CGAL::Surface_mesh_parameterization::Iterative_authalic_parameterizer_3< TriangleMesh_, BorderParameterizer_, SolverTraits_ > Class Template Reference

#include <CGAL/Surface_mesh_parameterization/Iterative_authalic_parameterizer_3.h>

Definition

template<class TriangleMesh_, class BorderParameterizer_ = Default, class SolverTraits_ = Default>
class CGAL::Surface_mesh_parameterization::Iterative_authalic_parameterizer_3< TriangleMesh_, BorderParameterizer_, SolverTraits_ >

The class Iterative_authalic_parameterizer_3 implements the Iterative Parameterization algorithm, as described by Jain et al.

[6].

This parameterization is a fixed border parameterization and is part of the authalic parameterization family, meaning that it aims to minimize area distortion between the input surface mesh and the parameterized output. More precisely, the approach used by this parameterizer is to iteratively redistribute the \( L_2\) stretch - as defined by Sander et al. [11] - over the mesh.

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 border vertices are not removed 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::Discrete_authalic_parameterizer_3<TriangleMesh, BorderParameterizer, SolverTraits>

CGAL::Surface_mesh_parameterization::ARAP_parameterizer_3<TriangleMesh, BorderParameterizer, SolverTraits>

Public Types
typedef Border_parameterizer_	Border_parameterizer
	Border parameterizer type.
typedef SolverTraits_	Solver_traits
	Solver traits type.
typedef TriangleMesh_	Triangle_mesh
	Triangle mesh type.

Public Member Functions
	Iterative_authalic_parameterizer_3 (Border_parameterizer border_parameterizer=Border_parameterizer(), Solver_traits sparse_la=Solver_traits())
	Constructor.
template<typename VertexUVmap , typename VertexIndexMap >
void	initialize_system_from_mesh_border (Matrix &A, Vector &Bu, Vector &Bv, const Triangle_mesh &tmesh, halfedge_descriptor bhd, VertexUVmap uvmap, VertexIndexMap vimap) const
	initializes A, Bu, and Bv after border parameterization.
template<typename VertexUVmap , typename VertexIndexMap , typename VertexParameterizedMap >
Error_code	parameterize (Triangle_mesh &tmesh, halfedge_descriptor bhd, VertexUVmap uvmap, VertexIndexMap vimap, VertexParameterizedMap vpmap, const unsigned int &iterations=15)
	computes a one-to-one mapping from a triangular 3D surface mesh to a piece of the 2D space.

Protected Member Functions
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).

Constructor & Destructor Documentation

Iterative_authalic_parameterizer_3()

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

CGAL::Surface_mesh_parameterization::Iterative_authalic_parameterizer_3< TriangleMesh_, BorderParameterizer_, SolverTraits_ >::Iterative_authalic_parameterizer_3	(	Border_parameterizer	border_parameterizer = Border_parameterizer(),
		Solver_traits	sparse_la = Solver_traits()
	)

Constructor.

Parameters

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

Member Function Documentation

initialize_system_from_mesh_border()

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

template<typename VertexUVmap , typename VertexIndexMap >

void CGAL::Surface_mesh_parameterization::Iterative_authalic_parameterizer_3< TriangleMesh_, BorderParameterizer_, SolverTraits_ >::initialize_system_from_mesh_border	(	Matrix &	A,
		Vector &	Bu,
		Vector &	Bv,
		const Triangle_mesh &	tmesh,
		halfedge_descriptor	bhd,
		VertexUVmap	uvmap,
		VertexIndexMap	vimap
	)		const

initializes A, Bu, and Bv after border parameterization.

Fill the border vertices' lines in both linear systems: "u = constant" and "v = constant".

Template Parameters

VertexUVmap	must be a model of ReadWritePropertyMap with boost::graph_traits<Triangle_mesh>::vertex_descriptor as key type and Point_2 (type deduced from Triangle_mesh using Kernel_traits) as value type.
VertexIndexMap	must be a model of ReadablePropertyMap with boost::graph_traits<Triangle_mesh>::vertex_descriptor as key type and a unique integer as value type.

Parameters

A	the matrix in both linear system
Bu	the right hand side vector in the linear system of x coordinates
Bv	the right hand side vector in the linear system of y coordinates
tmesh	a triangulated surface
bhd	a halfedge descriptor on the boundary of mesh
uvmap	an instantiation of the class VertexUVmap
vimap	an instantiation of the class VertexIndexMap

Precondition

Vertices must be indexed (vimap must be initialized).

A, Bu, and Bv must be allocated.

Border vertices must be parameterized.

parameterize()

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

template<typename VertexUVmap , typename VertexIndexMap , typename VertexParameterizedMap >

Error_code CGAL::Surface_mesh_parameterization::Iterative_authalic_parameterizer_3< TriangleMesh_, BorderParameterizer_, SolverTraits_ >::parameterize	(	Triangle_mesh &	tmesh,
		halfedge_descriptor	bhd,
		VertexUVmap	uvmap,
		VertexIndexMap	vimap,
		VertexParameterizedMap	vpmap,
		const unsigned int &	iterations = 15
	)

computes a one-to-one mapping from a triangular 3D surface mesh to a piece of the 2D space.

The mapping is piecewise linear (linear in each triangle). The result is the (u,v) pair image of each vertex of the 3D surface.

Template Parameters

VertexUVmap	must be a model of ReadWritePropertyMap with boost::graph_traits<Triangle_mesh>::vertex_descriptor as key type and Point_2 (type deduced from Triangle_mesh using Kernel_traits) as value type.
VertexIndexMap	must be a model of ReadablePropertyMap with boost::graph_traits<Triangle_mesh>::vertex_descriptor as key type and a unique integer as value type.
VertexParameterizedMap	must be a model of ReadWritePropertyMap with boost::graph_traits<Triangle_mesh>::vertex_descriptor as key type and a Boolean as value type.

Parameters

tmesh	a triangulated surface
bhd	a halfedge descriptor on the boundary of mesh
uvmap	an instantiation of the class VertexUVmap
vimap	an instantiation of the class VertexIndexMap
vpmap	an instantiation of the class VertexParameterizedMap
iterations	an integer number of iterations to run the parameterization

Precondition

tmesh must be a triangular mesh.

The mesh border must be mapped onto a convex polygon.

The vertices must be indexed (vimap must be initialized).