#include <CGAL/Surface_mesh_parameterization/ARAP_parameterizer_3.h>

Definition

template<class TriangleMesh_, class BorderParameterizer_ = Default, class SolverTraits_ = Default>
class CGAL::Surface_mesh_parameterization::ARAP_parameterizer_3< TriangleMesh_, BorderParameterizer_, SolverTraits_ >

The class ARAP_parameterizer_3 implements the Local/Global Approach to Mesh Parameterization [9].

This parameterization allows the user to prioritize angle preservation, shape preservation, or a balance of both. A parameter \(\lambda\) controls whether the priority is given to angle or to shape preservation: when \(\lambda=0\), the parameterization is as-similar-as-possible (ASAP) and is equivalent to the (conforming) LSCM parameterization. As \(\lambda\) grows, the shape preservation becomes more and more important, yielding, when \(\lambda\) goes to infinity, a parameterization that is as-rigid-as-possible (ARAP).

This is a free border parameterization. There is no need to map the border of the surface onto a convex polygon. When \(\lambda=0\), only two pinned vertices are needed to ensure a unique solution. When \(\lambda\) is non-null, the border does not need to be parameterized and a random vertex is pinned.

If flips are present in the initial parameterization, a post-processing step is applied using CGAL::Surface_mesh_parameterization::MVC_post_processor_3<TriangleMesh_, SolverTraits_> to attempt to obtain a valid final embedding.

A one-to-one mapping is not guaranteed.

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: Two_vertices_parameterizer_3<TriangleMesh_> CGAL::Surface_mesh_parameterization::Two_vertices_parameterizer_3 The class Two_vertices_parameterizer_3 parameterizes two extreme vertices of a 3D surface. Definition: Two_vertices_parameterizer_3.h:56
SolverTraits_	must be a model of `SparseLinearAlgebraTraits_d`. Note that the system is not symmetric. 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> > CGAL::Eigen_solver_traits 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: CGAL::Surface_mesh_parameterization::Fixed_border_parameterizer_3<TriangleMesh, BorderParameterizer, SolverTraits>; CGAL::Surface_mesh_parameterization::Iterative_authalic_parameterizer_3<TriangleMesh, BorderParameterizer, SolverTraits>

Public Types
typedef Border_parameterizer_	Border_parameterizer
	Border parameterizer type.

typedef SolverTraits_	Solver_traits
	Solver traits type.

typedef unspecified_type	NT
	Number type, deduced from the internal vertex point map of `Triangle_mesh`

typedef TriangleMesh_	Triangle_mesh
	Triangle mesh type.

typedef boost::graph_traits< Triangle_mesh >::halfedge_descriptor	halfedge_descriptor
	Mesh halfedge type.

Public Member Functions
template<typename FaceRange , typename VertexUVMap >
bool	is_one_to_one_mapping (const Triangle_mesh &mesh, const FaceRange &faces, const VertexUVMap uvmap) const
	returns whether the 3D -> 2D mapping is one-to-one.

template<typename VertexUVMap , typename VertexIndexMap , typename VertexParameterizedMap >
Error_code	parameterize (Triangle_mesh &mesh, halfedge_descriptor bhd, VertexUVMap uvmap, VertexIndexMap vimap, VertexParameterizedMap vpmap)
	computes a mapping from a triangular 3D surface mesh to a piece of the 2D space.

	ARAP_parameterizer_3 (NT lambda)
	Constructor taking only the parameter \(\lambda\).

	ARAP_parameterizer_3 (Border_parameterizer border_param=Border_parameterizer(), Solver_traits sparse_la=Solver_traits(), NT lambda=1000., unsigned int iterations=50, NT tolerance=1e-6)
	Default Constructor.

Constructor & Destructor Documentation

◆ ARAP_parameterizer_3()

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

CGAL::Surface_mesh_parameterization::ARAP_parameterizer_3< TriangleMesh_, BorderParameterizer_, SolverTraits_ >::ARAP_parameterizer_3	(	Border_parameterizer	border_param = `Border_parameterizer()`,
		Solver_traits	sparse_la = `Solver_traits()`,
		NT	lambda = `1000.`,
		unsigned int	iterations = `50`,
		NT	tolerance = `1e-6`
	)

Default Constructor.

Parameters

border_param	Object that maps the surface's border to the 2D space.
sparse_la	Traits object to access a sparse linear system.
lambda	Parameter to give importance to shape or angle preservation.
iterations	Maximal number of iterations in the energy minimization process.
tolerance	Minimal energy difference between two iterations for the minimization process to continue.

Member Function Documentation

◆ parameterize()

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

template<typename VertexUVMap , typename VertexIndexMap , typename VertexParameterizedMap >

Error_code CGAL::Surface_mesh_parameterization::ARAP_parameterizer_3< TriangleMesh_, BorderParameterizer_, SolverTraits_ >::parameterize	(	Triangle_mesh &	mesh,
		halfedge_descriptor	bhd,
		VertexUVMap	uvmap,
		VertexIndexMap	vimap,
		VertexParameterizedMap	vpmap
	)

computes a 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.