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 * \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 */
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 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
275protected:
276
277 Coefficients m_coeffs;
278};
279
280} // end namespace Eigen
281
282#endif // EIGEN_HYPERPLANE_H
283