// This file is part of Eigen, a lightweight C++ template library // for linear algebra. // // Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr> // Copyright (C) 2008 Benoit Jacob <jacob.benoit.1@gmail.com> // // This Source Code Form is subject to the terms of the Mozilla // Public License v. 2.0. If a copy of the MPL was not distributed // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
/** \geometry_module \ingroup Geometry_Module * * \class Hyperplane * * \brief A hyperplane * * A hyperplane is an affine subspace of dimension n-1 in a space of dimension n. * For example, a hyperplane in a plane is a line; a hyperplane in 3-space is a plane. * * \tparam _Scalar the scalar type, i.e., the type of the coefficients * \tparam _AmbientDim the dimension of the ambient space, can be a compile time value or Dynamic. * Notice that the dimension of the hyperplane is _AmbientDim-1. * * This class represents an hyperplane as the zero set of the implicit equation * \f$ n \cdot x + d = 0 \f$ where \f$ n \f$ is a unit normal vector of the plane (linear part) * and \f$ d \f$ is the distance (offset) to the origin.
*/ template <typename _Scalar, int _AmbientDim, int _Options> class Hyperplane
{ public:
EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(_Scalar,_AmbientDim==Dynamic ? Dynamic : _AmbientDim+1) enum {
AmbientDimAtCompileTime = _AmbientDim,
Options = _Options
}; typedef _Scalar Scalar; typedeftypename NumTraits<Scalar>::Real RealScalar; typedef Eigen::Index Index; ///< \deprecated since Eigen 3.3 typedef Matrix<Scalar,AmbientDimAtCompileTime,1> VectorType; typedef Matrix<Scalar,Index(AmbientDimAtCompileTime)==Dynamic
? Dynamic
: Index(AmbientDimAtCompileTime)+1,1,Options> Coefficients; typedef Block<Coefficients,AmbientDimAtCompileTime,1> NormalReturnType; typedefconst Block<const Coefficients,AmbientDimAtCompileTime,1> ConstNormalReturnType;
/** Default constructor without initialization */
EIGEN_DEVICE_FUNC inline Hyperplane() {}
/** Constructs a dynamic-size hyperplane with \a _dim the dimension
* of the ambient space */
EIGEN_DEVICE_FUNC inlineexplicit Hyperplane(Index _dim) : m_coeffs(_dim+1) {}
/** Construct a plane from its normal \a n and a point \a e onto the plane. * \warning the vector normal is assumed to be normalized.
*/
EIGEN_DEVICE_FUNC inline Hyperplane(const VectorType& n, const VectorType& e)
: m_coeffs(n.size()+1)
{
normal() = n;
offset() = -n.dot(e);
}
/** Constructs a plane from its normal \a n and distance to the origin \a d * such that the algebraic equation of the plane is \f$ n \cdot x + d = 0 \f$. * \warning the vector normal is assumed to be normalized.
*/
EIGEN_DEVICE_FUNC inline Hyperplane(const VectorType& n, const Scalar& d)
: m_coeffs(n.size()+1)
{
normal() = n;
offset() = d;
}
/** Constructs a hyperplane passing through the two points. If the dimension of the ambient space * is greater than 2, then there isn't uniqueness, so an arbitrary choice is made.
*/
EIGEN_DEVICE_FUNC staticinline Hyperplane Through(const VectorType& p0, const VectorType& p1)
{
Hyperplane result(p0.size());
result.normal() = (p1 - p0).unitOrthogonal();
result.offset() = -p0.dot(result.normal()); return result;
}
/** Constructs a hyperplane passing through the three points. The dimension of the ambient space * is required to be exactly 3.
*/
EIGEN_DEVICE_FUNC staticinline Hyperplane Through(const VectorType& p0, const VectorType& p1, const VectorType& p2)
{
EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(VectorType, 3)
Hyperplane result(p0.size());
VectorType v0(p2 - p0), v1(p1 - p0);
result.normal() = v0.cross(v1);
RealScalar norm = result.normal().norm(); if(norm <= v0.norm() * v1.norm() * NumTraits<RealScalar>::epsilon())
{
Matrix<Scalar,2,3> m; m << v0.transpose(), v1.transpose();
JacobiSVD<Matrix<Scalar,2,3> > svd(m, ComputeFullV);
result.normal() = svd.matrixV().col(2);
} else
result.normal() /= norm;
result.offset() = -p0.dot(result.normal()); return result;
}
/** Constructs a hyperplane passing through the parametrized line \a parametrized. * If the dimension of the ambient space is greater than 2, then there isn't uniqueness, * so an arbitrary choice is made.
*/ // FIXME to be consistent with the rest this could be implemented as a static Through function ??
EIGEN_DEVICE_FUNC explicit Hyperplane(const ParametrizedLine<Scalar, AmbientDimAtCompileTime>& parametrized)
{
normal() = parametrized.direction().unitOrthogonal();
offset() = -parametrized.origin().dot(normal());
}
EIGEN_DEVICE_FUNC ~Hyperplane() {}
/** \returns the dimension in which the plane holds */
EIGEN_DEVICE_FUNC inline Index dim() const { return AmbientDimAtCompileTime==Dynamic ? m_coeffs.size()-1 : Index(AmbientDimAtCompileTime); }
/** \returns the signed distance between the plane \c *this and a point \a p. * \sa absDistance()
*/
EIGEN_DEVICE_FUNC inline Scalar signedDistance(const VectorType& p) const { return normal().dot(p) + offset(); }
/** \returns the absolute distance between the plane \c *this and a point \a p. * \sa signedDistance()
*/
EIGEN_DEVICE_FUNC inline Scalar absDistance(const VectorType& p) const { return numext::abs(signedDistance(p)); }
/** \returns the projection of a point \a p onto the plane \c *this.
