1// This file is part of Eigen, a lightweight C++ template library
2// for linear algebra.
3//
4// Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
5// Copyright (C) 2008 Benoit Jacob <jacob.benoit.1@gmail.com>
6//
7// This Source Code Form is subject to the terms of the Mozilla
8// Public License v. 2.0. If a copy of the MPL was not distributed
9// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
10
11#ifndef EIGEN_HYPERPLANE_H
12#define EIGEN_HYPERPLANE_H
13
14namespace Eigen {
15
16/** \geometry_module \ingroup Geometry_Module
17  *
18  * \class Hyperplane
19  *
20  * \brief A hyperplane
21  *
22  * A hyperplane is an affine subspace of dimension n-1 in a space of dimension n.
23  * For example, a hyperplane in a plane is a line; a hyperplane in 3-space is a plane.
24  *
25  * \param _Scalar the scalar type, i.e., the type of the coefficients
26  * \param _AmbientDim the dimension of the ambient space, can be a compile time value or Dynamic.
27  *             Notice that the dimension of the hyperplane is _AmbientDim-1.
28  *
29  * This class represents an hyperplane as the zero set of the implicit equation
30  * \f$ n \cdot x + d = 0 \f$ where \f$ n \f$ is a unit normal vector of the plane (linear part)
31  * and \f$ d \f$ is the distance (offset) to the origin.
32  */
33template <typename _Scalar, int _AmbientDim, int _Options>
34class Hyperplane
35{
36public:
37  EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(_Scalar,_AmbientDim==Dynamic ? Dynamic : _AmbientDim+1)
38  enum {
39    AmbientDimAtCompileTime = _AmbientDim,
40    Options = _Options
41  };
42  typedef _Scalar Scalar;
43  typedef typename NumTraits<Scalar>::Real RealScalar;
44  typedef DenseIndex Index;
45  typedef Matrix<Scalar,AmbientDimAtCompileTime,1> VectorType;
46  typedef Matrix<Scalar,Index(AmbientDimAtCompileTime)==Dynamic
47                        ? Dynamic
48                        : Index(AmbientDimAtCompileTime)+1,1,Options> Coefficients;
49  typedef Block<Coefficients,AmbientDimAtCompileTime,1> NormalReturnType;
50  typedef const Block<const Coefficients,AmbientDimAtCompileTime,1> ConstNormalReturnType;
51
52  /** Default constructor without initialization */
53  inline Hyperplane() {}
54
55  template<int OtherOptions>
56  Hyperplane(const Hyperplane<Scalar,AmbientDimAtCompileTime,OtherOptions>& other)
57   : m_coeffs(other.coeffs())
58  {}
59
60  /** Constructs a dynamic-size hyperplane with \a _dim the dimension
61    * of the ambient space */
62  inline explicit Hyperplane(Index _dim) : m_coeffs(_dim+1) {}
63
64  /** Construct a plane from its normal \a n and a point \a e onto the plane.
65    * \warning the vector normal is assumed to be normalized.
66    */
67  inline Hyperplane(const VectorType& n, const VectorType& e)
68    : m_coeffs(n.size()+1)
69  {
70    normal() = n;
71    offset() = -n.dot(e);
72  }
73
74  /** Constructs a plane from its normal \a n and distance to the origin \a d
75    * such that the algebraic equation of the plane is \f$ n \cdot x + d = 0 \f$.
76    * \warning the vector normal is assumed to be normalized.
77    */
78  inline Hyperplane(const VectorType& n, const Scalar& d)
79    : m_coeffs(n.size()+1)
80  {
81    normal() = n;
82    offset() = d;
83  }
84
85  /** Constructs a hyperplane passing through the two points. If the dimension of the ambient space
86    * is greater than 2, then there isn't uniqueness, so an arbitrary choice is made.
87    */
88  static inline Hyperplane Through(const VectorType& p0, const VectorType& p1)
89  {
90    Hyperplane result(p0.size());
91    result.normal() = (p1 - p0).unitOrthogonal();
92    result.offset() = -p0.dot(result.normal());
93    return result;
94  }
95
96  /** Constructs a hyperplane passing through the three points. The dimension of the ambient space
97    * is required to be exactly 3.
