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// This file is part of Eigen, a lightweight C++ template library // for linear algebra. // // Copyright (C) 2008 Gael Guennebaud // Copyright (C) 2008 Benoit Jacob // // This Source Code Form is subject to the terms of the Mozilla // Public License v. 2.0. If a copy of the MPL was not distributed // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. #ifndef EIGEN_HYPERPLANE_H #define EIGEN_HYPERPLANE_H // IWYU pragma: private #include "./InternalHeaderCheck.h" namespace Eigen { /** \geometry_module \ingroup Geometry_Module * * \class Hyperplane * * \brief A hyperplane * * A hyperplane is an affine subspace of dimension n-1 in a space of dimension n. * For example, a hyperplane in a plane is a line; a hyperplane in 3-space is a plane. * * \tparam Scalar_ the scalar type, i.e., the type of the coefficients * \tparam AmbientDim_ the dimension of the ambient space, can be a compile time value or Dynamic. * Notice that the dimension of the hyperplane is AmbientDim_-1. * * This class represents an hyperplane as the zero set of the implicit equation * \f$ n \cdot x + d = 0 \f$ where \f$ n \f$ is a unit normal vector of the plane (linear part) * and \f$ d \f$ is the distance (offset) to the origin. */ template class Hyperplane { public: EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(Scalar_, AmbientDim_ == Dynamic ? Dynamic : AmbientDim_ + 1) enum { AmbientDimAtCompileTime = AmbientDim_, Options = Options_ }; typedef Scalar_ Scalar; typedef typename NumTraits::Real RealScalar; typedef Eigen::Index Index; ///< \deprecated since Eigen 3.3 typedef Matrix VectorType; typedef Matrix Coefficients; typedef Block NormalReturnType; typedef const Block ConstNormalReturnType; /** Default constructor without initialization */ EIGEN_DEVICE_FUNC inline Hyperplane() {} template EIGEN_DEVICE_FUNC Hyperplane(const Hyperplane& other) : m_coeffs(other.coeffs()) {} /** Constructs a dynamic-size hyperplane with \a _dim the dimension * of the ambient space */ EIGEN_DEVICE_FUNC inline explicit Hyperplane(Index _dim) : m_coeffs(_dim + 1) {} /** Construct a plane from its normal \a n and a point \a e onto the plane. * \warning the vector normal is assumed to be normalized. */ EIGEN_DEVICE_FUNC inline Hyperplane(const VectorType& n, const VectorType& e) : m_coeffs(n.size() + 1) { normal() = n; offset() = -n.dot(e); } /** Constructs a plane from its normal \a n and distance to the origin \a d * such that the algebraic equation of the plane is \f$ n \cdot x + d = 0 \f$. * \warning the vector normal is assumed to be normalized. */ EIGEN_DEVICE_FUNC inline Hyperplane(const VectorType& n, const Scalar& d) : m_coeffs(n.size() + 1) { normal() = n; offset() = d; } /** Constructs a hyperplane passing through the two points. If the dimension of the ambient space * is greater than 2, then there isn't uniqueness, so an arbitrary choice is made. */ EIGEN_DEVICE_FUNC static inline Hyperplane Through(const VectorType& p0, const VectorType& p1) { Hyperplane result(p0.size()); result.normal() = (p1 - p0).unitOrthogonal(); result.offset() = -p0.dot(result.normal()); return result; } /** Constructs a hyperplane passing through the three points. The dimension of the ambient space * is required to be exactly 3. */ EIGEN_DEVICE_FUNC static inline Hyperplane Through(const VectorType& p0, const VectorType& p1, const VectorType& p2) { EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(VectorType, 3) Hyperplane result(p0.size()); VectorType v0(p2 - p0), v1(p1 - p0); result.normal() = v0.cross(v1); RealScalar norm = result.normal().norm(); if (norm <= v0.norm() * v1.norm() * NumTraits::epsilon()) { Matrix m; m << v0.transpose(), v1.transpose(); JacobiSVD, ComputeFullV> svd(m); result.normal() = svd.matrixV().col(2); } else result.normal() /= norm; result.offset() = -p0.dot(result.normal()); return result; } /** Constructs a hyperplane passing through the parametrized line \a parametrized. * If the dimension of the ambient space is greater than 2, then there isn't uniqueness, * so an arbitrary choice is made. */ // FIXME to be consistent with the rest this could be implemented as a static Through function ?? EIGEN_DEVICE_FUNC explicit Hyperplane(const ParametrizedLine& parametrized) { normal() = parametrized.direction().unitOrthogonal(); offset() = -parametrized.origin().dot(normal()); } EIGEN_DEVICE_FUNC ~Hyperplane() {} /** \returns the dimension in which the plane holds */ EIGEN_DEVICE_FUNC inline Index dim() const { return AmbientDimAtCompileTime == Dynamic ? m_coeffs.size() - 1 : Index(AmbientDimAtCompileTime); } /** normalizes \c *this */ EIGEN_DEVICE_FUNC void normalize(void) { m_coeffs /= normal().norm(); } /** \returns the signed distance between the plane \c *this and a point \a p. * \sa absDistance() */ EIGEN_DEVICE_FUNC inline Scalar signedDistance(const VectorType& p) const { return normal().dot(p) + offset(); } /** \returns the absolute distance between the plane \c *this and a point \a p. * \sa signedDistance() */ EIGEN_DEVICE_FUNC inline Scalar absDistance(const VectorType& p) const { return numext::abs(signedDistance(p)); } /** \returns