CGAL::Barycentric_mapping_parameterizer_3< ParameterizationMesh_3, BorderParameterizer_3, SparseLinearAlgebraTraits_d > Class Template Reference

#include <CGAL/Barycentric_mapping_parameterizer_3.h>

Inherits from

CGAL::Fixed_border_parameterizer_3< ParameterizationMesh_3, BorderParameterizer_3, SparseLinearAlgebraTraits_d >.

Definition

template<class ParameterizationMesh_3, class BorderParameterizer_3 = Circular_border_arc_length_parameterizer_3<ParameterizationMesh_3>, class SparseLinearAlgebraTraits_d = Eigen_solver_traits<Eigen::BiCGSTAB<Eigen_sparse_matrix<double>::EigenType, Eigen::IncompleteLUT< double > > >>
class CGAL::Barycentric_mapping_parameterizer_3< ParameterizationMesh_3, BorderParameterizer_3, SparseLinearAlgebraTraits_d >

The class Barycentric_mapping_parameterizer_3 implements Tutte Barycentric Mapping algorithm

[9]. This algorithm is also called Tutte Uniform Weights by other authors.

One-to-one mapping is guaranteed if the surface's border is mapped to a convex polygon.

This class is used by the main parameterization algorithm Fixed_border_parameterizer_3::parameterize().

• It provides default BorderParameterizer_3 and SparseLinearAlgebraTraits_d template parameters that make sense.
• It implements compute_w_ij() to compute w_ij = (i,j) coefficient of matrix A for j neighbor vertex of i based on Tutte Barycentric Mapping method.
• It implements an optimized version of is_one_to_one_mapping().

Is Model Of:

ParameterizerTraits_3

Template Parameters

ParameterizationMesh_3	3D surface mesh.
BorderParameterizer_3	Strategy to parameterize the surface border.
SparseLinearAlgebraTraits_d	Traits class to solve a sparse linear system. Note: the system is not symmetric because Fixed_border_parameterizer_3 does not remove (yet) border vertices from the system.

See Also

CGAL::Parameterizer_traits_3<ParameterizationMesh_3>

CGAL::Fixed_border_parameterizer_3<ParameterizationMesh_3, BorderParameterizer_3, SparseLinearAlgebraTraits_d>

CGAL::Discrete_authalic_parameterizer_3<ParameterizationMesh_3, BorderParameterizer_3, SparseLinearAlgebraTraits_d>

CGAL::Discrete_conformal_map_parameterizer_3<ParameterizationMesh_3, BorderParameterizer_3, SparseLinearAlgebraTraits_d>

CGAL::LSCM_parameterizer_3<ParameterizationMesh_3, BorderParameterizer_3>

CGAL::Mean_value_coordinates_parameterizer_3<ParameterizationMesh_3, BorderParameterizer_3, SparseLinearAlgebraTraits_d>

Examples:

Surface_mesh_parameterization/polyhedron_ex_parameterization.cpp.

Public Member Functions
	Barycentric_mapping_parameterizer_3 (Border_param border_param=Border_param(), Sparse_LA sparse_la=Sparse_LA())
	Constructor. More...
Public Member Functions inherited from CGAL::Fixed_border_parameterizer_3< ParameterizationMesh_3, BorderParameterizer_3, SparseLinearAlgebraTraits_d >
	Fixed_border_parameterizer_3 (Border_param border_param=Border_param(), Sparse_LA sparse_la=Sparse_LA())
	Constructor. More...
virtual Error_code	parameterize (Adaptor &mesh)
	Compute a one-to-one mapping from a triangular 3D surface mesh to a piece of the 2D space. More...
Public Member Functions inherited from CGAL::Parameterizer_traits_3< ParameterizationMesh_3 >
virtual	~Parameterizer_traits_3 ()
	Destructor of base class should be virtual.

Protected Member Functions
virtual NT	compute_w_ij (const Adaptor &, Vertex_const_handle, Vertex_around_vertex_const_circulator)
	Compute w_ij = (i,j) coefficient of matrix A for j neighbor vertex of i. More...
virtual bool	is_one_to_one_mapping (const Adaptor &, const Matrix &, const Vector &, const Vector &)
	Check if 3D -> 2D mapping is one-to-one. More...
Protected Member Functions inherited from CGAL::Fixed_border_parameterizer_3< ParameterizationMesh_3, BorderParameterizer_3, SparseLinearAlgebraTraits_d >
virtual Error_code	check_parameterize_preconditions (Adaptor &mesh)
	Check parameterize() preconditions: More...
void	initialize_system_from_mesh_border (Matrix &A, Vector &Bu, Vector &Bv, const Adaptor &mesh)
	Initialize A, Bu and Bv after border parameterization. More...
virtual Error_code	setup_inner_vertex_relations (Matrix &A, Vector &Bu, Vector &Bv, const Adaptor &mesh, Vertex_const_handle vertex)
	Compute the line i of matrix A for i inner vertex: More...
void	set_mesh_uv_from_system (Adaptor &mesh, const Vector &Xu, const Vector &Xv)
	Copy Xu and Xv coordinates into the (u,v) pair of each surface vertex.
virtual Error_code	check_parameterize_postconditions (const Adaptor &mesh, const Matrix &A, const Vector &Bu, const Vector &Bv)
	Check parameterize() postconditions: More...
Border_param &	get_border_parameterizer ()
	Get the object that maps the surface's border onto a 2D space.
Sparse_LA &	get_linear_algebra_traits ()
	Get the sparse linear algebra (traits object to access the linear system).

