Eigen/src/Eigen2Support/Geometry/Hyperplane.h - mirror - Git at Google

 // This file is part of Eigen, a lightweight C++ template library
 // for linear algebra. Eigen itself is part of the KDE project.
 //
 // Copyright (C) 2008 Gael Guennebaud <g.gael@free.fr>
 // Copyright (C) 2008 Benoit Jacob <jacob.benoit.1@gmail.com>
 //
 // 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/.

 // no include guard, we'll include this twice from All.h from Eigen2Support, and it's internal anyway

 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.
   *
   * \param _Scalar the scalar type, i.e., the type of the coefficients
   * \param _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 <typename _Scalar, int _AmbientDim>
 class Hyperplane
 {
 public:
   EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(_Scalar,_AmbientDim==Dynamic ? Dynamic : _AmbientDim+1)
   enum { AmbientDimAtCompileTime = _AmbientDim };
   typedef _Scalar Scalar;
   typedef typename NumTraits<Scalar>::Real RealScalar;
   typedef Matrix<Scalar,AmbientDimAtCompileTime,1> VectorType;
   typedef Matrix<Scalar,int(AmbientDimAtCompileTime)==Dynamic
                         ? Dynamic
                         : int(AmbientDimAtCompileTime)+1,1> Coefficients;
   typedef Block<Coefficients,AmbientDimAtCompileTime,1> NormalReturnType;

   /** Default constructor without initialization */
   inline explicit Hyperplane() {}

   /** Constructs a dynamic-size hyperplane with \a _dim the dimension
     * of the ambient space */
   inline explicit Hyperplane(int _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.
     */
   inline Hyperplane(const VectorType& n, const VectorType& e)
     : m_coeffs(n.size()+1)
   {
     normal() = n;
     offset() = -e.eigen2_dot(n);
   }

   /** 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.
     */
   inline Hyperplane(const VectorType& n, 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.
     */
   static inline Hyperplane Through(const VectorType& p0, const VectorType& p1)
   {
     Hyperplane result(p0.size());
     result.normal() = (p1 - p0).unitOrthogonal();
     result.offset() = -result.normal().eigen2_dot(p0);
     return result;
   }

   /** Constructs a hyperplane passing through the three points. The dimension of the ambient space
     * is required to be exactly 3.
     */
   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());
     result.normal() = (p2 - p0).cross(p1 - p0).normalized();
     result.offset() = -result.normal().eigen2_dot(p0);
     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 consitent with the rest this could be implemented as a static Through function ??
   explicit Hyperplane(const ParametrizedLine<Scalar, AmbientDimAtCompileTime>& parametrized)
   {
     normal() = parametrized.direction().unitOrthogonal();
     offset() = -normal().eigen2_dot(parametrized.origin());
   }

   ~Hyperplane() {}

   /** \returns the dimension in which the plane holds */
   inline int dim() const { return int(AmbientDimAtCompileTime)==Dynamic ? m_coeffs.size()-1 : int(AmbientDimAtCompileTime); }

   /** normalizes \c *this */
   void normalize(void)
   {
     m_coeffs /= normal().norm();
   }

   /** \returns the signed distance between the plane \c *this and a point \a p.
     * \sa absDistance()
     */
   inline Scalar signedDistance(const VectorType& p) const { return p.eigen2_dot(normal()) + offset(); }

   /** \returns the absolute distance between the plane \c *this and a point \a p.
     * \sa signedDistance()
     */
   inline Scalar absDistance(const VectorType& p) const { return ei_abs(signedDistance(p)); }

