CGAL::Surface_mesh_parameterization::Barycentric_mapping_parameterizer_3< TriangleMesh_, BorderParameterizer_, SolverTraits_ > Class Template Reference

#include <CGAL/Surface_mesh_parameterization/Barycentric_mapping_parameterizer_3.h>

Inherits from

CGAL::Surface_mesh_parameterization::Fixed_border_parameterizer_3< TriangleMesh_, Default::Get< BorderParameterizer_, Circular_border_arc_length_parameterizer_3< TriangleMesh_ > >::type, Default::Get< SolverTraits_, SolverTraits_ >::type >.

Definition

template<typename TriangleMesh_, typename BorderParameterizer_ = Default, typename SolverTraits_ = Default>
class CGAL::Surface_mesh_parameterization::Barycentric_mapping_parameterizer_3< TriangleMesh_, BorderParameterizer_, SolverTraits_ >

The class Barycentric_mapping_parameterizer_3 implements Tutte Barycentric Mapping algorithm.

This algorithm is also called Tutte Uniform Weights by other authors [10].

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

This class is a strategy called by the main parameterization algorithm Fixed_border_parameterizer_3::parameterize() and it:

• provides the template parameters BorderParameterizer_ and SolverTraits_.
• implements compute_w_ij() to compute w_ij = (i,j), coefficient of the matrix A for j neighbor vertex of i, based on Tutte Barycentric Mapping method.

Is Model Of:

Parameterizer_3

Template Parameters

TriangleMesh_	must be a model of FaceGraph.
BorderParameterizer_	is a Strategy to parameterize the surface border and must be a model of Parameterizer_3. Default: Circular_border_arc_length_parameterizer_3<TriangleMesh_>
SolverTraits_	must be a model of SparseLinearAlgebraTraits_d. Note that the system is not symmetric because Fixed_border_parameterizer_3 does not remove border vertices from the system. Default: If Eigen 3.1 (or greater) is available and CGAL_EIGEN3_ENABLED is defined, then an overload of Eigen_solver_traits is provided as default parameter: CGAL::Eigen_solver_traits< Eigen::BiCGSTAB<Eigen_sparse_matrix<double>::EigenType, Eigen::IncompleteLUT< double > > >

See also

CGAL::Surface_mesh_parameterization::Fixed_border_parameterizer_3<TriangleMesh, BorderParameterizer, SolverTraits>

Public Types
typedef Border_parameterizer_	Border_parameterizer
typedef SolverTraits_	SolverTraits
typedef TriangleMesh_	TriangleMesh
Public Types inherited from CGAL::Surface_mesh_parameterization::Fixed_border_parameterizer_3< TriangleMesh_, Default::Get< BorderParameterizer_, Circular_border_arc_length_parameterizer_3< TriangleMesh_ > >::type, Default::Get< SolverTraits_, SolverTraits_ >::type >
typedef Border_parameterizer_	Border_parameterizer
typedef Default::Get< SolverTraits_, SolverTraits_ >::type	Solver_traits
typedef TriangleMesh_	TriangleMesh

Public Member Functions
	Barycentric_mapping_parameterizer_3 (Border_parameterizer border_param=Border_parameterizer(), Solver_traits sparse_la=Solver_traits())
	Constructor. More...
template<typename VertexUVMap , typename Faces_Container >
bool	is_one_to_one_mapping (const TriangleMesh &mesh, halfedge_descriptor bhd, const VertexUVMap uvmap) const
	Check if the 3D -> 2D mapping is one-to-one. More...
Public Member Functions inherited from CGAL::Surface_mesh_parameterization::Fixed_border_parameterizer_3< TriangleMesh_, Default::Get< BorderParameterizer_, Circular_border_arc_length_parameterizer_3< TriangleMesh_ > >::type, Default::Get< SolverTraits_, SolverTraits_ >::type >
	Fixed_border_parameterizer_3 (Border_parameterizer border_param=Border_parameterizer(), Solver_traits sparse_la=Solver_traits())
	Constructor. More...
virtual	~Fixed_border_parameterizer_3 ()
	Destructor of base class should be virtual.
Error_code	parameterize (TriangleMesh &mesh, halfedge_descriptor bhd, VertexUVmap uvmap, VertexIndexMap vimap, VertexParameterizedMap vpmap)
	Compute a one-to-one mapping from a triangular 3D surface mesh to a piece of the 2D space. More...

