Hyperplane.h

00001 // This file is part of Eigen, a lightweight C++ template library
00002 // for linear algebra.
00003 //
00004 // Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
00005 // Copyright (C) 2008 Benoit Jacob <jacob.benoit.1@gmail.com>
00006 //
00007 // This Source Code Form is subject to the terms of the Mozilla
00008 // Public License v. 2.0. If a copy of the MPL was not distributed
00009 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
00010 
00011 #ifndef EIGEN_HYPERPLANE_H
00012 #define EIGEN_HYPERPLANE_H
00013 
00014 namespace Eigen { 
00015 
00016 /** \geometry_module \ingroup Geometry_Module
00017   *
00018   * \class Hyperplane
00019   *
00020   * \brief A hyperplane
00021   *
00022   * A hyperplane is an affine subspace of dimension n-1 in a space of dimension n.
00023   * For example, a hyperplane in a plane is a line; a hyperplane in 3-space is a plane.
00024   *
00025   * \param _Scalar the scalar type, i.e., the type of the coefficients
00026   * \param _AmbientDim the dimension of the ambient space, can be a compile time value or Dynamic.
00027   *             Notice that the dimension of the hyperplane is _AmbientDim-1.
00028   *
00029   * This class represents an hyperplane as the zero set of the implicit equation
00030   * \f$ n \cdot x + d = 0 \f$ where \f$ n \f$ is a unit normal vector of the plane (linear part)
00031   * and \f$ d \f$ is the distance (offset) to the origin.
00032   */
00033 template <typename _Scalar, int _AmbientDim, int _Options>
00034 class Hyperplane 
00035 {
00036 public:
00037   EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(_Scalar,_AmbientDim==Dynamic ? Dynamic : _AmbientDim+1)
00038   enum {
00039     AmbientDimAtCompileTime = _AmbientDim,
00040     Options = _Options
00041   };
00042   typedef _Scalar Scalar;
00043   typedef typename NumTraits<Scalar>::Real RealScalar;
00044   typedef DenseIndex Index;
00045   typedef Matrix<Scalar,AmbientDimAtCompileTime,1>  VectorType ;
00046   typedef Matrix<Scalar,Index(AmbientDimAtCompileTime)==Dynamic
00047                         ? Dynamic
00048                         : Index(AmbientDimAtCompileTime)+1,1,Options> Coefficients ;
00049   typedef Block<Coefficients,AmbientDimAtCompileTime,1> NormalReturnType;
00050   typedef const Block<const Coefficients,AmbientDimAtCompileTime,1> ConstNormalReturnType;
00051 
00052   /** Default constructor without initialization */
00053   inline Hyperplane() {}
00054   
00055   template<int OtherOptions>
00056   Hyperplane(const Hyperplane<Scalar,AmbientDimAtCompileTime,OtherOptions> & other)
00057    : m_coeffs(other.coeffs ())
00058   {}
00059 
00060   /** Constructs a dynamic-size hyperplane with \a _dim the dimension
00061     * of the ambient space */
00062   inline explicit Hyperplane(Index _dim) : m_coeffs(_dim+1) {}
00063 
00064   /** Construct a plane from its normal \a n and a point \a e onto the plane.
00065     * \warning the vector normal is assumed to be normalized.
00066     */
00067   inline Hyperplane(const VectorType & n, const VectorType & e)
00068     : m_coeffs(n.size()+1)
00069   {
00070     normal () = n;
00071     offset () = -n.dot(e);
00072   }
00073 
00074   /** Constructs a plane from its normal \a n and distance to the origin \a d
00075     * such that the algebraic equation of the plane is \f$ n \cdot x + d = 0 \f$.
00076     * \warning the vector normal is assumed to be normalized.
00077     */
00078   inline Hyperplane(const VectorType & n, const Scalar& d)
00079     : m_coeffs(n.size()+1)
00080   {
00081     normal () = n;
00082     offset () = d;
00083   }
00084 
00085   /** Constructs a hyperplane passing through the two points. If the dimension of the ambient space
00086     * is greater than 2, then there isn't uniqueness, so an arbitrary choice is made.
00087     */
00088   static inline Hyperplane  Through(const VectorType & p0, const VectorType & p1)
00089   {
00090     Hyperplane  result(p0.size());
00091     result.normal () = (p1 - p0).unitOrthogonal();
00092     result.offset() = -p0.dot(result.normal());
00093     return result;
00094   }
00095 
00096   /** Constructs a hyperplane passing through the three points. The dimension of the ambient space
00097     * is required to be exactly 3.
