1 // This file is part of Eigen, a lightweight C++ template library 2 // for linear algebra. 3 // 4 // Copyright (C) 2008 Gael Guennebaud <gael.guennebaud (at) inria.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 #ifndef EIGEN_HYPERPLANE_H 12 #define EIGEN_HYPERPLANE_H 13 14 namespace Eigen { 15 16 /** \geometry_module \ingroup Geometry_Module 17 * 18 * \class Hyperplane 19 * 20 * \brief A hyperplane 21 * 22 * A hyperplane is an affine subspace of dimension n-1 in a space of dimension n. 23 * For example, a hyperplane in a plane is a line; a hyperplane in 3-space is a plane. 24 * 25 * \param _Scalar the scalar type, i.e., the type of the coefficients 26 * \param _AmbientDim the dimension of the ambient space, can be a compile time value or Dynamic. 27 * Notice that the dimension of the hyperplane is _AmbientDim-1. 28 * 29 * This class represents an hyperplane as the zero set of the implicit equation 30 * \f$ n \cdot x + d = 0 \f$ where \f$ n \f$ is a unit normal vector of the plane (linear part) 31 * and \f$ d \f$ is the distance (offset) to the origin. 32 */ 33 template <typename _Scalar, int _AmbientDim, int _Options> 34 class Hyperplane 35 { 36 public: 37 EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(_Scalar,_AmbientDim==Dynamic ? Dynamic : _AmbientDim+1) 38 enum { 39 AmbientDimAtCompileTime = _AmbientDim, 40 Options = _Options 41 }; 42 typedef _Scalar Scalar; 43 typedef typename NumTraits<Scalar>::Real RealScalar; 44 typedef DenseIndex Index; 45 typedef Matrix<Scalar,AmbientDimAtCompileTime,1> VectorType; 46 typedef Matrix<Scalar,Index(AmbientDimAtCompileTime)==Dynamic 47 ? Dynamic 48 : Index(AmbientDimAtCompileTime)+1,1,Options> Coefficients; 49 typedef Block<Coefficients,AmbientDimAtCompileTime,1> NormalReturnType; 50 typedef const Block<const Coefficients,AmbientDimAtCompileTime,1> ConstNormalReturnType; 51 52 /** Default constructor without initialization */ 53 inline Hyperplane() {} 54 55 template<int OtherOptions> 56 Hyperplane(const Hyperplane<Scalar,AmbientDimAtCompileTime,OtherOptions>& other) 57 : m_coeffs(other.coeffs()) 58 {} 59 60 /** Constructs a dynamic-size hyperplane with \a _dim the dimension 61 * of the ambient space */ 62 inline explicit Hyperplane(Index _dim) : m_coeffs(_dim+1) {} 63 64 /** Construct a plane from its normal \a n and a point \a e onto the plane. 65 * \warning the vector normal is assumed to be normalized. 66 */ 67 inline Hyperplane(const VectorType& n, const VectorType& e) 68 : m_coeffs(n.size()+1) 69 { 70 normal() = n; 71 offset() = -n.dot(e); 72 } 73 74 /** Constructs a plane from its normal \a n and distance to the origin \a d 75 * such that the algebraic equation of the plane is \f$ n \cdot x + d = 0 \f$. 76 * \warning the vector normal is assumed to be normalized. 77 */ 78 inline Hyperplane(const VectorType& n, const Scalar& d) 79 : m_coeffs(n.size()+1) 80 { 81 normal() = n; 82 offset() = d; 83 } 84 85 /** Constructs a hyperplane passing through the two points. If the dimension of the ambient space 86 * is greater than 2, then there isn't uniqueness, so an arbitrary choice is made. 87 */ 88 static inline Hyperplane Through(const VectorType& p0, const VectorType& p1) 89 { 90 Hyperplane result(p0.size()); 91 result.normal() = (p1 - p0).unitOrthogonal(); 92 result.offset() = -p0.dot(result.normal()); 93 return result; 94 } 95 96 /** Constructs a hyperplane passing through the three points. The dimension of the ambient space 97 * is required to be exactly 3. 