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    result.normal() = (p2 - p0).cross(p1 - p0).normalized();
104    result.offset() = -p0.dot(result.normal());
105    return result;
106  }
107
108  /** Constructs a hyperplane passing through the parametrized line \a parametrized.
109    * If the dimension of the ambient space is greater than 2, then there isn't uniqueness,
110    * so an arbitrary choice is made.
111    */
112  // FIXME to be consitent with the rest this could be implemented as a static Through function ??
113  explicit Hyperplane(const ParametrizedLine<Scalar, AmbientDimAtCompileTime>& parametrized)
114  {
115    normal() = parametrized.direction().unitOrthogonal();
116    offset() = -parametrized.origin().dot(normal());
117  }
118
119  ~Hyperplane() {}
120
121  /** \returns the dimension in which the plane holds */
122  inline Index dim() const { return AmbientDimAtCompileTime==Dynamic ? m_coeffs.size()-1 : Index(AmbientDimAtCompileTime); }
123
124  /** normalizes \c *this */
125  void normalize(void)
126  {
127    m_coeffs /= normal().norm();
128  }
129
130  /** \returns the signed distance between the plane \c *this and a point \a p.
131    * \sa absDistance()
132    */
133  inline Scalar signedDistance(const VectorType& p) const { return normal().dot(p) + offset(); }
134
135  /** \returns the absolute distance between the plane \c *this and a point \a p.
136    * \sa signedDistance()
137    */
138  inline Scalar absDistance(const VectorType& p) const { using std::abs; return abs(signedDistance(p)); }
139
140  /** \returns the projection of a point \a p onto the plane \c *this.
141    */
142  inline VectorType projection(const VectorType& p) const { return p - signedDistance(p) * normal(); }
143
144  /** \returns a constant reference to the unit normal vector of the plane, which corresponds
145    * to the linear part of the implicit equation.
146    */
147  inline ConstNormalReturnType normal() const { return ConstNormalReturnType(m_coeffs,0,0,dim(),1); }
148
149  /** \returns a non-constant reference to the unit normal vector of the plane, which corresponds
150    * to the linear part of the implicit equation.
151    */
152  inline NormalReturnType normal() { return NormalReturnType(m_coeffs,0,0,dim(),1); }
153
154  /** \returns the distance to the origin, which is also the "constant term" of the implicit equation
155    * \warning the vector normal is assumed to be normalized.
156    */
157  inline const Scalar& offset() const { return m_coeffs.coeff(dim()); }
158
159  /** \returns a non-constant reference to the distance to the origin, which is also the constant part
160    * of the implicit equation */
161  inline Scalar& offset() { return m_coeffs(dim()); }
162
163  /** \returns a constant reference to the coefficients c_i of the plane equation:
164    * \f$ c_0*x_0 + ... + c_{d-1}*x_{d-1} + c_d = 0 \f$
165    */
166  inline const Coefficients& coeffs() const { return m_coeffs; }
167
168  /** \returns a non-constant reference to the coefficients c_i of the plane equation:
169    * \f$ c_0*x_0 + ... + c_{d-1}*x_{d-1} + c_d = 0 \f$
170    */
171  inline Coefficients& coeffs() { return m_coeffs; }
172
173  /** \returns the intersection of *this with \a other.
174    *
175    * \warning The ambient space must be a plane, i.e. have dimension 2, so that \c *this and \a other are lines.
176    *
177    * \note If \a other is approximately parallel to *this, this method will return any point on *this.
178    */
179  VectorType intersection(const Hyperplane& other) const
180  {
181    using std::abs;
182    EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(VectorType, 2)
183    Scalar det = coeffs().coeff(0) * other.coeffs().coeff(1) - coeffs().coeff(1) * other.coeffs().coeff(0);
184    // since the line equations ax+by=c are normalized with a^2+b^2=1, the following tests
185    // whether the two lines are approximately parallel.
