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CGAL 6.0 - Planar Parameterization of Triangulated Surface Meshes
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#include <CGAL/Surface_mesh_parameterization/Barycentric_mapping_parameterizer_3.h>
The class Barycentric_mapping_parameterizer_3 implements Tutte Barycentric Mapping algorithm.
This algorithm is also called Tutte Uniform Weights by other authors [12].
A one-to-one mapping is guaranteed if the surface's border is mapped to a convex polygon.
This class is a strategy called by the main parameterization algorithm Fixed_border_parameterizer_3::parameterize() and it:
BorderParameterizer_ and SolverTraits_.w_ij, the (i,j)-coefficient of the matrix A for j neighbor vertex of i, based on Tutte Barycentric Mapping method.Parameterizer_3 | 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: This class parameterizes the border of a 3D surface onto a circle, with an arc-length parameterizatio... Definition: Circular_border_parameterizer_3.h:244 |
| 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: Eigen::BiCGSTAB<Eigen_sparse_matrix<double>::EigenType,
Eigen::IncompleteLUT< double > > >
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CGAL::Surface_mesh_parameterization::Fixed_border_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. | |
| typedef boost::graph_traits< Triangle_mesh >::halfedge_descriptor | halfedge_descriptor |
| Mesh halfedge type. | |
Public Types inherited from CGAL::Surface_mesh_parameterization::Fixed_border_parameterizer_3< TriangleMesh_, BorderParameterizer_, SolverTraits_ > | |
| 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. | |
| 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 | |
| Barycentric_mapping_parameterizer_3 (Border_parameterizer border_param=Border_parameterizer(), Solver_traits sparse_la=Solver_traits()) | |
| Constructor. | |
| template<typename VertexUVMap > | |
| bool | is_one_to_one_mapping (const Triangle_mesh &mesh, halfedge_descriptor bhd, const VertexUVMap uvmap) const |
| returns whether the 3D -> 2D mapping is one-to-one. | |
Public Member Functions inherited from CGAL::Surface_mesh_parameterization::Fixed_border_parameterizer_3< TriangleMesh_, BorderParameterizer_, SolverTraits_ > | |
| Fixed_border_parameterizer_3 (Border_parameterizer border_param=Border_parameterizer(), Solver_traits sparse_la=Solver_traits()) | |
| Constructor. | |
| virtual | ~Fixed_border_parameterizer_3 () |
| Destructor of base class should be virtual. | |
| template<typename VertexUVmap , typename VertexIndexMap , typename VertexParameterizedMap > | |
| 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. | |
Protected Member Functions | |
| virtual NT | compute_w_ij (const Triangle_mesh &, vertex_descriptor, Vertex_around_target_circulator< Triangle_mesh >) const |
computes w_ij, the coefficient of matrix A for j neighbor vertex of i. | |
Protected Member Functions inherited from CGAL::Surface_mesh_parameterization::Fixed_border_parameterizer_3< TriangleMesh_, BorderParameterizer_, SolverTraits_ > | |
| template<typename VertexUVmap , typename VertexIndexMap > | |
| 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. | |
| 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 =0 |
computes w_ij, coefficient of matrix A for j neighbor vertex of i. | |
| template<typename VertexIndexMap > | |
| 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: | |
| 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). | |
| CGAL::Surface_mesh_parameterization::Barycentric_mapping_parameterizer_3< TriangleMesh_, BorderParameterizer_, SolverTraits_ >::Barycentric_mapping_parameterizer_3 | ( | Border_parameterizer | border_param = Border_parameterizer(), |
| Solver_traits | sparse_la = Solver_traits() |
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| ) |
Constructor.
| border_param | Object that maps the surface's border to 2D space. |
| sparse_la | Traits object to access a sparse linear system. |
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protectedvirtual |
computes w_ij, the coefficient of matrix A for j neighbor vertex of i.
In the Tutte Barycentric Mapping algorithm, we have w_ij = 1, for j neighbor vertex of i.
| bool CGAL::Surface_mesh_parameterization::Barycentric_mapping_parameterizer_3< TriangleMesh_, BorderParameterizer_, SolverTraits_ >::is_one_to_one_mapping | ( | const Triangle_mesh & | mesh, |
| halfedge_descriptor | bhd, | ||
| const VertexUVMap | uvmap | ||
| ) | const |
returns whether the 3D -> 2D mapping is one-to-one.
Theorem: A one-to-one mapping is guaranteed if all w_ij coefficients are > 0 (for j vertex neighbor of i) and if the surface border is mapped onto a 2D convex polygon. Here, all w_ij coefficients are equal to 1 (for j vertex neighbor of i), thus a valid embedding is guaranteed if the surface border is mapped onto a 2D convex polygon.