98 */ 99 static inline Hyperplane Through(const VectorType& p0, const VectorType& p1, const VectorType& p2) 100 { 101 EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(VectorType, 3) 102 Hyperplane result(p0.size()); 103 VectorType v0(p2 - p0), v1(p1 - p0); 104 result.normal() = v0.cross(v1); 105 RealScalar norm = result.normal().norm(); 106 if(norm <= v0.norm() * v1.norm() * NumTraits<RealScalar>::epsilon()) 107 { 108 Matrix<Scalar,2,3> m; m << v0.transpose(), v1.transpose(); 109 JacobiSVD<Matrix<Scalar,2,3> > svd(m, ComputeFullV); 110 result.normal() = svd.matrixV().col(2); 111 } 112 else 113 result.normal() /= norm; 114 result.offset() = -p0.dot(result.normal()); 115 return result; 116 } 117 118 /** Constructs a hyperplane passing through the parametrized line \a parametrized. 119 * If the dimension of the ambient space is greater than 2, then there isn't uniqueness, 120 * so an arbitrary choice is made. 121 */ 122 // FIXME to be consitent with the rest this could be implemented as a static Through function ?? 123 explicit Hyperplane(const ParametrizedLine<Scalar, AmbientDimAtCompileTime>& parametrized) 124 { 125 normal() = parametrized.direction().unitOrthogonal(); 126 offset() = -parametrized.origin().dot(normal()); 127 } 128 129 ~Hyperplane() {} 130 131 /** \returns the dimension in which the plane holds */ 132 inline Index dim() const { return AmbientDimAtCompileTime==Dynamic ? m_coeffs.size()-1 : Index(AmbientDimAtCompileTime); } 133 134 /** normalizes \c *this */ 135 void normalize(void) 136 { 137 m_coeffs /= normal().norm(); 138 } 139 140 /** \returns the signed distance between the plane \c *this and a point \a p. 141 * \sa absDistance() 142 */ 143 inline Scalar signedDistance(const VectorType& p) const { return normal().dot(p) + offset(); } 144 145 /** \returns the absolute distance between the plane \c *this and a point \a p. 146 * \sa signedDistance() 147 */ 148 inline Scalar absDistance(const VectorType& p) const { using std::abs; return abs(signedDistance(p)); } 149 150 /** \returns the projection of a point \a p onto the plane \c *this. 151 */ 152 inline VectorType projection(const VectorType& p) const { return p - signedDistance(p) * normal(); } 153 154 /** \returns a constant reference to the unit normal vector of the plane, which corresponds 155 * to the linear part of the implicit equation. 156 */ 157 inline ConstNormalReturnType normal() const { return ConstNormalReturnType(m_coeffs,0,0,dim(),1); } 158 159 /** \returns a non-constant reference to the unit normal vector of the plane, which corresponds 160 * to the linear part of the implicit equation. 161 */ 162 inline NormalReturnType normal() { return NormalReturnType(m_coeffs,0,0,dim(),1); } 163 164 /** \returns the distance to the origin, which is also the "constant term" of the implicit equation 165 * \warning the vector normal is assumed to be normalized. 166 */ 167 inline const Scalar& offset() const { return m_coeffs.coeff(dim()); } 168 169 /** \returns a non-constant reference to the distance to the origin, which is also the constant part 170 * of the implicit equation */ 171 inline Scalar& offset() { return m_coeffs(dim()); } 172 173 /** \returns a constant reference to the coefficients c_i of the plane equation: 174 * \f$ c_0*x_0 + ... + c_{d-1}*x_{d-1} + c_d = 0 \f$ 175 */ 176 inline const Coefficients& coeffs() const { return m_coeffs; } 177 178 /** \returns a non-constant reference to the coefficients c_i of the plane equation: 179 * \f$ c_0*x_0 + ... + c_{d-1}*x_{d-1} + c_d = 0 \f$ 180 */ 181 inline Coefficients& coeffs() { return m_coeffs; } 182 183 /** \returns the intersection of *this with \a other. 184 * 185 * \warning The ambient space must be a plane, i.e. have dimension 2, so that \c *this and \a other are lines. 186 * 187 * \note If \a other is approximately parallel to *this, this method will return any point on *this. 188 */ 189 VectorType intersection(const Hyperplane& other) const 190 { 191 using std::abs; 192 EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(VectorType, 2) 193 Scalar det = coeffs().coeff(0) * other.coeffs().coeff(1) - coeffs().coeff(1) * other.coeffs().coeff(0); 194 // since the line equations ax+by=c are normalized with a^2+b^2=1, the following tests 195 // whether the two lines are approximately parallel. 