\section{Cutting a Mesh \label{sec:Cutting-a-Mesh}}

\subsection{Computing a Cut Graph}

All surface parameterization methods proposed in this package only
deal with meshes which are homeomorphic (topologically equivalent) to discs.
Nevertheless meshes with arbitrary topology and number of connected components car be parameterized, provided that the user specifies a cut graph (an oriented list of vertices), which is the border of a topological disc. If no cut graph is
provided as input, the longest border already
in the input mesh is taken as default border, all other borders being considered as holes. Note that only the inside part (i.e., one connected component) of the given border is parameterized.

% cut graph
\begin{center}
    \label{Surface_mesh_parameterization-fig-cut}
    % Image
    \begin{ccTexOnly}
        \includegraphics[width=0.8\textwidth]{Surface_mesh_parameterization/cut}
    \end{ccTexOnly}
    \begin{ccHtmlOnly}
        <img width="80%" border=0 src="./cut.png"><P>
    \end{ccHtmlOnly}
    % Title
    \begin{figure}[h]
        \caption{Cut Graph}
    \end{figure}
\end{center}

This package does not provide any algorithm to transform an arbitrary mesh
into a topological disk, the user being responsible
for generating such a cut graph. Nevertheless, we provide in
\ccc{polyhedron_ex_parameterization.cpp} 
a simple cutting algorithm for the sake of completeness.


\subsection{Applying a Cut}

The surface parameterization classes in this package only \emph{directly} support
 surfaces which are homeomorphic to disks (models of
\ccc{ParameterizationMesh_3}). This software design simplifies the
implementation of all new parameterization methods.

The \ccc{CGAL::Parameterization_mesh_patch_3<ParameterizationPatchableMesh_3>}
class is responsible for \emph{virtually} cutting
a patch in a \ccc{ParameterizationPatchableMesh_3} mesh.
The resulting patch is a topological
disk (if the cut graph is correct)
and provides a \ccc{ParameterizationMesh_3} interface. It can be used as
parameter of \ccc{CGAL::parameterize()}.

\ccc{ParameterizationPatchableMesh_3} inherits from concept \ccc{ParameterizationMesh_3},
thus is a concept for a 3D surface mesh.
\ccc{ParameterizationPatchableMesh_3} adds the ability to support patches and
virtual seams. \emph{Patches} are a subset of a 3D mesh.
\emph{Virtual seams} behave exactly as if the surface was cut along a certain graph.

The \ccc{ParameterizationMesh_3} interface with the Polyhedron is both a model of
\ccc{ParameterizationMesh_3} and \ccc{ParameterizationPatchableMesh_3}: \\
\ccc{CGAL::Parameterization_polyhedron_adaptor_3<Polyhedron_3_>}

Note that this class is a decorator which adds {\em on the fly}
the necessary fields to unmodified \cgal\ data structures (using STL
maps). For better performances, it is recommended to use \cgal\ data
structures enriched with the proper fields. See \ccc{Polyhedron_ex}
class in \ccc{polyhedron_ex_parameterization.cpp} example.


\subsection{Cutting a Mesh Example}

\ccc{Mesh_cutting_parameterization.cpp} \emph{virtually} cuts a
\ccc{CGAL::Polyhedron_3<Traits>} mesh
to make it a topological disk, then applies the default parameterization:

\ccIncludeExampleCode{Surface_mesh_parameterization/Mesh_cutting_parameterization.cpp}