CGAL::Surface_mesh_parameterization::Orbifold_Tutte_parameterizer_3< SeamMesh, SolverTraits_ > Class Template Reference

#include <CGAL/Surface_mesh_parameterization/Orbifold_Tutte_parameterizer_3.h>

Definition

template<typename SeamMesh, typename SolverTraits_ = Default>
class CGAL::Surface_mesh_parameterization::Orbifold_Tutte_parameterizer_3< SeamMesh, SolverTraits_ >

The class Orbifold_Tutte_parameterizer_3 implements Orbifold Tutte Planar Embeddings

[1].

This is a borderless parameterization. A one-to-one mapping is guaranteed.

The main function of the class Orbifold_Tutte_parameterizer_3 is parameterize(), to which the user provides a Seam_mesh with marked edges (the seams) and a set of vertices of the mesh (the cones). The choice of cones influences the resulting parameterization, but not the choice of the seam path between these cones.

The example orbifold.cpp shows how to select cones on the input mesh and automatically construct the seams and the cones on the Seam_mesh.

Is Model Of:

Parameterizer_3

Template Parameters

SeamMesh

must be a Seam_mesh, with underlying mesh any model of FaceListGraph and HalfedgeListGraph.

SolverTraits_

must be a model of SparseLinearAlgebraTraits_d. 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::SparseLU<Eigen_sparse_matrix<double>::EigenType> > Moreover, if SparseSuite solvers are available, which is greatly preferable for speed, then the default parameter is: CGAL::Eigen_solver_traits< Eigen::UmfPackLU<Eigen_sparse_matrix<double>::EigenType> >

See Also

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

CGAL::Surface_mesh_parameterization::Barycentric_mapping_parameterizer_3<TriangleMesh, BorderParameterizer, SolverTraits>

CGAL::Surface_mesh_parameterization::Discrete_authalic_parameterizer_3<TriangleMesh, BorderParameterizer, SolverTraits>

CGAL::Surface_mesh_parameterization::Discrete_conformal_map_parameterizer_3<TriangleMesh, BorderParameterizer, SolverTraits>

CGAL::Surface_mesh_parameterization::LSCM_parameterizer_3<TriangleMesh, BorderParameterizer>

CGAL::Surface_mesh_parameterization::Mean_value_coordinates_parameterizer_3<TriangleMesh, BorderParameterizer, SolverTraits>

Public Types
typedef SolverTraits_	Solver_traits

Public Member Functions
template<typename ConeMap , typename VertexIndexMap , typename VertexUVMap >
Error_code	parameterize (SeamMesh &mesh, halfedge_descriptor bhd, ConeMap cmap, VertexUVMap uvmap, VertexIndexMap vimap) const
	Compute a one-to-one mapping from a triangular 3D surface mesh to a piece of the 2D space. More...
	Orbifold_Tutte_parameterizer_3 (const Orbifold_type orb_type=Square, const Weight_type weight_type=Cotangent)
	Constructor.

Member Function Documentation

template<typename SeamMesh , typename SolverTraits_ = Default>

template<typename ConeMap , typename VertexIndexMap , typename VertexUVMap >

Error_code CGAL::Surface_mesh_parameterization::Orbifold_Tutte_parameterizer_3< SeamMesh, SolverTraits_ >::parameterize	(	SeamMesh &	mesh,
		halfedge_descriptor	bhd,
		ConeMap	cmap,
		VertexUVMap	uvmap,
		VertexIndexMap	vimap
	)		const

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

Parameters

mesh	a Seam_mesh parameterized by any model of a FaceGraph
bhd	a halfedge on the border of the seam mesh
cmap	a mapping of the vertex_descriptors of mesh that are cones to their respective Cone_type.
uvmap	an instanciation of the class VertexUVmap.
vimap	an instanciation of the class VertexIndexMap.

Precondition

mesh must be a triangular mesh.

The underlying mesh of mesh is a topological ball.

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

The cones are vertices of mesh and their number is adapted to the orbifold type (4 for types I, II or III and 6 for type IV).

The seam edges form a set of segments that contains the different cones, and starts and ends at two different cones.

The seam edges form a set of segments that is homotopic to a line. Specifically, the paths between cones must not self intersect or intersect other paths (see Figure below).

Figure 64.1 Bad (left) and good (right) seam paths. The seam edges are shown in dark red. Cones are marked in yellow and blue.