Protected Member Functions
virtual NT	compute_w_ij (const TriangleMesh &, vertex_descriptor, vertex_around_target_circulator) const
	Compute w_ij = (i,j), coefficient of matrix A for j neighbor vertex of i. More...
Protected Member Functions inherited from CGAL::Surface_mesh_parameterization::Fixed_border_parameterizer_3< TriangleMesh_, Default::Get< BorderParameterizer_, Circular_border_arc_length_parameterizer_3< TriangleMesh_ > >::type, Default::Get< SolverTraits_, SolverTraits_ >::type >
void	initialize_system_from_mesh_border (Matrix &A, Vector &Bu, Vector &Bv, const TriangleMesh &mesh, halfedge_descriptor bhd, VertexUVmap uvmap, VertexIndexMap vimap) const
	Initialize A, Bu and Bv after border parameterization. More...
Error_code	setup_inner_vertex_relations (Matrix &A, Vector &, Vector &, const TriangleMesh &mesh, vertex_descriptor vertex, VertexIndexMap vimap) const
	Compute the line i of matrix A for i inner vertex: More...
Border_parameterizer &	get_border_parameterizer ()
	Get the object that maps the surface's border onto a 2D space.
Solver_traits &	get_linear_algebra_traits ()
	Get the sparse linear algebra (traits object to access the linear system).

Additional Inherited Members
Protected Types inherited from CGAL::Surface_mesh_parameterization::Fixed_border_parameterizer_3< TriangleMesh_, Default::Get< BorderParameterizer_, Circular_border_arc_length_parameterizer_3< TriangleMesh_ > >::type, Default::Get< SolverTraits_, SolverTraits_ >::type >
typedef internal::Kernel_traits< TriangleMesh >::Kernel	Kernel
typedef internal::Kernel_traits< TriangleMesh >::PPM	PPM
typedef Kernel::FT	NT
typedef Kernel::Point_2	Point_2
typedef Kernel::Point_3	Point_3
typedef Kernel::Vector_3	Vector_3
typedef Solver_traits::Vector	Vector
typedef Solver_traits::Matrix	Matrix

Constructor & Destructor Documentation

Barycentric_mapping_parameterizer_3()

template<typename TriangleMesh_ , typename BorderParameterizer_ = Default, typename SolverTraits_ = Default>

CGAL::Surface_mesh_parameterization::Barycentric_mapping_parameterizer_3< TriangleMesh_, BorderParameterizer_, SolverTraits_ >::Barycentric_mapping_parameterizer_3	(	Border_parameterizer	border_param = Border_parameterizer(),
		Solver_traits	sparse_la = Solver_traits()
	)

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

compute_w_ij()

template<typename TriangleMesh_ , typename BorderParameterizer_ = Default, typename SolverTraits_ = Default>

virtual NT CGAL::Surface_mesh_parameterization::Barycentric_mapping_parameterizer_3< TriangleMesh_, BorderParameterizer_, SolverTraits_ >::compute_w_ij ( const TriangleMesh &  , vertex_descriptor  , vertex_around_target_circulator    ) const

protectedvirtual

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

In the Tutte Barycentric Mapping algorithm, we have w_ij = 1, for j neighbor vertex of i.

Implements CGAL::Surface_mesh_parameterization::Fixed_border_parameterizer_3< TriangleMesh_, Default::Get< BorderParameterizer_, Circular_border_arc_length_parameterizer_3< TriangleMesh_ > >::type, Default::Get< SolverTraits_, SolverTraits_ >::type >.

is_one_to_one_mapping()

template<typename TriangleMesh_ , typename BorderParameterizer_ = Default, typename SolverTraits_ = Default>

template<typename VertexUVMap , typename Faces_Container >

bool CGAL::Surface_mesh_parameterization::Barycentric_mapping_parameterizer_3< TriangleMesh_, BorderParameterizer_, SolverTraits_ >::is_one_to_one_mapping	(	const TriangleMesh &	mesh,
		halfedge_descriptor	bhd,
		const VertexUVMap	uvmap
	)		const

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

Theorem: A 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. Here, all w_ij coefficients = 1 (for j vertex neighbor of i), thus a valid embedding is guaranteed if the surface border is mapped onto a 2D convex polygon.