the projection of a point \a p onto the plane \c *this. */ EIGEN_DEVICE_FUNC inline VectorType projection(const VectorType& p) const { return p - signedDistance(p) * normal(); } /** \returns a constant reference to the unit normal vector of the plane, which corresponds * to the linear part of the implicit equation. */ EIGEN_DEVICE_FUNC inline ConstNormalReturnType normal() const { return ConstNormalReturnType(m_coeffs, 0, 0, dim(), 1); } /** \returns a non-constant reference to the unit normal vector of the plane, which corresponds * to the linear part of the implicit equation. */ EIGEN_DEVICE_FUNC inline NormalReturnType normal() { return NormalReturnType(m_coeffs, 0, 0, dim(), 1); } /** \returns the distance to the origin, which is also the "constant term" of the implicit equation * \warning the vector normal is assumed to be normalized. */ EIGEN_DEVICE_FUNC inline const Scalar& offset() const { return m_coeffs.coeff(dim()); } /** \returns a non-constant reference to the distance to the origin, which is also the constant part * of the implicit equation */ EIGEN_DEVICE_FUNC inline Scalar& offset() { return m_coeffs(dim()); } /** \returns a constant reference to the coefficients c_i of the plane equation: * \f$ c_0*x_0 + ... + c_{d-1}*x_{d-1} + c_d = 0 \f$ */ EIGEN_DEVICE_FUNC inline const Coefficients& coeffs() const { return m_coeffs; } /** \returns a non-constant reference to the coefficients c_i of the plane equation: * \f$ c_0*x_0 + ... + c_{d-1}*x_{d-1} + c_d = 0 \f$ */ EIGEN_DEVICE_FUNC inline Coefficients& coeffs() { return m_coeffs; } /** \returns the intersection of *this with \a other. * * \warning The ambient space must be a plane, i.e. have dimension 2, so that \c *this and \a other are lines. * * \note If \a other is approximately parallel to *this, this method will return any point on *this. */ EIGEN_DEVICE_FUNC VectorType intersection(const Hyperplane& other) const { EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(VectorType, 2) Scalar det = coeffs().coeff(0) * other.coeffs().coeff(1) - coeffs().coeff(1) * other.coeffs().coeff(0); // since the line equations ax+by=c are normalized with a^2+b^2=1, the following tests // whether the two lines are approximately parallel. if (internal::isMuchSmallerThan(det, Scalar(1))) { // special case where the two lines are approximately parallel. // Pick any point on the first line. if (numext::abs(coeffs().coeff(1)) > numext::abs(coeffs().coeff(0))) return VectorType(coeffs().coeff(1), -coeffs().coeff(2) / coeffs().coeff(1) - coeffs().coeff(0)); else return VectorType(-coeffs().coeff(2) / coeffs().coeff(0) - coeffs().coeff(1), coeffs().coeff(0)); } else { // general case Scalar invdet = Scalar(1) / det; return VectorType( invdet * (coeffs().coeff(1) * other.coeffs().coeff(2) - other.coeffs().coeff(1) * coeffs().coeff(2)), invdet * (other.coeffs().coeff(0) * coeffs().coeff(2) - coeffs().coeff(0) * other.coeffs().coeff(2))); } } /** Applies the transformation matrix \a mat to \c *this and returns a reference to \c *this. * * \param mat the Dim x Dim transformation matrix * \param traits specifies whether the matrix \a mat represents an #Isometry * or a more generic #Affine transformation. The default is #Affine. */ template EIGEN_DEVICE_FUNC inline Hyperplane& transform(const MatrixBase& mat, TransformTraits traits = Affine) { if (traits == Affine) { normal() = mat.inverse().transpose() * normal(); m_coeffs /= normal().norm(); } else if (traits == Isometry) normal() = mat * normal(); else { eigen_assert(0 && "invalid traits value in Hyperplane::transform()"); } return *this; } /** Applies the transformation \a t to \c *this and returns a reference to \c *this. * * \param t the transformation of dimension Dim * \param traits specifies whether the transformation \a t represents an #Isometry * or a more generic #Affine transformation. The default is #Affine. * Other kind of transformations are not supported. */ template EIGEN_DEVICE_FUNC inline Hyperplane& transform(const Transform& t, TransformTraits traits = Affine) { transform(t.linear(), traits); offset() -= normal().dot(t.translation()); return *this; } /** \returns \c *this with scalar type casted to \a NewScalarType * * Note that if \a NewScalarType is equal to the current scalar type of \c *this * then this function smartly returns a const reference to \c *this. */ template EIGEN_DEVICE_FUNC inline typename internal::cast_return_type >::type cast() const { return typename internal::cast_return_type >::type(*this); } /** Copy constructor with scalar type conversion */ template EIGEN_DEVICE_FUNC inline explicit Hyperplane( const Hyperplane& other) { m_coeffs = other.coeffs().template cast(); } /** \returns \c true if \c *this is approximately equal to \a other, within the precision * determined by \a prec. * * \sa MatrixBase::isApprox() */ template EIGEN_DEVICE_FUNC bool isApprox( const Hyperplane& other, const typename NumTraits::Real& prec = NumTraits::dummy_precision()) const { return m_coeffs.isApprox(other.m_coeffs, prec); } protected: Coefficients m_coeffs; }; } // end namespace Eigen #endif // EIGEN_HYPERPLANE_H