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 
14 namespace 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   * \tparam _Scalar the scalar type, i.e., the type of the coefficients
26   * \tparam _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   */
33 template <typename _Scalar, int _AmbientDim, int _Options>
34 class Hyperplane
35 {
36 public:
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 Eigen::Index Index; ///< \deprecated since Eigen 3.3
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   EIGEN_DEVICE_FUNC inline Hyperplane() {}
54 
55   template<int OtherOptions>
56   EIGEN_DEVICE_FUNC 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   EIGEN_DEVICE_FUNC 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   EIGEN_DEVICE_FUNC 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   EIGEN_DEVICE_FUNC 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   EIGEN_DEVICE_FUNC 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   EIGEN_DEVICE_FUNC 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   EIGEN_DEVICE_FUNC explicit Hyperplane(const ParametrizedLine<Scalar, AmbientDimAtCompileTime>& parametrized)
124   {
125     normal() = parametrized.direction().unitOrthogonal();
126     offset() = -parametrized.origin().dot(normal());
127   }
128 
129   EIGEN_DEVICE_FUNC ~Hyperplane() {}
130 
131   /** \returns the dimension in which the plane holds */
132   EIGEN_DEVICE_FUNC inline Index dim() const { return AmbientDimAtCompileTime==Dynamic ? m_coeffs.size()-1 : Index(AmbientDimAtCompileTime); }
133 
134   /** normalizes \c *this */
135   EIGEN_DEVICE_FUNC 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   EIGEN_DEVICE_FUNC 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   EIGEN_DEVICE_FUNC inline Scalar absDistance(const VectorType& p) const { return numext::abs(signedDistance(p)); }
149 
150   /** \returns the projection of a point \a p onto the plane \c *this.
151     */
152   EIGEN_DEVICE_FUNC 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   EIGEN_DEVICE_FUNC 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   EIGEN_DEVICE_FUNC 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   EIGEN_DEVICE_FUNC 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   EIGEN_DEVICE_FUNC 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   EIGEN_DEVICE_FUNC 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   EIGEN_DEVICE_FUNC 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   EIGEN_DEVICE_FUNC VectorType intersection(const Hyperplane& other) const
190   {
191     EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(VectorType, 2)
192     Scalar det = coeffs().coeff(0) * other.coeffs().coeff(1) - coeffs().coeff(1) * other.coeffs().coeff(0);
193     // since the line equations ax+by=c are normalized with a^2+b^2=1, the following tests
194     // whether the two lines are approximately parallel.
195     if(internal::isMuchSmallerThan(det, Scalar(1)))
196     {   // special case where the two lines are approximately parallel. Pick any point on the first line.
197         if(numext::abs(coeffs().coeff(1))>numext::abs(coeffs().coeff(0)))
198             return VectorType(coeffs().coeff(1), -coeffs().coeff(2)/coeffs().coeff(1)-coeffs().coeff(0));
199         else
200             return VectorType(-coeffs().coeff(2)/coeffs().coeff(0)-coeffs().coeff(1), coeffs().coeff(0));
201     }
202     else
203     {   // general case
204         Scalar invdet = Scalar(1) / det;
205         return VectorType(invdet*(coeffs().coeff(1)*other.coeffs().coeff(2)-other.coeffs().coeff(1)*coeffs().coeff(2)),
206                           invdet*(other.coeffs().coeff(0)*coeffs().coeff(2)-coeffs().coeff(0)*other.coeffs().coeff(2)));
207     }
208   }
209 
210   /** Applies the transformation matrix \a mat to \c *this and returns a reference to \c *this.
211     *
212     * \param mat the Dim x Dim transformation matrix
213     * \param traits specifies whether the matrix \a mat represents an #Isometry
214     *               or a more generic #Affine transformation. The default is #Affine.
215     */
216   template<typename XprType>
217   EIGEN_DEVICE_FUNC inline Hyperplane& transform(const MatrixBase<XprType>& mat, TransformTraits traits = Affine)
218   {
219     if (traits==Affine)
220     {
221       normal() = mat.inverse().transpose() * normal();
222       m_coeffs /= normal().norm();
223     }
224     else if (traits==Isometry)
225       normal() = mat * normal();
226     else
227     {
228       eigen_assert(0 && "invalid traits value in Hyperplane::transform()");
229     }
230     return *this;
231   }
232 
233   /** Applies the transformation \a t to \c *this and returns a reference to \c *this.
234     *
235     * \param t the transformation of dimension Dim
236     * \param traits specifies whether the transformation \a t represents an #Isometry
237     *               or a more generic #Affine transformation. The default is #Affine.
238     *               Other kind of transformations are not supported.
239     */
240   template<int TrOptions>
241   EIGEN_DEVICE_FUNC inline Hyperplane& transform(const Transform<Scalar,AmbientDimAtCompileTime,Affine,TrOptions>& t,
242                                 TransformTraits traits = Affine)
243   {
244     transform(t.linear(), traits);
245     offset() -= normal().dot(t.translation());
246     return *this;
247   }
248 
249   /** \returns \c *this with scalar type casted to \a NewScalarType
250     *
251     * Note that if \a NewScalarType is equal to the current scalar type of \c *this
252     * then this function smartly returns a const reference to \c *this.
253     */
254   template<typename NewScalarType>
255   EIGEN_DEVICE_FUNC inline typename internal::cast_return_type<Hyperplane,
256            Hyperplane<NewScalarType,AmbientDimAtCompileTime,Options> >::type cast() const
257   {
258     return typename internal::cast_return_type<Hyperplane,
259                     Hyperplane<NewScalarType,AmbientDimAtCompileTime,Options> >::type(*this);
260   }
261 
262   /** Copy constructor with scalar type conversion */
263   template<typename OtherScalarType,int OtherOptions>
264   EIGEN_DEVICE_FUNC inline explicit Hyperplane(const Hyperplane<OtherScalarType,AmbientDimAtCompileTime,OtherOptions>& other)
265   { m_coeffs = other.coeffs().template cast<Scalar>(); }
266 
267   /** \returns \c true if \c *this is approximately equal to \a other, within the precision
268     * determined by \a prec.
269     *
270     * \sa MatrixBase::isApprox() */
271   template<int OtherOptions>
272   EIGEN_DEVICE_FUNC bool isApprox(const Hyperplane<Scalar,AmbientDimAtCompileTime,OtherOptions>& other, const typename NumTraits<Scalar>::Real& prec = NumTraits<Scalar>::dummy_precision()) const
273   { return m_coeffs.isApprox(other.m_coeffs, prec); }
274 
275 protected:
276 
277   Coefficients m_coeffs;
278 };
279 
280 } // end namespace Eigen
281 
282 #endif // EIGEN_HYPERPLANE_H
283