Eigen/src/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@math.jussieu.fr>
 //
 // Eigen is free software; you can redistribute it and/or
 // modify it under the terms of the GNU Lesser General Public
 // License as published by the Free Software Foundation; either
 // version 3 of the License, or (at your option) any later version.
 //
 // Alternatively, you can redistribute it and/or
 // modify it under the terms of the GNU General Public License as
 // published by the Free Software Foundation; either version 2 of
 // the License, or (at your option) any later version.
 //
 // Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
 // WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
 // FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
 // GNU General Public License for more details.
 //
 // You should have received a copy of the GNU Lesser General Public
 // License and a copy of the GNU General Public License along with
 // Eigen. If not, see <http://www.gnu.org/licenses/>.

 #ifndef EIGEN_HYPERPLANE_H
 #define EIGEN_HYPERPLANE_H

 /** \geometry_module \ingroup GeometryModule
   *
   * \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
   #ifdef EIGEN_VECTORIZE
   : public ei_with_aligned_operator_new<_Scalar,_AmbientDim==Dynamic ? Dynamic : _AmbientDim+1>
   #endif
 {
   public:

     enum { AmbientDimAtCompileTime = _AmbientDim };
     typedef _Scalar Scalar;
     typedef typename NumTraits<Scalar>::Real RealScalar;
     typedef Matrix<Scalar,AmbientDimAtCompileTime,1> VectorType;
     typedef Matrix<Scalar,AmbientDimAtCompileTime==Dynamic
                           ? Dynamic
                           : AmbientDimAtCompileTime+1,1> Coefficients;
     typedef Block<Coefficients,AmbientDimAtCompileTime,1> NormalReturnType;

     /** Default constructor without initialization */
     inline explicit Hyperplane(int _dim = AmbientDimAtCompileTime) : 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.dot(n);
     }

     /** Constructs a plane from its normal \a n and distance to the origin \a d.
       * \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().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().dot(p0);
       return result;
     }

     Hyperplane(const ParametrizedLine<Scalar, AmbientDimAtCompileTime>& parametrized)
     {
       normal() = parametrized.direction().unitOrthogonal();
       offset() = -normal().dot(parametrized.origin());
     }

     ~Hyperplane() {}

     /** \returns the dimension in which the plane holds */
     inline int dim() const { return AmbientDimAtCompileTime==Dynamic ? m_coeffs.size()-1 : 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.
       */
     inline Scalar signedDistance(const VectorType& p) const { return p.dot(normal()) + offset(); }

     /** \returns the absolute distance between the plane \c *this and a point \a p.
       */
     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(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 *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)));
       }
     }

     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;
     }

     inline Hyperplane& transform(const Transform<Scalar,AmbientDimAtCompileTime>& t,
                                  TransformTraits traits = Affine)
     {
       transform(t.linear(), traits);
       offset() -= t.translation().dot(normal());
       return *this;
     }

 protected:

     Coefficients m_coeffs;
 };

 #endif // EIGEN_HYPERPLANE_H
	// 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@math.jussieu.fr>
	//
	// Eigen is free software; you can redistribute it and/or
	// modify it under the terms of the GNU Lesser General Public
	// License as published by the Free Software Foundation; either
	// version 3 of the License, or (at your option) any later version.
	//
	// Alternatively, you can redistribute it and/or
	// modify it under the terms of the GNU General Public License as
	// published by the Free Software Foundation; either version 2 of
	// the License, or (at your option) any later version.
	//
	// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
	// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
	// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
	// GNU General Public License for more details.
	//
	// You should have received a copy of the GNU Lesser General Public
	// License and a copy of the GNU General Public License along with
	// Eigen. If not, see <http://www.gnu.org/licenses/>.

	#ifndef EIGEN_HYPERPLANE_H
	#define EIGEN_HYPERPLANE_H

	/** \geometry_module \ingroup GeometryModule
	*
	* \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
	#ifdef EIGEN_VECTORIZE
	: public ei_with_aligned_operator_new<_Scalar,_AmbientDim==Dynamic ? Dynamic : _AmbientDim+1>
	#endif
	{
	public:

	enum { AmbientDimAtCompileTime = _AmbientDim };
	typedef _Scalar Scalar;
	typedef typename NumTraits<Scalar>::Real RealScalar;
	typedef Matrix<Scalar,AmbientDimAtCompileTime,1> VectorType;
	typedef Matrix<Scalar,AmbientDimAtCompileTime==Dynamic
	? Dynamic
	: AmbientDimAtCompileTime+1,1> Coefficients;
	typedef Block<Coefficients,AmbientDimAtCompileTime,1> NormalReturnType;

	/** Default constructor without initialization */
	inline explicit Hyperplane(int _dim = AmbientDimAtCompileTime) : 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.dot(n);
	}

	/** Constructs a plane from its normal \a n and distance to the origin \a d.
	* \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().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().dot(p0);
	return result;
	}

	Hyperplane(const ParametrizedLine<Scalar, AmbientDimAtCompileTime>& parametrized)
	{
	normal() = parametrized.direction().unitOrthogonal();
	offset() = -normal().dot(parametrized.origin());
	}

	~Hyperplane() {}

	/** \returns the dimension in which the plane holds */
	inline int dim() const { return AmbientDimAtCompileTime==Dynamic ? m_coeffs.size()-1 : 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.
	*/
	inline Scalar signedDistance(const VectorType& p) const { return p.dot(normal()) + offset(); }

	/** \returns the absolute distance between the plane \c *this and a point \a p.
	*/
	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(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 *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)));
	}
	}

	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;
	}

	inline Hyperplane& transform(const Transform<Scalar,AmbientDimAtCompileTime>& t,
	TransformTraits traits = Affine)
	{
	transform(t.linear(), traits);
	offset() -= t.translation().dot(normal());
	return *this;
	}

	protected:

	Coefficients m_coeffs;
	};

	#endif // EIGEN_HYPERPLANE_H