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CGAL 4.13.2 - Planar Parameterization of Triangulated Surface Meshes
List of all members | Public Types | Public Member Functions | Protected Member Functions
CGAL::Surface_mesh_parameterization::Discrete_authalic_parameterizer_3< TriangleMesh_, BorderParameterizer_, SolverTraits_ > Class Template Reference
Triangulated Surface Mesh Parameterization Reference » Surface Parameterization Methods

#include <CGAL/Surface_mesh_parameterization/Discrete_authalic_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<class TriangleMesh_, class BorderParameterizer_ = Default, class SolverTraits_ = Default>
class CGAL::Surface_mesh_parameterization::Discrete_authalic_parameterizer_3< TriangleMesh_, BorderParameterizer_, SolverTraits_ >

The class Discrete_authalic_parameterizer_3 implements the Discrete Authalic Parameterization algorithm.

This method is sometimes called DAP or just Authalic parameterization [2].

DAP is a weak area-preserving parameterization. It is a compromise between area-preserving and angle-preserving.

A one-to-one mapping is guaranteed if the surface's border is mapped onto 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 Discrete Authalic Parameterization algorithm.
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_ Solver_traits
 
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

 Discrete_authalic_parameterizer_3 (Border_parameterizer border_param=Border_parameterizer(), Solver_traits sparse_la=Solver_traits())
 Constructor. 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 &mesh, vertex_descriptor main_vertex_v_i, vertex_around_target_circulator neighbor_vertex_v_j) 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

◆ Discrete_authalic_parameterizer_3()

template<class TriangleMesh_ , class BorderParameterizer_ = Default, class SolverTraits_ = Default>
CGAL::Surface_mesh_parameterization::Discrete_authalic_parameterizer_3< TriangleMesh_, BorderParameterizer_, SolverTraits_ >::Discrete_authalic_parameterizer_3 ( Border_parameterizer  border_param = Border_parameterizer(),
Solver_traits  sparse_la = Solver_traits() 
)

Constructor.

Parameters
border_paramObject that maps the surface's border to 2D space.
sparse_laTraits object to access a sparse linear system.

Member Function Documentation

◆ compute_w_ij()

template<class TriangleMesh_ , class BorderParameterizer_ = Default, class SolverTraits_ = Default>
virtual NT CGAL::Surface_mesh_parameterization::Discrete_authalic_parameterizer_3< TriangleMesh_, BorderParameterizer_, SolverTraits_ >::compute_w_ij ( const TriangleMesh &  mesh,
vertex_descriptor  main_vertex_v_i,
vertex_around_target_circulator  neighbor_vertex_v_j 
) const
protectedvirtual

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

Parameters
mesha triangulated surface.
main_vertex_v_ithe vertex of mesh with index i
neighbor_vertex_v_jthe vertex of mesh with index j

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 >.