1 // This file is part of Eigen, a lightweight C++ template library 2 // for linear algebra. Eigen itself is part of the KDE project. 3 // 4 // Copyright (C) 2008 Gael Guennebaud <g.gael (at) free.fr> 5 // Copyright (C) 2008 Benoit Jacob <jacob.benoit.1 (at) gmail.com> 6 // 7 // This Source Code Form is subject to the terms of the Mozilla 8 // Public License v. 2.0. If a copy of the MPL was not distributed 9 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. 10 11 // no include guard, we'll include this twice from All.h from Eigen2Support, and it's internal anyway 12 13 namespace Eigen { 14 15 /** \geometry_module \ingroup Geometry_Module 16 * 17 * \class Hyperplane 18 * 19 * \brief A hyperplane 20 * 21 * A hyperplane is an affine subspace of dimension n-1 in a space of dimension n. 22 * For example, a hyperplane in a plane is a line; a hyperplane in 3-space is a plane. 23 * 24 * \param _Scalar the scalar type, i.e., the type of the coefficients 25 * \param _AmbientDim the dimension of the ambient space, can be a compile time value or Dynamic. 26 * Notice that the dimension of the hyperplane is _AmbientDim-1. 27 * 28 * This class represents an hyperplane as the zero set of the implicit equation 29 * \f$ n \cdot x + d = 0 \f$ where \f$ n \f$ is a unit normal vector of the plane (linear part) 30 * and \f$ d \f$ is the distance (offset) to the origin. 31 */ 32 template <typename _Scalar, int _AmbientDim> 33 class Hyperplane 34 { 35 public: 36 EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(_Scalar,_AmbientDim==Dynamic ? Dynamic : _AmbientDim+1) 37 enum { AmbientDimAtCompileTime = _AmbientDim }; 38 typedef _Scalar Scalar; 39 typedef typename NumTraits<Scalar>::Real RealScalar; 40 typedef Matrix<Scalar,AmbientDimAtCompileTime,1> VectorType; 41 typedef Matrix<Scalar,int(AmbientDimAtCompileTime)==Dynamic 42 ? Dynamic 43 : int(AmbientDimAtCompileTime)+1,1> Coefficients; 44 typedef Block<Coefficients,AmbientDimAtCompileTime,1> NormalReturnType; 45 46 /** Default constructor without initialization */ 47 inline explicit Hyperplane() {} 48 49 /** Constructs a dynamic-size hyperplane with \a _dim the dimension 50 * of the ambient space */ 51 inline explicit Hyperplane(int _dim) : m_coeffs(_dim+1) {} 52 53 /** Construct a plane from its normal \a n and a point \a e onto the plane. 54 * \warning the vector normal is assumed to be normalized. 55 */ 56 inline Hyperplane(const VectorType& n, const VectorType& e) 57 : m_coeffs(n.size()+1) 58 { 59 normal() = n; 60 offset() = -e.eigen2_dot(n); 61 } 62 63 /** Constructs a plane from its normal \a n and distance to the origin \a d 64 * such that the algebraic equation of the plane is \f$ n \cdot x + d = 0 \f$. 65 * \warning the vector normal is assumed to be normalized. 66 */ 67 inline Hyperplane(const VectorType& n, Scalar d) 68 : m_coeffs(n.size()+1) 69 { 70 normal() = n; 71 offset() = d; 72 } 73 74 /** Constructs a hyperplane passing through the two points. If the dimension of the ambient space 75 * is greater than 2, then there isn't uniqueness, so an arbitrary choice is made. 76 */ 77 static inline Hyperplane Through(const VectorType& p0, const VectorType& p1) 78 { 79 Hyperplane result(p0.size()); 80 result.normal() = (p1 - p0).unitOrthogonal(); 81 result.offset() = -result.normal().eigen2_dot(p0); 82 return result; 83 } 84 85 /** Constructs a hyperplane passing through the three points. The dimension of the ambient space 86 * is required to be exactly 3. 