00098     */
00099   static inline Hyperplane  Through(const VectorType & p0, const VectorType & p1, const VectorType & p2)
00100   {
00101     EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(VectorType , 3)
00102     Hyperplane  result(p0.size());
00103     VectorType  v0(p2 - p0), v1(p1 - p0);
00104     result.normal() = v0.cross(v1);
00105     RealScalar norm = result.normal().norm();
00106     if(norm <= v0.norm() * v1.norm() * NumTraits<RealScalar>::epsilon())
00107     {
00108       Matrix<Scalar,2,3> m; m << v0.transpose(), v1.transpose();
00109       JacobiSVD<Matrix<Scalar,2,3> > svd(m, ComputeFullV);
00110       result.normal() = svd.matrixV ().col(2);
00111     }
00112     else
00113       result.normal() /= norm;
00114     result.offset() = -p0.dot(result.normal());
00115     return result;
00116   }
00117 
00118   /** Constructs a hyperplane passing through the parametrized line \a parametrized.
00119     * If the dimension of the ambient space is greater than 2, then there isn't uniqueness,
00120     * so an arbitrary choice is made.
00121     */
00122   // FIXME to be consitent with the rest this could be implemented as a static Through function ??
00123   explicit Hyperplane(const ParametrizedLine<Scalar, AmbientDimAtCompileTime> & parametrized)
00124   {
00125     normal () = parametrized.direction().unitOrthogonal();
00126     offset () = -parametrized.origin().dot(normal ());
00127   }
00128 
00129   ~Hyperplane () {}
00130 
00131   /** \returns the dimension in which the plane holds */
00132   inline Index dim () const { return AmbientDimAtCompileTime==Dynamic ? m_coeffs.size()-1 : Index(AmbientDimAtCompileTime); }
00133 
00134   /** normalizes \c *this */
00135   void normalize(void)
00136   {
00137     m_coeffs /= normal ().norm();
00138   }
00139 
00140   /** \returns the signed distance between the plane \c *this and a point \a p.
00141     * \sa absDistance()
00142     */
00143   inline Scalar signedDistance (const VectorType & p) const { return normal ().dot(p) + offset (); }
00144 
00145   /** \returns the absolute distance between the plane \c *this and a point \a p.
00146     * \sa signedDistance()
00147     */
00148   inline Scalar absDistance (const VectorType & p) const { using std::abs; return abs(signedDistance (p)); }
00149 
00150   /** \returns the projection of a point \a p onto the plane \c *this.
00151     */
00152   inline VectorType  projection (const VectorType & p) const { return p - signedDistance (p) * normal (); }
00153 
00154   /** \returns a constant reference to the unit normal vector of the plane, which corresponds
00155     * to the linear part of the implicit equation.
00156     */
00157   inline ConstNormalReturnType normal () const { return ConstNormalReturnType(m_coeffs,0,0,dim (),1); }
00158 
00159   /** \returns a non-constant reference to the unit normal vector of the plane, which corresponds
00160     * to the linear part of the implicit equation.
00161     */
00162   inline NormalReturnType normal () { return NormalReturnType(m_coeffs,0,0,dim (),1); }
00163 
00164   /** \returns the distance to the origin, which is also the "constant term" of the implicit equation
00165     * \warning the vector normal is assumed to be normalized.
00166     */
00167   inline const Scalar& offset () const { return m_coeffs.coeff(dim ()); }
00168 
00169   /** \returns a non-constant reference to the distance to the origin, which is also the constant part
00170     * of the implicit equation */
00171   inline Scalar& offset () { return m_coeffs(dim ()); }
00172 
00173   /** \returns a constant reference to the coefficients c_i of the plane equation:
00174     * \f$ c_0*x_0 + ... + c_{d-1}*x_{d-1} + c_d = 0 \f$
00175     */
00176   inline const Coefficients & coeffs () const { return m_coeffs; }
00177 
00178   /** \returns a non-constant reference to the coefficients c_i of the plane equation:
00179     * \f$ c_0*x_0 + ... + c_{d-1}*x_{d-1} + c_d = 0 \f$
00180     */
00181   inline Coefficients & coeffs () { return m_coeffs; }
00182 
00183   /** \returns the intersection of *this with \a other.
00184     *
00185     * \warning The ambient space must be a plane, i.e. have dimension 2, so that \c *this and \a other are lines.
00186     *
00187     * \note If \a other is approximately parallel to *this, this method will return any point on *this.