   /** \returns the projection of a point \a p onto the plane \c *this.
     */
   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.
     */
   inline const NormalReturnType normal() const { return NormalReturnType(*const_cast<Coefficients*>(&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.
     */
   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.
     */
   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 */
   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$
     */
   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$
     */
   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.
     */
   VectorType intersection(const Hyperplane& other)
   {
     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(ei_isMuchSmallerThan(det, Scalar(1)))
     {   // special case where the two lines are approximately parallel. Pick any point on the first line.
         if(ei_abs(coeffs().coeff(1))>ei_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<typename XprType>
   inline Hyperplane& transform(const MatrixBase<XprType>& mat, TransformTraits traits = Affine)
   {
     if (traits==Affine)
       normal() = mat.inverse().transpose() * normal();
     else if (traits==Isometry)
       normal() = mat * normal();
     else
     {
       ei_assert("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.
     */
   inline Hyperplane& transform(const Transform<Scalar,AmbientDimAtCompileTime>& t,
                                 TransformTraits traits = Affine)
   {
     transform(t.linear(), traits);
     offset() -= t.translation().eigen2_dot(normal());
     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<typename NewScalarType>
   inline typename internal::cast_return_type<Hyperplane,
            Hyperplane<NewScalarType,AmbientDimAtCompileTime> >::type cast() const
   {
     return typename internal::cast_return_type<Hyperplane,
                     Hyperplane<NewScalarType,AmbientDimAtCompileTime> >::type(*this);
   }

   /** Copy constructor with scalar type conversion */
   template<typename OtherScalarType>
   inline explicit Hyperplane(const Hyperplane<OtherScalarType,AmbientDimAtCompileTime>& other)
   { m_coeffs = other.coeffs().template cast<Scalar>(); }

   /** \returns \c true if \c *this is approximately equal to \a other, within the precision
     * determined by \a prec.
     *
     * \sa MatrixBase::isApprox() */
   bool isApprox(const Hyperplane& other, typename NumTraits<Scalar>::Real prec = precision<Scalar>()) const
   { return m_coeffs.isApprox(other.m_coeffs, prec); }

 protected:

   Coefficients m_coeffs;
 };

 } // end namespace Eigen
	// This file is part of Eigen, a lightweight C++ template library
	// for linear algebra. Eigen itself is part of the KDE project.
	//
	// Copyright (C) 2008 Gael Guennebaud <g.gael@free.fr>
	// Copyright (C) 2008 Benoit Jacob <jacob.benoit.1@gmail.com>
	//
	// 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/.

	// no include guard, we'll include this twice from All.h from Eigen2Support, and it's internal anyway

	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.
	*
	* \param _Scalar the scalar type, i.e., the type of the coefficients
	* \param _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 <typename _Scalar, int _AmbientDim>
	class Hyperplane
	{
	public:
	EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(_Scalar,_AmbientDim==Dynamic ? Dynamic : _AmbientDim+1)
	enum { AmbientDimAtCompileTime = _AmbientDim };
	typedef _Scalar Scalar;
	typedef typename NumTraits<Scalar>::Real RealScalar;
	typedef Matrix<Scalar,AmbientDimAtCompileTime,1> VectorType;
	typedef Matrix<Scalar,int(AmbientDimAtCompileTime)==Dynamic
	? Dynamic
	: int(AmbientDimAtCompileTime)+1,1> Coefficients;
	typedef Block<Coefficients,AmbientDimAtCompileTime,1> NormalReturnType;

	/** Default constructor without initialization */
	inline explicit Hyperplane() {}

	/** Constructs a dynamic-size hyperplane with \a _dim the dimension
	* of the ambient space */
	inline explicit Hyperplane(int _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.
	*/
	inline Hyperplane(const VectorType& n, const VectorType& e)
	: m_coeffs(n.size()+1)
	{
	normal() = n;
	offset() = -e.eigen2_dot(n);
	}

	/** 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.
	*/
	inline Hyperplane(const VectorType& n, 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.
	*/
	static inline Hyperplane Through(const VectorType& p0, const VectorType& p1)
	{
	Hyperplane result(p0.size());
	result.normal() = (p1 - p0).unitOrthogonal();
	result.offset() = -result.normal().eigen2_dot(p0);
	return result;
	}

