\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)

CGAL 4.8 - Planar Parameterization of Triangulated Surface Meshes

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Mesh
Triangulated Surface Mesh Parameterization Reference

##

The general definition of input meshes handled directly by `CGAL::parameterize()` is:

- Model of `ParameterizationMesh_3`.
- Triangulated.
- 2-manifold.
- Oriented.
- Homeomorphic to a disc (may have holes).

This package provides a model of the `ParameterizationMesh_3` concept to access `CGAL::Polyhedron_3<Traits>`:

`CGAL::Parameterization_polyhedron_adaptor_3<Polyhedron_3_>`

The meshes supported indirectly by the package can be of any genus and have any number of connected components. If it is not a topological disc, the input mesh has to come with a description of a cutting path (an oriented list of vertices) which is the border of a topological disc. If no cutting path is given as input, we assume that the surface border is the longest border already in the input mesh (the other borders will be considered as holes).

The `CGAL::Parameterization_mesh_patch_3<ParameterizationPatchableMesh_3>` class is responsible for virtually cutting a patch in a `ParameterizationPatchableMesh_3` mesh. The resulting patch is a topological disk (if the input cutting path is correct) and provides a `ParameterizationMesh_3` interface. It can be used as parameter of `CGAL::parameterize()`.

Note that this way the user is responsible for cutting a closed mesh of arbitrary genus (even a topological disc with an intricate seam cut), as long as this condition is fulfilled.

The package provides an interface with `CGAL::Polyhedron_3<Traits>`:

`CGAL::Parameterization_polyhedron_adaptor_3<Polyhedron_3_>`

## Classes
class | CGAL::Parameterization_polyhedron_adaptor_3< Polyhedron_3_ >
| `Parameterization_polyhedron_adaptor_3` is an adaptor class to access to a Polyhedron 3D mesh using the `ParameterizationPatchableMesh_3` interface. More...
class | CGAL::Parameterization_mesh_patch_3< ParameterizationPatchableMesh_3 >
| Parameterization_mesh_patch_3 is a Decorator class to virtually cut a patch in a ParameterizationPatchableMesh_3 3D surface. More...