CGAL::LSCM_parameterizer_3<ParameterizationMesh_3, BorderParameterizer_3, SparseLinearAlgebraTraits_d>

Definition

The class LSCM_parameterizer_3 implements the Least Squares Conformal Maps (LSCM) parameterization [LPRM02].

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

This is a free border parameterization. No need to map the surface's border onto a convex polygon (only two pinned vertices are needed to ensure a unique solution), but one-to-one mapping is NOT guaranteed.

#include <CGAL/LSCM_parameterizer_3.h>

Is Model for the Concepts

Model of the ParameterizerTraits_3 concept.

Design Pattern

LSCM_parameterizer_3<ParameterizationMesh_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$$<
class ParameterizationMesh_3,
class BorderParameterizer_3 = Two_vertices_parameterizer_3<ParameterizationMesh_3>,
class SparseLinearAlgebraTraits_d = OpenNL::SymmetricLinearSolverTraits<typename ParameterizationMesh_3::NT>$$>
class LSCM_parameterizer_3;

Types

Creation

Parameters:

border_param

Object that maps the surface's border to 2D space

sparse_la

Traits object to access a sparse linear system

Operations

Parameterizer_traits_3<Adaptor>::Error_code
	param.parameterize ( Adaptor& mesh)
		Compute a one-to-one mapping from a triangular 3D surface 'mesh' to a piece of the 2D space. The mapping is linear by pieces (linear in each triangle). The result is the (u,v) pair image of each vertex of the 3D surface. Preconditions: 'mesh' must be a surface with one connected component. 'mesh' must be a triangular mesh.
Parameterizer_traits_3<Adaptor>::Error_code
	param.check_parameterize_preconditions ( Adaptor& mesh)
		Check parameterize() preconditions: 'mesh' must be a surface with one connected component. 'mesh' must be a triangular mesh.
void	param.initialize_system_from_mesh_border ( LeastSquaresSolver& solver, Adaptor mesh)
		Initialize A$$ X = B linear system after (at least two) border vertices are parameterized. Preconditions: vertices must be indexed. X and B must be allocated and empty. (at least two) border vertices must be parameterized.
void	param.project_triangle ( Point_3 p0, Point_3 p1, Point_3 p2, Point_2& z0, Point_2& z1, Point_2& z2)
		Utility for setup_triangle_relations(): Computes the coordinates of the vertices of a triangle in a local 2D orthonormal basis of the triangle's plane.
Parameterizer_traits_3<Adaptor>::Error_code
	param.setup_triangle_relations ( LeastSquaresSolver& solver, Adaptor mesh, Facet_const_handle facet)
		Create two lines in the linear system per triangle (one for u, one for v). Preconditions: vertices must be indexed.
void	param.set_mesh_uv_from_system ( Adaptor& mesh, LeastSquaresSolver solver)
		Copy X coordinates into the (u,v) pair of each vertex.
Parameterizer_traits_3<Adaptor>::Error_code
	param.check_parameterize_postconditions ( Adaptor mesh, LeastSquaresSolver solver)
		Check parameterize() postconditions: 3D -$$> 2D mapping is one-to-one.
bool	param.is_one_to_one_mapping ( Adaptor mesh, LeastSquaresSolver solver)
		Check if 3D -$$> 2D mapping is one-to-one.
Border_param&	param.get_border_parameterizer ()	Get the object that maps the surface's border onto a 2D space.
Sparse_LA&	param.get_linear_algebra_traits ()
		Get the sparse linear algebra (traits object to access the linear system).

See Also

CGAL::Parameterizer_traits_3<ParameterizationMesh_3>
CGAL::Fixed_border_parameterizer_3<ParameterizationMesh_3, BorderParameterizer_3, SparseLinearAlgebraTraits_d>
CGAL::Barycentric_mapping_parameterizer_3<ParameterizationMesh_3, BorderParameterizer_3, SparseLinearAlgebraTraits_d>
CGAL::Discrete_authalic_parameterizer_3<ParameterizationMesh_3, BorderParameterizer_3, SparseLinearAlgebraTraits_d>
CGAL::Discrete_conformal_map_parameterizer_3<ParameterizationMesh_3, BorderParameterizer_3, SparseLinearAlgebraTraits_d>
CGAL::Mean_value_coordinates_parameterizer_3<ParameterizationMesh_3, BorderParameterizer_3, SparseLinearAlgebraTraits_d>