#include <CGAL/Surface_mesh_parameterization/Discrete_authalic_parameterizer_3.h>

Inherits from

Definition

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, the (i,j)-coefficient of the matrix A for j neighbor vertex of i, based on Discrete Authalic Parameterization algorithm.

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>; CGAL::Surface_mesh_parameterization::ARAP_parameterizer_3<TriangleMesh, BorderParameterizer, SolverTraits>; CGAL::Surface_mesh_parameterization::Iterative_authalic_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.

typedef boost::graph_traits< Triangle_mesh >::vertex_descriptor	vertex_descriptor
	Mesh vertex type.

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
	Border parameterizer type.

typedef Default::Get< SolverTraits_, SolverTraits_ >::type	Solver_traits
	Solver traits type.

typedef TriangleMesh_	Triangle_mesh
	Triangle mesh type.

typedef boost::graph_traits< Triangle_mesh >::vertex_descriptor	vertex_descriptor
	Mesh vertex type.

typedef boost::graph_traits< Triangle_mesh >::halfedge_descriptor	halfedge_descriptor
	Mesh halfedge type.

typedef Solver_traits::Vector	Vector
	Solver vector type.

typedef Solver_traits::Matrix	Matrix
	Solver matrix type.

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 (Triangle_mesh &mesh, halfedge_descriptor bhd, VertexUVmap uvmap, VertexIndexMap vimap, VertexParameterizedMap vpmap)
	computes 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 Triangle_mesh &mesh, vertex_descriptor main_vertex_v_i, Vertex_around_target_circulator< Triangle_mesh > neighbor_vertex_v_j) const
	computes `w_ij`, the (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 Triangle_mesh &mesh, halfedge_descriptor bhd, VertexUVmap uvmap, VertexIndexMap vimap) const
	initializes `A`, `Bu` and `Bv` after border parameterization. More...

Error_code	setup_inner_vertex_relations (Matrix &A, Vector &, Vector &, const Triangle_mesh &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).

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

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 Triangle_mesh &	mesh,
		vertex_descriptor	main_vertex_v_i,
		Vertex_around_target_circulator< Triangle_mesh >	neighbor_vertex_v_j
	)		const

protectedvirtual

computes w_ij, the (i,j), coefficient of matrix A for j neighbor vertex of i.

Parameters

mesh	a triangulated surface.
main_vertex_v_i	the vertex of `mesh` with index `i`
neighbor_vertex_v_j	the vertex of `mesh` with index `j`