*/
EIGEN_DEVICE_FUNC inline VectorType projection(const VectorType& p) const { return p - signedDistance(p) * normal(); }
/** \returns a constant reference to the unit normal vector of the plane, which corresponds * to the linear part of the implicit equation.
*/
EIGEN_DEVICE_FUNC inline ConstNormalReturnType normal() const { return ConstNormalReturnType(m_coeffs,0,0,dim(),1); }
/** \returns a non-constant reference to the unit normal vector of the plane, which corresponds * to the linear part of the implicit equation.
*/
EIGEN_DEVICE_FUNC inline NormalReturnType normal() { return NormalReturnType(m_coeffs,0,0,dim(),1); }
/** \returns the distance to the origin, which is also the "constant term" of the implicit equation * \warning the vector normal is assumed to be normalized.
*/
EIGEN_DEVICE_FUNC inlineconst Scalar& offset() const { return m_coeffs.coeff(dim()); }
/** \returns a non-constant reference to the distance to the origin, which is also the constant part
* of the implicit equation */
EIGEN_DEVICE_FUNC inline Scalar& offset() { return m_coeffs(dim()); }
/** \returns a constant reference to the coefficients c_i of the plane equation: * \f$ c_0*x_0 + ... + c_{d-1}*x_{d-1} + c_d = 0 \f$
*/
EIGEN_DEVICE_FUNC inlineconst Coefficients& coeffs() const { return m_coeffs; }
/** \returns a non-constant reference to the coefficients c_i of the plane equation: * \f$ c_0*x_0 + ... + c_{d-1}*x_{d-1} + c_d = 0 \f$
*/
EIGEN_DEVICE_FUNC inline Coefficients& coeffs() { return m_coeffs; }
/** \returns the intersection of *this with \a other. * * \warning The ambient space must be a plane, i.e. have dimension 2, so that \c *this and \a other are lines. * * \note If \a other is approximately parallel to *this, this method will return any point on *this.
*/
EIGEN_DEVICE_FUNC VectorType intersection(const Hyperplane& other) const
{
EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(VectorType, 2)
Scalar det = coeffs().coeff(0) * other.coeffs().coeff(1) - coeffs().coeff(1) * other.coeffs().coeff(0); // since the line equations ax+by=c are normalized with a^2+b^2=1, the following tests // whether the two lines are approximately parallel. if(internal::isMuchSmallerThan(det, Scalar(1)))
{ // special case where the two lines are approximately parallel. Pick any point on the first line. if(numext::abs(coeffs().coeff(1))>numext::abs(coeffs().coeff(0))) return VectorType(coeffs().coeff(1), -coeffs().coeff(2)/coeffs().coeff(1)-coeffs().coeff(0)); else return VectorType(-coeffs().coeff(2)/coeffs().coeff(0)-coeffs().coeff(1), coeffs().coeff(0));
} else
{ // general case
Scalar invdet = Scalar(1) / det; return VectorType(invdet*(coeffs().coeff(1)*other.coeffs().coeff(2)-other.coeffs().coeff(1)*coeffs().coeff(2)),
invdet*(other.coeffs().coeff(0)*coeffs().coeff(2)-coeffs().coeff(0)*other.coeffs().coeff(2)));
}
}
/** Applies the transformation matrix \a mat to \c *this and returns a reference to \c *this. * * \param mat the Dim x Dim transformation matrix * \param traits specifies whether the matrix \a mat represents an #Isometry * or a more generic #Affine transformation. The default is #Affine.
*/ template<typename XprType>
EIGEN_DEVICE_FUNC inline Hyperplane& transform(const MatrixBase<XprType>& mat, TransformTraits traits = Affine)
{ if (traits==Affine)
{
normal() = mat.inverse().transpose() * normal();
m_coeffs /= normal().norm();
} elseif (traits==Isometry)
normal() = mat * normal(); else
{
eigen_assert(0 && "invalid traits value in Hyperplane::transform()");
} return *this;
}
/** Applies the transformation \a t to \c *this and returns a reference to \c *this. * * \param t the transformation of dimension Dim * \param traits specifies whether the transformation \a t represents an #Isometry * or a more generic #Affine transformation. The default is #Affine. * Other kind of transformations are not supported.
*/ template<int TrOptions>
EIGEN_DEVICE_FUNC inline Hyperplane& transform(const Transform<Scalar,AmbientDimAtCompileTime,Affine,TrOptions>& t,
TransformTraits traits = Affine)
{
transform(t.linear(), traits);
offset() -= normal().dot(t.translation()); return *this;
}
/** \returns \c *this with scalar type casted to \a NewScalarType * * Note that if \a NewScalarType is equal to the current scalar type of \c *this * then this function smartly returns a const reference to \c *this.
*/ template<typename NewScalarType>
EIGEN_DEVICE_FUNC inlinetypename internal::cast_return_type<Hyperplane,
Hyperplane<NewScalarType,AmbientDimAtCompileTime,Options> >::type cast() const
{ returntypename internal::cast_return_type<Hyperplane,
Hyperplane<NewScalarType,AmbientDimAtCompileTime,Options> >::type(*this);
}
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