hyperplane() {}
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54 |
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55 | template<int OtherOptions>
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56 | Hyperplane(const Hyperplane<Scalar,AmbientDimAtCompileTime,OtherOptions>& other)
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57 | : m_coeffs(other.coeffs())
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58 | {}
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59 |
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60 | /** Constructs a dynamic-size hyperplane with \a _dim the dimension
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61 | * of the ambient space */
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62 | inline explicit Hyperplane(Index _dim) : m_coeffs(_dim+1) {}
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63 |
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64 | /** Construct a plane from its normal \a n and a point \a e onto the plane.
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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, const VectorType& e)
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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() = -n.dot(e);
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72 | }
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73 |
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74 | /** Constructs a plane from its normal \a n and distance to the origin \a d
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75 | * such that the algebraic equation of the plane is \f$ n \cdot x + d = 0 \f$.
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76 | * \warning the vector normal is assumed to be normalized.
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77 | */
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78 | inline Hyperplane(const VectorType& n, const Scalar& d)
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79 | : m_coeffs(n.size()+1)
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80 | {
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81 | normal() = n;
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82 | offset() = d;
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83 | }
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84 |
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85 | /** Constructs a hyperplane passing through the two points. If the dimension of the ambient space
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86 | * is greater than 2, then there isn't uniqueness, so an arbitrary choice is made.
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87 | */
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88 | static inline Hyperplane Through(const VectorType& p0, const VectorType& p1)
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89 | {
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90 | Hyperplane result(p0.size());
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91 | result.normal() = (p1 - p0).unitOrthogonal();
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92 | result.offset() = -p0.dot(result.normal());
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93 | return result;
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94 | }
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95 |
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96 | /** Constructs a hyperplane passing through the three points. The dimension of the ambient space
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97 | * is required to be exactly 3.
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98 | */
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99 | static inline Hyperplane Through(const VectorType& p0, const VectorType& p1, const VectorType& p2)
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100 | {
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101 | EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(VectorType, 3)
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102 | Hyperplane result(p0.size());
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103 | VectorType v0(p2 - p0), v1(p1 - p0);
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104 | result.normal() = v0.cross(v1);
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105 | RealScalar norm = result.normal().norm();
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106 | if(norm <= v0.norm() * v1.norm() * NumTraits<RealScalar>::epsilon())
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107 | {
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108 | Matrix<Scalar,2,3> m; m << v0.transpose(), v1.transpose();
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109 | JacobiSVD<Matrix<Scalar,2,3> > svd(m, ComputeFullV);
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110 | result.normal() = svd.matrixV().col(2);
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111 | }
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112 | else
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113 | result.normal() /= norm;
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114 | result.offset() = -p0.dot(result.normal());
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115 | return result;
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116 | }
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117 |
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118 | /** Constructs a hyperplane passing through the parametrized line \a parametrized.
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119 | * If the dimension of the ambient space is greater than 2, then there isn't uniqueness,
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120 | * so an arbitrary choice is made.
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121 | */
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122 | // FIXME to be consitent with the rest this could be implemented as a static Through function ??
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123 | explicit Hyperplane(const ParametrizedLine<Scalar, AmbientDimAtCompileTime>& parametrized)
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124 | {
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125 | normal() = parametrized.direction().unitOrthogonal();
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126 | offset() = -parametrized.origin().dot(normal());
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127 | }
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128 |
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129 | ~Hyperplane() {}
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130 |
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131 | /** \returns the dimension in which the plane holds */
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132 | inline Index dim() const { return AmbientDimAtCompileTime==Dynamic ? m_coeffs.size()-1 : Index(AmbientDimAtCompileTime); }
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133 |
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134 | /** normalizes \c *this */
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135 | void normalize(void)
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136 | {
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137 | m_coeffs /= normal().norm();
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138 | }
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139 |
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140 | /** \returns the signed distance between the plane \c *this and a point \a p.
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141 | * \sa absDistance()
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142 | */
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143 | inline Scalar signedDistance(const VectorType& p) const { return normal().dot(p) + offset(); }
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144 |
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145 | /** \returns the absolute distance between the plane \c *this and a point \a p.
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146 | * \sa signedDistance()
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147 | */
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148 | inline Scalar absDistance(const VectorType& p) const { using std::abs; return abs(signedDistance(p)); }
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149 |
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150 | /** \returns the projection of a point \a p onto the plane \c *this.
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151 | */
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152 | inline VectorType projection(const VectorType& p) const { return p - signedDistance(p) * normal(); }
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153 |
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154 | /** \returns a constant reference to the unit normal vector of the plane, which corresponds
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155 | * to the linear part of the implicit equation.
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156 | */
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157 | inline ConstNormalReturnType normal() const { return ConstNormalReturnType(m_coeffs,0,0,dim(),1); }
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158 |
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159 | /** \returns a non-constant reference to the unit normal vector of the plane, which corresponds
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160 | * to the linear part of the implicit equation.
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161 | */
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162 | inline NormalReturnType normal() { return NormalReturnType(m_coeffs,0,0,dim(),1); }
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163 |
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164 | /** \returns the distance to the origin, which is also the "constant term" of the implicit equation
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165 | * \warning the vector normal is assumed to be normalized.