00188     */
00189   VectorType  intersection (const Hyperplane & other) const
00190   {
00191     using std::abs;
00192     EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(VectorType , 2)
00193     Scalar det = coeffs ().coeff(0) * other.coeffs ().coeff(1) - coeffs ().coeff(1) * other.coeffs ().coeff(0);
00194     // since the line equations ax+by=c are normalized with a^2+b^2=1, the following tests
00195     // whether the two lines are approximately parallel.
00196     if(internal::isMuchSmallerThan(det, Scalar(1)))
00197     {   // special case where the two lines are approximately parallel. Pick any point on the first line.
00198         if(abs(coeffs ().coeff(1))>abs(coeffs ().coeff(0)))
00199             return VectorType (coeffs ().coeff(1), -coeffs ().coeff(2)/coeffs ().coeff(1)-coeffs ().coeff(0));
00200         else
00201             return VectorType (-coeffs ().coeff(2)/coeffs ().coeff(0)-coeffs ().coeff(1), coeffs ().coeff(0));
00202     }
00203     else
00204     {   // general case
00205         Scalar invdet = Scalar(1) / det;
00206         return VectorType (invdet*(coeffs ().coeff(1)*other.coeffs ().coeff(2)-other.coeffs ().coeff(1)*coeffs ().coeff(2)),
00207                           invdet*(other.coeffs ().coeff(0)*coeffs ().coeff(2)-coeffs ().coeff(0)*other.coeffs ().coeff(2)));
00208     }
00209   }
00210 
00211   /** Applies the transformation matrix \a mat to \c *this and returns a reference to \c *this.
00212     *
00213     * \param mat the Dim x Dim transformation matrix
00214     * \param traits specifies whether the matrix \a mat represents an #Isometry
00215     *               or a more generic #Affine transformation. The default is #Affine.
00216     */
00217   template<typename XprType>
00218   inline Hyperplane & transform(const MatrixBase<XprType>& mat, TransformTraits traits = Affine)
00219   {
00220     if (traits==Affine)
00221       normal () = mat.inverse ().transpose() * normal ();
00222     else if (traits==Isometry)
00223       normal () = mat * normal ();
00224     else
00225     {
00226       eigen_assert(0 && "invalid traits value in Hyperplane::transform()");
00227     }
00228     return *this;
00229   }
00230 
00231   /** Applies the transformation \a t to \c *this and returns a reference to \c *this.
00232     *
00233     * \param t the transformation of dimension Dim
00234     * \param traits specifies whether the transformation \a t represents an #Isometry
00235     *               or a more generic #Affine transformation. The default is #Affine.
00236     *               Other kind of transformations are not supported.
00237     */
00238   template<int TrOptions>
00239   inline Hyperplane & transform(const Transform<Scalar,AmbientDimAtCompileTime,Affine,TrOptions> & t,
00240                                 TransformTraits traits = Affine)
00241   {
00242     transform(t.linear (), traits);
00243     offset () -= normal ().dot(t.translation ());
00244     return *this;
00245   }
00246 
00247   /** \returns \c *this with scalar type casted to \a NewScalarType
00248     *
00249     * Note that if \a NewScalarType is equal to the current scalar type of \c *this
00250     * then this function smartly returns a const reference to \c *this.
00251     */
00252   template<typename NewScalarType>
00253   inline typename internal::cast_return_type<Hyperplane,
00254            Hyperplane<NewScalarType,AmbientDimAtCompileTime,Options>  >::type cast () const
00255   {
00256     return typename internal::cast_return_type<Hyperplane,
00257                     Hyperplane<NewScalarType,AmbientDimAtCompileTime,Options>  >::type(*this);
00258   }
00259 
00260   /** Copy constructor with scalar type conversion */
00261   template<typename OtherScalarType,int OtherOptions>
00262   inline explicit Hyperplane(const Hyperplane<OtherScalarType,AmbientDimAtCompileTime,OtherOptions> & other)
00263   { m_coeffs = other.coeffs ().template cast<Scalar>(); }
00264 
00265   /** \returns \c true if \c *this is approximately equal to \a other, within the precision
00266     * determined by \a prec.
00267     *
00268     * \sa MatrixBase::isApprox() */
00269   template<int OtherOptions>
00270   bool isApprox (const Hyperplane<Scalar,AmbientDimAtCompileTime,OtherOptions> & other, const typename NumTraits<Scalar>::Real& prec = NumTraits<Scalar>::dummy_precision()) const
00271   { return m_coeffs.isApprox(other.m_coeffs, prec); }
00272 
00273 protected:
00274 
00275   Coefficients m_coeffs;
00276 };
00277 
00278 } // end namespace Eigen
00279 
00280 #endif // EIGEN_HYPERPLANE_H