98    */
99  static inline Hyperplane Through(const VectorType& p0, const VectorType& p1, const VectorType& p2)
100  {
101    EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(VectorType, 3)
102    Hyperplane result(p0.size());
103    VectorType v0(p2 - p0), v1(p1 - p0);
104    result.normal() = v0.cross(v1);
105    RealScalar norm = result.normal().norm();
106    if(norm <= v0.norm() * v1.norm() * NumTraits<RealScalar>::epsilon())
107    {
108      Matrix<Scalar,2,3> m; m << v0.transpose(), v1.transpose();
109      JacobiSVD<Matrix<Scalar,2,3> > svd(m, ComputeFullV);
110      result.normal() = svd.matrixV().col(2);
111    }
112    else
113      result.normal() /= norm;
114    result.offset() = -p0.dot(result.normal());
115    return result;
116  }
117
118  /** Constructs a hyperplane passing through the parametrized line \a parametrized.
119    * If the dimension of the ambient space is greater than 2, then there isn't uniqueness,
120    * so an arbitrary choice is made.
121    */
122  // FIXME to be consitent with the rest this could be implemented as a static Through function ??
123  explicit Hyperplane(const ParametrizedLine<Scalar, AmbientDimAtCompileTime>& parametrized)
124  {
125    normal() = parametrized.direction().unitOrthogonal();
126    offset() = -parametrized.origin().dot(normal());
127  }
128
129  ~Hyperplane() {}
130
131  /** \returns the dimension in which the plane holds */
132  inline Index dim() const { return AmbientDimAtCompileTime==Dynamic ? m_coeffs.size()-1 : Index(AmbientDimAtCompileTime); }
133
134  /** normalizes \c *this */
135  void normalize(void)
136  {
137    m_coeffs /= normal().norm();
138  }
139
140  /** \returns the signed distance between the plane \c *this and a point \a p.
141    * \sa absDistance()
142    */
143  inline Scalar signedDistance(const VectorType& p) const { return normal().dot(p) + offset(); }
144
145  /** \returns the absolute distance between the plane \c *this and a point \a p.
146    * \sa signedDistance()
147    */
148  inline Scalar absDistance(const VectorType& p) const { using std::abs; return abs(signedDistance(p)); }
149
150  /** \returns the projection of a point \a p onto the plane \c *this.
151    */
152  inline VectorType projection(const VectorType& p) const { return p - signedDistance(p) * normal(); }
153
154  /** \returns a constant reference to the unit normal vector of the plane, which corresponds
155    * to the linear part of the implicit equation.
156    */
157  inline ConstNormalReturnType normal() const { return ConstNormalReturnType(m_coeffs,0,0,dim(),1); }
158
159  /** \returns a non-constant reference to the unit normal vector of the plane, which corresponds
160    * to the linear part of the implicit equation.
161    */
162  inline NormalReturnType normal() { return NormalReturnType(m_coeffs,0,0,dim(),1); }
163
164  /** \returns the distance to the origin, which is also the "constant term" of the implicit equation
165    * \warning the vector normal is assumed to be normalized.
166    */
167  inline const Scalar& offset() const { return m_coeffs.coeff(dim()); }
168
169  /** \returns a non-constant reference to the distance to the origin, which is also the constant part
170    * of the implicit equation */
171  inline Scalar& offset() { return m_coeffs(dim()); }
172
173  /** \returns a constant reference to the coefficients c_i of the plane equation:
174    * \f$ c_0*x_0 + ... + c_{d-1}*x_{d-1} + c_d = 0 \f$
175    */
176  inline const Coefficients& coeffs() const { return m_coeffs; }
177
178  /** \returns a non-constant reference to the coefficients c_i of the plane equation:
179    * \f$ c_0*x_0 + ... + c_{d-1}*x_{d-1} + c_d = 0 \f$
180    */
181  inline Coefficients& coeffs() { return m_coeffs; }
182
183  /** \returns the intersection of *this with \a other.
184    *
185    * \warning The ambient space must be a plane, i.e. have dimension 2, so that \c *this and \a other are lines.
186    *
187    * \note If \a other is approximately parallel to *this, this method will return any point on *this.