87 */ 88 static inline Hyperplane Through(const VectorType& p0, const VectorType& p1, const VectorType& p2) 89 { 90 EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(VectorType, 3) 91 Hyperplane result(p0.size()); 92 result.normal() = (p2 - p0).cross(p1 - p0).normalized(); 93 result.offset() = -result.normal().eigen2_dot(p0); 94 return result; 95 } 96 97 /** Constructs a hyperplane passing through the parametrized line \a parametrized. 98 * If the dimension of the ambient space is greater than 2, then there isn't uniqueness, 99 * so an arbitrary choice is made. 100 */ 101 // FIXME to be consitent with the rest this could be implemented as a static Through function ?? 102 explicit Hyperplane(const ParametrizedLine<Scalar, AmbientDimAtCompileTime>& parametrized) 103 { 104 normal() = parametrized.direction().unitOrthogonal(); 105 offset() = -normal().eigen2_dot(parametrized.origin()); 106 } 107 108 ~Hyperplane() {} 109 110 /** \returns the dimension in which the plane holds */ 111 inline int dim() const { return int(AmbientDimAtCompileTime)==Dynamic ? m_coeffs.size()-1 : int(AmbientDimAtCompileTime); } 112 113 /** normalizes \c *this */ 114 void normalize(void) 115 { 116 m_coeffs /= normal().norm(); 117 } 118 119 /** \returns the signed distance between the plane \c *this and a point \a p. 120 * \sa absDistance() 121 */ 122 inline Scalar signedDistance(const VectorType& p) const { return p.eigen2_dot(normal()) + offset(); } 123 124 /** \returns the absolute distance between the plane \c *this and a point \a p. 125 * \sa signedDistance() 126 */ 127 inline Scalar absDistance(const VectorType& p) const { return ei_abs(signedDistance(p)); } 128 129 /** \returns the projection of a point \a p onto the plane \c *this. 130 */ 131 inline VectorType projection(const VectorType& p) const { return p - signedDistance(p) * normal(); } 132 133 /** \returns a constant reference to the unit normal vector of the plane, which corresponds 134 * to the linear part of the implicit equation. 135 */ 136 inline const NormalReturnType normal() const { return NormalReturnType(*const_cast<Coefficients*>(&m_coeffs),0,0,dim(),1); } 137 138 /** \returns a non-constant reference to the unit normal vector of the plane, which corresponds 139 * to the linear part of the implicit equation. 140 */ 141 inline NormalReturnType normal() { return NormalReturnType(m_coeffs,0,0,dim(),1); } 142 143 /** \returns the distance to the origin, which is also the "constant term" of the implicit equation 144 * \warning the vector normal is assumed to be normalized. 145 */ 146 inline const Scalar& offset() const { return m_coeffs.coeff(dim()); } 147 148 /** \returns a non-constant reference to the distance to the origin, which is also the constant part 149 * of the implicit equation */ 150 inline Scalar& offset() { return m_coeffs(dim()); } 151 152 /** \returns a constant reference to the coefficients c_i of the plane equation: 153 * \f$ c_0*x_0 + ... + c_{d-1}*x_{d-1} + c_d = 0 \f$ 154 */ 155 inline const Coefficients& coeffs() const { return m_coeffs; } 156 157 /** \returns a non-constant reference to the coefficients c_i of the plane equation: 158 * \f$ c_0*x_0 + ... + c_{d-1}*x_{d-1} + c_d = 0 \f$ 159 */ 160 inline Coefficients& coeffs() { return m_coeffs; } 161 162 /** \returns the intersection of *this with \a other. 163 * 164 * \warning The ambient space must be a plane, i.e. have dimension 2, so that \c *this and \a other are lines. 