Additional Inherited Members
Public Types inherited from CGAL::Fixed_border_parameterizer_3< ParameterizationMesh_3, BorderParameterizer_3, SparseLinearAlgebraTraits_d >
typedef BorderParameterizer_3	Border_param
	Export BorderParameterizer_3 template parameter.
typedef SparseLinearAlgebraTraits_d	Sparse_LA
	Export SparseLinearAlgebraTraits_d template parameter.
Static Public Member Functions inherited from CGAL::Parameterizer_traits_3< ParameterizationMesh_3 >
static const char *	get_error_message (int error_code)
	Get message corresponding to an error code. More...
Protected Types inherited from CGAL::Parameterizer_traits_3< ParameterizationMesh_3 >
typedef Adaptor::NT	NT
typedef Adaptor::Point_2	Point_2
typedef Adaptor::Point_3	Point_3
typedef Adaptor::Vector_2	Vector_2
typedef Adaptor::Vector_3	Vector_3
typedef Adaptor::Facet	Facet
typedef Adaptor::Facet_handle	Facet_handle
typedef Adaptor::Facet_const_handle	Facet_const_handle
typedef Adaptor::Facet_iterator	Facet_iterator
typedef Adaptor::Facet_const_iterator	Facet_const_iterator
typedef Adaptor::Vertex	Vertex
typedef Adaptor::Vertex_handle	Vertex_handle
typedef Adaptor::Vertex_const_handle	Vertex_const_handle
typedef Adaptor::Vertex_iterator	Vertex_iterator
typedef Adaptor::Vertex_const_iterator	Vertex_const_iterator
typedef Adaptor::Border_vertex_iterator	Border_vertex_iterator
typedef Adaptor::Border_vertex_const_iterator	Border_vertex_const_iterator
typedef Adaptor::Vertex_around_facet_circulator	Vertex_around_facet_circulator
typedef Adaptor::Vertex_around_facet_const_circulator	Vertex_around_facet_const_circulator
typedef Adaptor::Vertex_around_vertex_circulator	Vertex_around_vertex_circulator
typedef Adaptor::Vertex_around_vertex_const_circulator	Vertex_around_vertex_const_circulator

Constructor & Destructor Documentation

template<class ParameterizationMesh_3 , class BorderParameterizer_3 = Circular_border_arc_length_parameterizer_3<ParameterizationMesh_3>, class SparseLinearAlgebraTraits_d = Eigen_solver_traits<Eigen::BiCGSTAB<Eigen_sparse_matrix<double>::EigenType, Eigen::IncompleteLUT< double > > >>

CGAL::Barycentric_mapping_parameterizer_3< ParameterizationMesh_3, BorderParameterizer_3, SparseLinearAlgebraTraits_d >::Barycentric_mapping_parameterizer_3	(	Border_param	border_param = Border_param(),
		Sparse_LA	sparse_la = Sparse_LA()
	)

Constructor.

Parameters

border_param	object that maps the surface's border to 2D space.
sparse_la	Traits object to access a sparse linear system.

Member Function Documentation

template<class ParameterizationMesh_3 , class BorderParameterizer_3 = Circular_border_arc_length_parameterizer_3<ParameterizationMesh_3>, class SparseLinearAlgebraTraits_d = Eigen_solver_traits<Eigen::BiCGSTAB<Eigen_sparse_matrix<double>::EigenType, Eigen::IncompleteLUT< double > > >>

virtual NT CGAL::Barycentric_mapping_parameterizer_3< ParameterizationMesh_3, BorderParameterizer_3, SparseLinearAlgebraTraits_d >::compute_w_ij ( const Adaptor &  , Vertex_const_handle  , Vertex_around_vertex_const_circulator    )

protectedvirtual

Compute w_ij = (i,j) coefficient of matrix A for j neighbor vertex of i.

Tutte Barycentric Mapping algorithm is the most simple one: w_ij = 1 for j neighbor vertex of i.

Implements CGAL::Fixed_border_parameterizer_3< ParameterizationMesh_3, BorderParameterizer_3, SparseLinearAlgebraTraits_d >.

template<class ParameterizationMesh_3 , class BorderParameterizer_3 = Circular_border_arc_length_parameterizer_3<ParameterizationMesh_3>, class SparseLinearAlgebraTraits_d = Eigen_solver_traits<Eigen::BiCGSTAB<Eigen_sparse_matrix<double>::EigenType, Eigen::IncompleteLUT< double > > >>

virtual bool CGAL::Barycentric_mapping_parameterizer_3< ParameterizationMesh_3, BorderParameterizer_3, SparseLinearAlgebraTraits_d >::is_one_to_one_mapping ( const Adaptor &  , const Matrix &  , const Vector &  , const Vector &    )

protectedvirtual

Check if 3D -> 2D mapping is one-to-one.

Theorem: one-to-one mapping is guaranteed if all w_ij coefficients are > 0 (for j vertex neighbor of i) and if the surface border is mapped onto a 2D convex polygon. All w_ij coefficients = 1 (for j vertex neighbor of i), thus mapping is guaranteed if the surface border is mapped onto a 2D convex polygon.

Reimplemented from CGAL::Fixed_border_parameterizer_3< ParameterizationMesh_3, BorderParameterizer_3, SparseLinearAlgebraTraits_d >.