Mean_value_coordinates_parameterizer_3<ParameterizationMesh_3, BorderParameterizer_3, SparseLinearAlgebraTraits

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CGAL::Mean_value_coordinates_parameterizer_3<ParameterizationMesh_3, BorderParameterizer_3, SparseLinearAlgebraTraits_d>

Definition

The class Mean_value_coordinates_parameterizer_3 implements Floater Mean Value Coordinates parameterization [Flo03a]. This method is sometimes called simply Floater parameterization.

This is a conformal parameterization, i.e. it attempts to preserve angles.

One-to-one mapping is guaranteed if the surface's border is mapped to a convex polygon.

This class is a Strategy [GHJV95] called by the main parameterization algorithm Fixed_border_parameterizer_3::parameterize(). It provides default BorderParameterizer_3 and SparseLinearAlgebraTraits_d template parameters that make sense. It implements compute_w_ij() to compute w_ij = (i, j) coefficient of matrix A for j neighbor vertex of i based on Floater Mean Value Coordinates parameterization. It implements an optimized version of is_one_to_one_mapping().

#include <CGAL/Mean_value_coordinates_parameterizer_3.h>

Inherits From

Fixed_border_parameterizer_3<ParameterizationMesh_3, BorderParameterizer_3, SparseLinearAlgebraTraits_d>

Is Model for the Concepts

Model of the ParameterizerTraits_3 concept.

Design Pattern

Mean_value_coordinates_parameterizer_3 class is a Strategy [GHJV95]: it implements a strategy of surface parameterization for models of ParameterizationMesh_3.

Parameters

The full template declaration is:

template<class ParameterizationMesh_3, class BorderParameterizer_3 = Circular_border_arc_length_parameterizer_3<ParameterizationMesh_3>, class SparseLinearAlgebraTraits_d = OpenNL::DefaultLinearSolverTraits<typename ParameterizationMesh_3::NT>>
class Mean_value_coordinates_parameterizer_3;

Creation

Mean_value_coordinates_parameterizer_3<ParameterizationMesh_3, BorderParameterizer_3, SparseLinearAlgebraTraits_d> param ( Border_param border_param = Border_param(),
Sparse_LA sparse_la = Sparse_LA());

Constructor.

Parameters:

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

Operations

virtual NT

param.compute_w_ij ( Adaptor mesh,
Vertex_const_handle main_vertex_v_i,
Vertex_around_vertex_const_circulator neighbor_vertex_v_j)

Compute w_ij = (i, j) coefficient of matrix A for j neighbor vertex of i.

virtual bool

param.is_one_to_one_mapping ( Adaptor, Matrix, Vector, Vector)

Check if 3D -> 2D mapping is one-to-one.
Theorem: 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. Floater formula above implies that w_ij > 0 (for j vertex neighbor of i), thus mapping is guaranteed if the surface border is mapped onto a 2D convex polygon.

Example

See Simple_parameterization.cpp example.

Next: ParameterizationMesh_3