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% | Reference manual page: Discrete_authalic_parameterizer_3.tex
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% | 21.09.2005   Laurent Saboret, Pierre Alliez
% | Package: Parameterization
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\RCSdef{\RCSDiscreteauthalicparameterizerRev}{$Id$}
\RCSdefDate{\RCSDiscreteauthalicparameterizerDate}{$Date$}
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\ccDefinition

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The class Discrete\_authalic\_parameterizer\_3 implements the Discrete Authalic Parameterization algorithm \cite{cgal:dma-ipsm-02}. This method is sometimes called {\em DAP} or just {\em Authalic parameterization}.

DAP is a weak area-preserving parameterization. It is a compromise between area-preserving and angle-preserving.

1 to 1 mapping is guaranteed if surface's border is mapped onto a convex polygon.

As all parameterization algorithms of the package, this class is usually called via the global function parameterize().

This class is a Strategy \cite{cgal:ghjv-dpero-95} called by the main parameterization algorithm Fixed\_border\_parameterizer\_3::parameterize(). Discrete\_authalic\_parameterizer\_3:\begin{itemize}
\item provides default BorderParameterizer\_3 and SparseLinearAlgebraTraits\_d template parameters that make sense.\item implements compute\_w\_ij() to compute w\_ij = (i,j) coefficient of matrix A for j neighbor vertex of i based on Discrete Authalic Parameterization algorithm.\end{itemize}

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\ccInclude{CGAL/Discrete_authalic_parameterizer_3.h}


\ccIsModel

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Model of the ParameterizerTraits\_3 concept.

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\ccParameters

The full template declaration is:

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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 Discrete\_authalic\_parameterizer\_3;

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\ccTypes

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\ccCreation
\ccCreationVariable{param}  %% variable name used by \ccMethod below

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\ccConstructor{Discrete_authalic_parameterizer_3 (Border_param border_param = Border_param(), Sparse_LA sparse_la = Sparse_LA());}
{
Constructor.
}
\ccGlue
\begin{description}
\item[Parameters: ]
\begin{description}
\item[border\_param]Object that maps the surface's border to 2D space. \item[sparse\_la]Traits object to access a sparse linear system. \end{description}
\end{description}
\ccGlue

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\ccOperations

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\ccMethod{virtual NT compute_w_ij (const 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.
}
\ccGlue

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\ccSeeAlso

\ccRefIdfierPage{CGAL::Parameterizer_traits_3}  \\
\ccRefIdfierPage{CGAL::Fixed_border_parameterizer_3}  \\
\ccRefIdfierPage{CGAL::Barycentric_mapping_parameterizer_3}  \\
\ccRefIdfierPage{CGAL::Discrete_conformal_map_parameterizer_3}  \\
\ccRefIdfierPage{CGAL::LSCM_parameterizer_3}  \\
\ccRefIdfierPage{CGAL::Mean_value_coordinates_parameterizer_3}  \\


\ccExample

See \ccc{Authalic_parameterization.C} example.


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