196 if(internal::isMuchSmallerThan(det, Scalar(1))) 197 { // special case where the two lines are approximately parallel. Pick any point on the first line. 198 if(abs(coeffs().coeff(1))>abs(coeffs().coeff(0))) 199 return VectorType(coeffs().coeff(1), -coeffs().coeff(2)/coeffs().coeff(1)-coeffs().coeff(0)); 200 else 201 return VectorType(-coeffs().coeff(2)/coeffs().coeff(0)-coeffs().coeff(1), coeffs().coeff(0)); 202 } 203 else 204 { // general case 205 Scalar invdet = Scalar(1) / det; 206 return VectorType(invdet*(coeffs().coeff(1)*other.coeffs().coeff(2)-other.coeffs().coeff(1)*coeffs().coeff(2)), 207 invdet*(other.coeffs().coeff(0)*coeffs().coeff(2)-coeffs().coeff(0)*other.coeffs().coeff(2))); 208 } 209 } 210 211 /** Applies the transformation matrix \a mat to \c *this and returns a reference to \c *this. 212 * 213 * \param mat the Dim x Dim transformation matrix 214 * \param traits specifies whether the matrix \a mat represents an #Isometry 215 * or a more generic #Affine transformation. The default is #Affine. 216 */ 217 template<typename XprType> 218 inline Hyperplane& transform(const MatrixBase<XprType>& mat, TransformTraits traits = Affine) 219 { 220 if (traits==Affine) 221 normal() = mat.inverse().transpose() * normal(); 222 else if (traits==Isometry) 223 normal() = mat * normal(); 224 else 225 { 226 eigen_assert(0 && "invalid traits value in Hyperplane::transform()"); 227 } 228 return *this; 229 } 230 231 /** Applies the transformation \a t to \c *this and returns a reference to \c *this. 232 * 233 * \param t the transformation of dimension Dim 234 * \param traits specifies whether the transformation \a t represents an #Isometry 235 * or a more generic #Affine transformation. The default is #Affine. 236 * Other kind of transformations are not supported. 237 */ 238 template<int TrOptions> 239 inline Hyperplane& transform(const Transform<Scalar,AmbientDimAtCompileTime,Affine,TrOptions>& t, 240 TransformTraits traits = Affine) 241 { 242 transform(t.linear(), traits); 243 offset() -= normal().dot(t.translation()); 244 return *this; 245 } 246 247 /** \returns \c *this with scalar type casted to \a NewScalarType 248 * 249 * Note that if \a NewScalarType is equal to the current scalar type of \c *this 250 * then this function smartly returns a const reference to \c *this. 251 */ 252 template<typename NewScalarType> 253 inline typename internal::cast_return_type<Hyperplane, 254 Hyperplane<NewScalarType,AmbientDimAtCompileTime,Options> >::type cast() const 255 { 256 return typename internal::cast_return_type<Hyperplane, 257 Hyperplane<NewScalarType,AmbientDimAtCompileTime,Options> >::type(*this); 258 } 259 260 /** Copy constructor with scalar type conversion */ 261 template<typename OtherScalarType,int OtherOptions> 262 inline explicit Hyperplane(const Hyperplane<OtherScalarType,AmbientDimAtCompileTime,OtherOptions>& other) 263 { m_coeffs = other.coeffs().template cast<Scalar>(); } 264 265 /** \returns \c true if \c *this is approximately equal to \a other, within the precision 266 * determined by \a prec. 267 * 268 * \sa MatrixBase::isApprox() */ 269 template<int OtherOptions> 270 bool isApprox(const Hyperplane<Scalar,AmbientDimAtCompileTime,OtherOptions>& other, const typename NumTraits<Scalar>::Real& prec = NumTraits<Scalar>::dummy_precision()) const 271 { return m_coeffs.isApprox(other.m_coeffs, prec); } 272 273 protected: 274 275 Coefficients m_coeffs; 276 }; 277 278 } // end namespace Eigen 279 280 #endif // EIGEN_HYPERPLANE_H 281
