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.

Some helper functions related to the class Orbifold_Tutte_parameterizer_3 (for example to read and compute paths between cones) can be found here .

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.

Attention

The global function CGAL::Surface_mesh_parameterization::parameterize() cannot be used with this parameterizer. Users should use this class's member function parameterize() instead.

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 SuiteSparse 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

Orbifold Helper Functions

Public Types
typedef SolverTraits_	Solver_traits
	Solver traits type.
typedef boost::graph_traits< SeamMesh >::halfedge_descriptor	halfedge_descriptor
	Mesh halfedge type.

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

Constructor & Destructor Documentation

Orbifold_Tutte_parameterizer_3()

template<typename SeamMesh , typename SolverTraits_ = Default>

CGAL::Surface_mesh_parameterization::Orbifold_Tutte_parameterizer_3< SeamMesh, SolverTraits_ >::Orbifold_Tutte_parameterizer_3	(	const Orbifold_type	orb_type = Square,
		const Weight_type	weight_type = Cotangent
	)

Constructor of the parameterizer.

The arguments allow to select the desired orbifold and weight types.

Member Function Documentation

parameterize()

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,
		const ConeMap &	cmap,
		VertexUVMap	uvmap,
		VertexIndexMap	vimap
	)		const

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

ConeMap	must be a model of AssociativeContainer with key type boost::graph_traits<Seam_mesh>::vertex_descriptor and Cone_type as value type.
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.

Parameters

mesh	a Seam_mesh parameterized by any model of a FaceListGraph and HalfedgeListGraph
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 classification.
uvmap	an instantiation of the class VertexUVmap.
vimap	an instantiation 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 73.1 Invalid (left) and valid (right) seam paths. Seam edges are drawn in teal. Cones are marked in yellow and blue.