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166 | */
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167 | inline const Scalar& offset() const { return m_coeffs.coeff(dim()); }
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168 |
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169 | /** \returns a non-constant reference to the distance to the origin, which is also the constant part
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170 | * of the implicit equation */
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171 | inline Scalar& offset() { return m_coeffs(dim()); }
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172 |
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173 | /** \returns a constant reference to the coefficients c_i of the plane equation:
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174 | * \f$ c_0*x_0 + ... + c_{d-1}*x_{d-1} + c_d = 0 \f$
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175 | */
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176 | inline const Coefficients& coeffs() const { return m_coeffs; }
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177 |
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178 | /** \returns a non-constant reference to the coefficients c_i of the plane equation:
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179 | * \f$ c_0*x_0 + ... + c_{d-1}*x_{d-1} + c_d = 0 \f$
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180 | */
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181 | inline Coefficients& coeffs() { return m_coeffs; }
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182 |
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183 | /** \returns the intersection of *this with \a other.
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184 | *
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185 | * \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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186 | *
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187 | * \note If \a other is approximately parallel to *this, this method will return any point on *this.
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188 | */
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189 | VectorType intersection(const Hyperplane& other) const
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190 | {
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191 | using std::abs;
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192 | EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(VectorType, 2)
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193 | Scalar det = coeffs().coeff(0) * other.coeffs().coeff(1) - coeffs().coeff(1) * other.coeffs().coeff(0);
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194 | // since the line equations ax+by=c are normalized with a^2+b^2=1, the following tests
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195 | // whether the two lines are approximately parallel.
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196 | if(internal::isMuchSmallerThan(det, Scalar(1)))
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197 | { // special case where the two lines are approximately parallel. Pick any point on the first line.
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198 | if(abs(coeffs().coeff(1))>abs(coeffs().coeff(0)))
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199 | return VectorType(coeffs().coeff(1), -coeffs().coeff(2)/coeffs().coeff(1)-coeffs().coeff(0));
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200 | else
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201 | return VectorType(-coeffs().coeff(2)/coeffs().coeff(0)-coeffs().coeff(1), coeffs().coeff(0));
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202 | }
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203 | else
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204 | { // general case
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205 | Scalar invdet = Scalar(1) / det;
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206 | return VectorType(invdet*(coeffs().coeff(1)*other.coeffs().coeff(2)-other.coeffs().coeff(1)*coeffs().coeff(2)),
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207 | invdet*(other.coeffs().coeff(0)*coeffs().coeff(2)-coeffs().coeff(0)*other.coeffs().coeff(2)));
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208 | }
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209 | }
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210 |
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211 | /** Applies the transformation matrix \a mat to \c *this and returns a reference to \c *this.
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212 | *
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213 | * \param mat the Dim x Dim transformation matrix
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214 | * \param traits specifies whether the matrix \a mat represents an #Isometry
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215 | * or a more generic #Affine transformation. The default is #Affine.
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216 | */
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217 | template<typename XprType>
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218 | inline Hyperplane& transform(const MatrixBase<XprType>& mat, TransformTraits traits = Affine)
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219 | {
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220 | if (traits==Affine)
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221 | normal() = mat.inverse().transpose() * normal();
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222 | else if (traits==Isometry)
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223 | normal() = mat * normal();
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224 | else
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225 | {
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226 | eigen_assert(0 && "invalid traits value in Hyperplane::transform()");
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227 | }
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228 | return *this;
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229 | }
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230 |
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231 | /** Applies the transformation \a t to \c *this and returns a reference to \c *this.
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232 | *
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233 | * \param t the transformation of dimension Dim
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234 | * \param traits specifies whether the transformation \a t represents an #Isometry
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235 | * or a more generic #Affine transformation. The default is #Affine.
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236 | * Other kind of transformations are not supported.
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237 | */
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238 | template<int TrOptions>
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239 | inline Hyperplane& transform(const Transform<Scalar,AmbientDimAtCompileTime,Affine,TrOptions>& t,
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240 | TransformTraits traits = Affine)
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241 | {
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242 | transform(t.linear(), traits);
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243 | offset() -= normal().dot(t.translation());
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244 | return *this;
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245 | }
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246 |
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247 | /** \returns \c *this with scalar type casted to \a NewScalarType
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248 | *
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249 | * Note that if \a NewScalarType is equal to the current scalar type of \c *this
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250 | * then this function smartly returns a const reference to \c *this.
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251 | */
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252 | template<typename NewScalarType>
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253 | inline typename internal::cast_return_type<Hyperplane,
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254 | Hyperplane<NewScalarType,AmbientDimAtCompileTime,Options> >::type cast() const
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255 | {
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256 | return typename internal::cast_return_type<Hyperplane,
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257 | Hyperplane<NewScalarType,AmbientDimAtCompileTime,Options> >::type(*this);
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258 | }
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259 |
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260 | /** Copy constructor with scalar type conversion */
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261 | template<typename OtherScalarType,int OtherOptions>
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262 | inline explicit Hyperplane(const Hyperplane<OtherScalarType,AmbientDimAtCompileTime,OtherOptions>& other)
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263 | { m_coeffs = other.coeffs().template cast<Scalar>(); }
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264 |
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265 | /** \returns \c true if \c *this is approximately equal to \a other, within the precision
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266 | * determined by \a prec.
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267 | *
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268 | * \sa MatrixBase::isApprox() */
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269 | template<int OtherOptions>
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270 | bool isApprox(const Hyperplane<Scalar,AmbientDimAtCompileTime,OtherOptions>& other, const typename NumTraits<Scalar>::Real& prec = NumTraits<Scalar>::dummy_precision()) const
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271 | { return m_coeffs.isApprox(other.m_coeffs, prec); }
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272 |
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273 | protected:
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274 |
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275 | Coefficients m_coeffs;
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276 | };
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277 |
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278 | } // end namespace Eigen
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279 |
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280 | #endif // EIGEN_HYPERPLANE_H
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