[136] | 1 | // This file is part of Eigen, a lightweight C++ template library
|
---|
| 2 | // for linear algebra.
|
---|
| 3 | //
|
---|
| 4 | // Copyright (C) 2008 Gael Guennebaud <g.gael@free.fr>
|
---|
| 5 | // Copyright (C) 2008 Benoit Jacob <jacob.benoit.1@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 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
|
---|