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