CGAL 6.2 - Planar Parameterization of Triangulated Surface Meshes

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CGAL::Surface_mesh_parameterization::Fixed_border_parameterizer_3< TriangleMesh_, BorderParameterizer_, SolverTraits_ > Class Template Referenceabstract
Reference Manual » Surface Parameterization Methods

`#include <CGAL/Surface_mesh_parameterization/Fixed_border_parameterizer_3.h>`

Inherited by 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_ >, and CGAL::Surface_mesh_parameterization::Mean_value_coordinates_parameterizer_3< TriangleMesh_, BorderParameterizer_, SolverTraits_ >.

## Definition

template<typename TriangleMesh_, class BorderParameterizer_ = Default, class SolverTraits_ = Default>
class CGAL::Surface_mesh_parameterization::Fixed_border_parameterizer_3< TriangleMesh_, BorderParameterizer_, SolverTraits_ >

The class `Fixed_border_parameterizer_3` is the base class of fixed border parameterization methods (Tutte, Floater, ...).

A one-to-one mapping is guaranteed if the border of the surface is mapped onto a convex polygon.

This class is a pure virtual class and thus cannot be instantiated. Nevertheless, it implements most of the parameterization algorithm `parameterize()`. Subclasses are Strategies [5] that modify the behavior of this algorithm:

- They provide the template parameters `BorderParameterizer_` and `SolverTraits_`.
- They implement `compute_w_ij()` to compute `w_ij`, the `(i,j)`-coefficient of matrix `A` for `j` neighbor vertex of `i`.
  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_> CGAL::Surface_mesh_parameterization::Circular_border_arc_length_parameterizer_3 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: CGAL::Eigen_solver_traits< Eigen::BiCGSTAB<Eigen_sparse_matrix<double>::EigenType, Eigen::IncompleteLUT< double > > > CGAL::Eigen_solver_traits

See also
  `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::Mean_value_coordinates_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.
typedef Solver_traits::Vector | Vector
| Solver vector type.
typedef Solver_traits::Matrix | Matrix
| Solver matrix type.

## Public Member Functions
| 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
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).

## Constructor & Destructor Documentation

## ◆ Fixed_border_parameterizer_3()

template<typename TriangleMesh_ , class BorderParameterizer_ = Default, class SolverTraits_ = Default>

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.

Parameters

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

## Member Function Documentation

## ◆ compute_w_ij()

template<typename TriangleMesh_ , class BorderParameterizer_ = Default, class SolverTraits_ = Default>

virtual NT CGAL::Surface_mesh_parameterization::Fixed_border_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
| protected pure virtual

computes `w_ij`, coefficient of matrix `A` for `j` neighbor vertex of `i`.

Implementation note: Subclasses must at least implement compute_w_ij().

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`

Implemented in 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_ >, and CGAL::Surface_mesh_parameterization::Mean_value_coordinates_parameterizer_3< TriangleMesh_, BorderParameterizer_, SolverTraits_ >.

## ◆ initialize_system_from_mesh_border()

template<typename TriangleMesh_ , class BorderParameterizer_ = Default, class SolverTraits_ = Default>
template<typename VertexUVmap , typename VertexIndexMap >

void CGAL::Surface_mesh_parameterization::Fixed_border_parameterizer_3< TriangleMesh_, BorderParameterizer_, SolverTraits_ >::initialize_system_from_mesh_border | ( | Matrix & | A,
|  | Vector & | Bu,
|  | Vector & | Bv,
|  | const Triangle_mesh & | mesh,
|  | halfedge_descriptor | bhd,
|  | VertexUVmap | uvmap,
|  | VertexIndexMap | vimap
| ) |  | const
| protected

initializes `A`, `Bu` and `Bv` after border parameterization.

Fill the border vertices' lines in both linear systems: "u = constant" and "v = constant".

Template Parameters

  VertexUVmap | must be a model of `ReadWritePropertyMap` with `boost::graph_traits<Triangle_mesh>::vertex_descriptor` as key type and Point_2 (type deduced from `Triangle_mesh` using `Kernel_traits`) as value type.
  VertexIndexMap | must be a model of `ReadablePropertyMap` with `boost::graph_traits<Triangle_mesh>::vertex_descriptor` as key type and a unique integer as value type.

Parameters

  A | the matrix in both linear system
  Bu | the right hand side vector in the linear system of x coordinates
  Bv | the right hand side vector in the linear system of y coordinates
  mesh | a triangulated surface.
  bhd | a halfedge descriptor on the boundary of `mesh`.
  uvmap | an instantiation of the class `VertexUVmap`.
  vimap | an instantiation of the class `VertexIndexMap`.

Precondition
  Vertices must be indexed (`vimap` must be initialized).
  `A`, `Bu`, and `Bv` must be allocated.
  Border vertices must be parameterized.

## ◆ parameterize()

template<typename TriangleMesh_ , class BorderParameterizer_ = Default, class SolverTraits_ = Default>
template<typename VertexUVmap , typename VertexIndexMap , typename VertexParameterizedMap >

Error_code CGAL::Surface_mesh_parameterization::Fixed_border_parameterizer_3< TriangleMesh_, BorderParameterizer_, SolverTraits_ >::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.

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

Parameters

  mesh | a triangulated surface.
  bhd | a halfedge descriptor on the boundary of `mesh`.
  uvmap | an instantiation of the class `VertexUVmap`.
  vimap | an instantiation of the class `VertexIndexMap`.
  vpmap | an instantiation of the class `VertexParameterizedMap`.

Precondition
  `mesh` must be a triangular mesh.
  The mesh border must be mapped onto a convex polygon.
  The vertices must be indexed (`vimap` must be initialized)

## ◆ setup_inner_vertex_relations()

template<typename TriangleMesh_ , class BorderParameterizer_ = Default, class SolverTraits_ = Default>
template<typename VertexIndexMap >

Error_code CGAL::Surface_mesh_parameterization::Fixed_border_parameterizer_3< TriangleMesh_, BorderParameterizer_, SolverTraits_ >::setup_inner_vertex_relations | ( | Matrix & | A,
|  | Vector & | ,
|  | Vector & | ,
|  | const Triangle_mesh & | mesh,
|  | vertex_descriptor | vertex,
|  | VertexIndexMap | vimap
| ) |  | const
| protected

Compute the line i of matrix A for i inner vertex:

- call compute_w_ij() to compute the A coefficient w_ij for each neighbor v_j.
- compute w_ii = - sum of w_ijs.

Precondition
  Vertices must be indexed.
  Vertex i mustn't be already parameterized.
  Line i of A must contain only zeros.