165 * 166 * \note If \a other is approximately parallel to *this, this method will return any point on *this. 167 */ 168 VectorType intersection(const Hyperplane& other) 169 { 170 EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(VectorType, 2) 171 Scalar det = coeffs().coeff(0) * other.coeffs().coeff(1) - coeffs().coeff(1) * other.coeffs().coeff(0); 172 // since the line equations ax+by=c are normalized with a^2+b^2=1, the following tests 173 // whether the two lines are approximately parallel. 174 if(ei_isMuchSmallerThan(det, Scalar(1))) 175 { // special case where the two lines are approximately parallel. Pick any point on the first line. 176 if(ei_abs(coeffs().coeff(1))>ei_abs(coeffs().coeff(0))) 177 return VectorType(coeffs().coeff(1), -coeffs().coeff(2)/coeffs().coeff(1)-coeffs().coeff(0)); 178 else 179 return VectorType(-coeffs().coeff(2)/coeffs().coeff(0)-coeffs().coeff(1), coeffs().coeff(0)); 180 } 181 else 182 { // general case 183 Scalar invdet = Scalar(1) / det; 184 return VectorType(invdet*(coeffs().coeff(1)*other.coeffs().coeff(2)-other.coeffs().coeff(1)*coeffs().coeff(2)), 185 invdet*(other.coeffs().coeff(0)*coeffs().coeff(2)-coeffs().coeff(0)*other.coeffs().coeff(2))); 186 } 187 } 188 189 /** Applies the transformation matrix \a mat to \c *this and returns a reference to \c *this. 190 * 191 * \param mat the Dim x Dim transformation matrix 192 * \param traits specifies whether the matrix \a mat represents an Isometry 193 * or a more generic Affine transformation. The default is Affine. 194 */ 195 template<typename XprType> 196 inline Hyperplane& transform(const MatrixBase<XprType>& mat, TransformTraits traits = Affine) 197 { 198 if (traits==Affine) 199 normal() = mat.inverse().transpose() * normal(); 200 else if (traits==Isometry) 201 normal() = mat * normal(); 202 else 203 { 204 ei_assert("invalid traits value in Hyperplane::transform()"); 205 } 206 return *this; 207 } 208 209 /** Applies the transformation \a t to \c *this and returns a reference to \c *this. 210 * 211 * \param t the transformation of dimension Dim 212 * \param traits specifies whether the transformation \a t represents an Isometry 213 * or a more generic Affine transformation. The default is Affine. 214 * Other kind of transformations are not supported. 215 */ 216 inline Hyperplane& transform(const Transform<Scalar,AmbientDimAtCompileTime>& t, 217 TransformTraits traits = Affine) 218 { 219 transform(t.linear(), traits); 220 offset() -= t.translation().eigen2_dot(normal()); 221 return *this; 222 } 223 224 /** \returns \c *this with scalar type casted to \a NewScalarType 225 * 226 * Note that if \a NewScalarType is equal to the current scalar type of \c *this 227 * then this function smartly returns a const reference to \c *this. 228 */ 229 template<typename NewScalarType> 230 inline typename internal::cast_return_type<Hyperplane, 231 Hyperplane<NewScalarType,AmbientDimAtCompileTime> >::type cast() const 232 { 233 return typename internal::cast_return_type<Hyperplane, 234 Hyperplane<NewScalarType,AmbientDimAtCompileTime> >::type(*this); 235 } 236 237 /** Copy constructor with scalar type conversion */ 238 template<typename OtherScalarType> 239 inline explicit Hyperplane(const Hyperplane<OtherScalarType,AmbientDimAtCompileTime>& other) 240 { m_coeffs = other.coeffs().template cast<Scalar>(); } 241 242 /** \returns \c true if \c *this is approximately equal to \a other, within the precision 243 * determined by \a prec. 244 * 245 * \sa MatrixBase::isApprox() */ 246 bool isApprox(const Hyperplane& other, typename NumTraits<Scalar>::Real prec = precision<Scalar>()) const 247 { return m_coeffs.isApprox(other.m_coeffs, prec); } 248 249 protected: 250 251 Coefficients m_coeffs; 252 }; 253 254 } // end namespace Eigen 255