186    if(internal::isMuchSmallerThan(det, Scalar(1)))
187    {   // special case where the two lines are approximately parallel. Pick any point on the first line.
188        if(abs(coeffs().coeff(1))>abs(coeffs().coeff(0)))
189            return VectorType(coeffs().coeff(1), -coeffs().coeff(2)/coeffs().coeff(1)-coeffs().coeff(0));
190        else
191            return VectorType(-coeffs().coeff(2)/coeffs().coeff(0)-coeffs().coeff(1), coeffs().coeff(0));
192    }
193    else
194    {   // general case
195        Scalar invdet = Scalar(1) / det;
196        return VectorType(invdet*(coeffs().coeff(1)*other.coeffs().coeff(2)-other.coeffs().coeff(1)*coeffs().coeff(2)),
197                          invdet*(other.coeffs().coeff(0)*coeffs().coeff(2)-coeffs().coeff(0)*other.coeffs().coeff(2)));
198    }
199  }
200
201  /** Applies the transformation matrix \a mat to \c *this and returns a reference to \c *this.
202    *
203    * \param mat the Dim x Dim transformation matrix
204    * \param traits specifies whether the matrix \a mat represents an #Isometry
205    *               or a more generic #Affine transformation. The default is #Affine.
206    */
207  template<typename XprType>
208  inline Hyperplane& transform(const MatrixBase<XprType>& mat, TransformTraits traits = Affine)
209  {
210    if (traits==Affine)
211      normal() = mat.inverse().transpose() * normal();
212    else if (traits==Isometry)
213      normal() = mat * normal();
214    else
215    {
216      eigen_assert(0 && "invalid traits value in Hyperplane::transform()");
217    }
218    return *this;
219  }
220
221  /** Applies the transformation \a t to \c *this and returns a reference to \c *this.
222    *
223    * \param t the transformation of dimension Dim
224    * \param traits specifies whether the transformation \a t represents an #Isometry
225    *               or a more generic #Affine transformation. The default is #Affine.
226    *               Other kind of transformations are not supported.
227    */
228  template<int TrOptions>
229  inline Hyperplane& transform(const Transform<Scalar,AmbientDimAtCompileTime,Affine,TrOptions>& t,
230                                TransformTraits traits = Affine)
231  {
232    transform(t.linear(), traits);
233    offset() -= normal().dot(t.translation());
234    return *this;
235  }
236
237  /** \returns \c *this with scalar type casted to \a NewScalarType
238    *
239    * Note that if \a NewScalarType is equal to the current scalar type of \c *this
240    * then this function smartly returns a const reference to \c *this.
241    */
242  template<typename NewScalarType>
243  inline typename internal::cast_return_type<Hyperplane,
244           Hyperplane<NewScalarType,AmbientDimAtCompileTime,Options> >::type cast() const
245  {
246    return typename internal::cast_return_type<Hyperplane,
247                    Hyperplane<NewScalarType,AmbientDimAtCompileTime,Options> >::type(*this);
248  }
249
250  /** Copy constructor with scalar type conversion */
251  template<typename OtherScalarType,int OtherOptions>
252  inline explicit Hyperplane(const Hyperplane<OtherScalarType,AmbientDimAtCompileTime,OtherOptions>& other)
253  { m_coeffs = other.coeffs().template cast<Scalar>(); }
254
255  /** \returns \c true if \c *this is approximately equal to \a other, within the precision
256    * determined by \a prec.
257    *
258    * \sa MatrixBase::isApprox() */
259  template<int OtherOptions>
260  bool isApprox(const Hyperplane<Scalar,AmbientDimAtCompileTime,OtherOptions>& other, const typename NumTraits<Scalar>::Real& prec = NumTraits<Scalar>::dummy_precision()) const
261  { return m_coeffs.isApprox(other.m_coeffs, prec); }
262
263protected:
264
265  Coefficients m_coeffs;
266};
267
268} // end namespace Eigen
269
270#endif // EIGEN_HYPERPLANE_H
271