	/** Constructs a hyperplane passing through the three points. The dimension of the ambient space
	* is required to be exactly 3.
	*/
	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());
	result.normal() = (p2 - p0).cross(p1 - p0).normalized();
	result.offset() = -result.normal().eigen2_dot(p0);
	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 consitent with the rest this could be implemented as a static Through function ??
	explicit Hyperplane(const ParametrizedLine<Scalar, AmbientDimAtCompileTime>& parametrized)
	{
	normal() = parametrized.direction().unitOrthogonal();
	offset() = -normal().eigen2_dot(parametrized.origin());
	}

	~Hyperplane() {}

	/** \returns the dimension in which the plane holds */
	inline int dim() const { return int(AmbientDimAtCompileTime)==Dynamic ? m_coeffs.size()-1 : int(AmbientDimAtCompileTime); }

	/** normalizes \c this /
	void normalize(void)
	{
	m_coeffs /= normal().norm();
	}

	/** \returns the signed distance between the plane \c *this and a point \a p.
	* \sa absDistance()
	*/
	inline Scalar signedDistance(const VectorType& p) const { return p.eigen2_dot(normal()) + offset(); }

	/** \returns the absolute distance between the plane \c *this and a point \a p.
	* \sa signedDistance()
	*/
	inline Scalar absDistance(const VectorType& p) const { return ei_abs(signedDistance(p)); }

	/** \returns the projection of a point \a p onto the plane \c *this.
	*/
	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.
	*/
	inline const NormalReturnType normal() const { return NormalReturnType(const_cast<Coefficients>(&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.
	*/
	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.
	*/
	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 */
	inline Scalar& offset() { return m_coeffs(dim()); }

	/** \returns a constant reference to the coefficients c_i of the plane equation:
	* \f$ c_0x_0 + ... + c_{d-1}x_{d-1} + c_d = 0 \f$
	*/
	inline const Coefficients& coeffs() const { return m_coeffs; }

	/** \returns a non-constant reference to the coefficients c_i of the plane equation:
	* \f$ c_0x_0 + ... + c_{d-1}x_{d-1} + c_d = 0 \f$
	*/
	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.
	*/
	VectorType intersection(const Hyperplane& other)
	{
	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(ei_isMuchSmallerThan(det, Scalar(1)))
	{ // special case where the two lines are approximately parallel. Pick any point on the first line.
	if(ei_abs(coeffs().coeff(1))>ei_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<typename XprType>
	inline Hyperplane& transform(const MatrixBase<XprType>& mat, TransformTraits traits = Affine)
	{
	if (traits==Affine)
	normal() = mat.inverse().transpose() * normal();
	else if (traits==Isometry)
	normal() = mat * normal();
	else
	{
	ei_assert("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.
	*/
	inline Hyperplane& transform(const Transform<Scalar,AmbientDimAtCompileTime>& t,
	TransformTraits traits = Affine)
	{
	transform(t.linear(), traits);
	offset() -= t.translation().eigen2_dot(normal());
	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<typename NewScalarType>
	inline typename internal::cast_return_type<Hyperplane,
	Hyperplane<NewScalarType,AmbientDimAtCompileTime> >::type cast() const
	{
	return typename internal::cast_return_type<Hyperplane,
	Hyperplane<NewScalarType,AmbientDimAtCompileTime> >::type(*this);
	}

	/** Copy constructor with scalar type conversion */
	template<typename OtherScalarType>
	inline explicit Hyperplane(const Hyperplane<OtherScalarType,AmbientDimAtCompileTime>& other)
	{ m_coeffs = other.coeffs().template cast<Scalar>(); }

	/** \returns \c true if \c *this is approximately equal to \a other, within the precision
	* determined by \a prec.
	*
	* \sa MatrixBase::isApprox() */
	bool isApprox(const Hyperplane& other, typename NumTraits<Scalar>::Real prec = precision<Scalar>()) const
	{ return m_coeffs.isApprox(other.m_coeffs, prec); }

	protected:

	Coefficients m_coeffs;
	};

	} // end namespace Eigen