188    */
189  VectorType intersection(const Hyperplane& other) const
190  {
191    using std::abs;
192    EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(VectorType, 2)
193    Scalar det = coeffs().coeff(0) * other.coeffs().coeff(1) - coeffs().coeff(1) * other.coeffs().coeff(0);
194    // since the line equations ax+by=c are normalized with a^2+b^2=1, the following tests
195    // whether the two lines are approximately parallel.
196    if(internal::isMuchSmallerThan(det, Scalar(1)))
197    {   // special case where the two lines are approximately parallel. Pick any point on the first line.
198        if(abs(coeffs().coeff(1))>abs(coeffs().coeff(0)))
199            return VectorType(coeffs().coeff(1), -coeffs().coeff(2)/coeffs().coeff(1)-coeffs().coeff(0));
200        else
201            return VectorType(-coeffs().coeff(2)/coeffs().coeff(0)-coeffs().coeff(1), coeffs().coeff(0));
202    }
203    else
204    {   // general case
205        Scalar invdet = Scalar(1) / det;
206        return VectorType(invdet*(coeffs().coeff(1)*other.coeffs().coeff(2)-other.coeffs().coeff(1)*coeffs().coeff(2)),
207                          invdet*(other.coeffs().coeff(0)*coeffs().coeff(2)-coeffs().coeff(0)*other.coeffs().coeff(2)));
208    }
209  }
210
211  /** Applies the transformation matrix \a mat to \c *this and returns a reference to \c *this.
212    *
213    * \param mat the Dim x Dim transformation matrix
214    * \param traits specifies whether the matrix \a mat represents an #Isometry
215    *               or a more generic #Affine transformation. The default is #Affine.
216    */
217  template<typename XprType>
218  inline Hyperplane& transform(const MatrixBase<XprType>& mat, TransformTraits traits = Affine)
219  {
220    if (traits==Affine)
221      normal() = mat.inverse().transpose() * normal();
222    else if (traits==Isometry)
223      normal() = mat * normal();
224    else
225    {
226      eigen_assert(0 && "invalid traits value in Hyperplane::transform()");
227    }
228    return *this;
229  }
230
231  /** Applies the transformation \a t to \c *this and returns a reference to \c *this.
232    *
233    * \param t the transformation of dimension Dim
234    * \param traits specifies whether the transformation \a t represents an #Isometry
235    *               or a more generic #Affine transformation. The default is #Affine.
236    *               Other kind of transformations are not supported.
237    */
238  template<int TrOptions>
239  inline Hyperplane& transform(const Transform<Scalar,AmbientDimAtCompileTime,Affine,TrOptions>& t,
240                                TransformTraits traits = Affine)
241  {
242    transform(t.linear(), traits);
243    offset() -= normal().dot(t.translation());
244    return *this;
245  }
246
247  /** \returns \c *this with scalar type casted to \a NewScalarType
248    *
249    * Note that if \a NewScalarType is equal to the current scalar type of \c *this
250    * then this function smartly returns a const reference to \c *this.
251    */
252  template<typename NewScalarType>
253  inline typename internal::cast_return_type<Hyperplane,
254           Hyperplane<NewScalarType,AmbientDimAtCompileTime,Options> >::type cast() const
255  {
256    return typename internal::cast_return_type<Hyperplane,
257                    Hyperplane<NewScalarType,AmbientDimAtCompileTime,Options> >::type(*this);
258  }
259
260  /** Copy constructor with scalar type conversion */
261  template<typename OtherScalarType,int OtherOptions>
262  inline explicit Hyperplane(const Hyperplane<OtherScalarType,AmbientDimAtCompileTime,OtherOptions>& other)
263  { m_coeffs = other.coeffs().template cast<Scalar>(); }
264
265  /** \returns \c true if \c *this is approximately equal to \a other, within the precision
266    * determined by \a prec.
267    *
268    * \sa MatrixBase::isApprox() */
269  template<int OtherOptions>
270  bool isApprox(const Hyperplane<Scalar,AmbientDimAtCompileTime,OtherOptions>& other, const typename NumTraits<Scalar>::Real& prec = NumTraits<Scalar>::dummy_precision()) const
271  { return m_coeffs.isApprox(other.m_coeffs, prec); }
272
273protected:
274
275  Coefficients m_coeffs;
276};
277
278} // end namespace Eigen
279
280#endif // EIGEN_HYPERPLANE_H
281