/// \defgroup PkgAlphaShapes2Ref 2D Alpha Shapes Reference
/// \defgroup PkgAlphaShapes2Concepts Concepts
/// \ingroup PkgAlphaShapes2Ref

/*!
\addtogroup PkgAlphaShapes2Ref

\cgalPkgDescriptionBegin{2D Alpha Shapes,PkgAlphaShapes2}
\cgalPkgPicture{alpha-detail.png}
\cgalPkgSummaryBegin
\cgalPkgAuthors{Tran Kai Frank Da}
\cgalPkgDesc{This package offers  a data structure encoding the whole family of alpha-complexes related to a given 2D Delaunay or regular triangulation. In particular, the data structure allows to retrieve   the alpha-complex for any alpha value, the whole spectrum of critical alpha values and a filtration on the triangulation faces (this filtration is based on the first  alpha value for which each face is included on the alpha-complex).}
\cgalPkgManuals{Chapter_2D_Alpha_Shapes,PkgAlphaShapes2Ref}
\cgalPkgSummaryEnd
\cgalPkgShortInfoBegin
\cgalPkgSince{2.1}
\cgalPkgDependsOn{\ref PkgTriangulation2}
\cgalPkgBib{cgal:d-as2}
\cgalPkgLicense{\ref licensesGPL  "GPL"}
\cgalPkgDemo{2D Alpha Shapes,alpha_shapes_2.zip}
\cgalPkgShortInfoEnd
\cgalPkgDescriptionEnd

This chapter presents a framework for alpha shapes. The description is based on
the articles \cgalCite{em-tdas-94}, \cgalCite{e-was-92}. Alpha shapes are
the generalization of the convex hull of a point set. Let \f$ S\f$ be a finite set of
points in \f$ \mathbb{R}^d\f$, \f$ d = 2,3\f$ and \f$ \alpha\f$ a parameter with \f$ 0 \leq \alpha \leq
\infty\f$. For \f$ \alpha = \infty\f$, the \f$ \alpha\f$-shape is the convex hull of \f$ S\f$. As 
\f$ \alpha\f$ decreases, the \f$ \alpha\f$-shape shrinks and develops cavities, as soon as 
a sphere of radius \f$ \sqrt{\alpha}\f$ can be put inside.
Finally, for \f$ \alpha = 0\f$, the \f$ \alpha\f$-shape is the set \f$ S\f$ itself.

We distinguish two versions of alpha shapes, one is based on the Delaunay
triangulation and the other on its generalization, the regular triangulation,
replacing the natural distance by the power to weighted points. The metric used
determines an underlying triangulation of the alpha shape and thus, the version
computed. 
The <I>basic alpha shape</I> (cf.  \ref I1_SectClassicAS2D) is associated with the Delaunay triangulation
(cf. Section \ref Section_2D_Triangulations_Delaunay "Delaunay Triangulations"). 
The <I>weighted alpha shape</I> (cf.  \ref I1_SectWeightedAS2D ) is associated with the regular triangulation
(cf. Section \ref Section_2D_Triangulations_Regular "Regular Triangulations").

There is a close connection between alpha shapes and the underlying
triangulations. More precisely, the \f$ \alpha\f$-complex of \f$ S\f$ is a
subcomplex of this triangulation of \f$ S\f$, containing the \f$ \alpha\f$-exposed
\f$ k\f$-simplices, \f$ 0 \leq k \leq d\f$. A simplex is \f$ \alpha\f$-exposed, if there is an
open disk (resp. ball) of radius \f$ \sqrt{\alpha}\f$ through the vertices of the
simplex that does not contain any other point of \f$ S\f$, for the metric used in
the computation of the underlying triangulation.  The corresponding
\f$ \alpha\f$-shape is defined as the underlying interior space of the
\f$ \alpha\f$-complex.

In general, an \f$ \alpha\f$-complex is a non-connected and non-pure polytope,
meaning that a \f$ k\f$-simplex, with \f$ 0 \leq k \leq d-1\f$, is not necessarily
adjacent to a \f$ (k+1)\f$-simplex.

The \f$ \alpha\f$-shapes of \f$ S\f$ form a discrete family, even though they
are defined for all real numbers \f$ \alpha\f$ with \f$ 0 \leq \alpha
\leq \infty\f$. Thus, we can represent the entire family of \f$ \alpha\f$-shapes
of \f$ S\f$ by the underlying triangulation of \f$ S\f$. In this representation
each \f$ k\f$-simplex of the underlying triangulation is associated with an
interval that specifies for which values of \f$ \alpha\f$ the \f$ k\f$-simplex
belongs to the \f$ \alpha\f$-shape. Relying on this result, the family of
\f$ \alpha\f$-shapes can be computed efficiently and relatively
easily. Furthermore, we can select an appropriate \f$ \alpha\f$-shape from a
finite number of different \f$ \alpha\f$-shapes and corresponding
\f$ \alpha\f$-values.

\cgalClassifedRefPages

\cgalCRPSection{Concepts}
- `AlphaShapeTraits_2`
- `AlphaShapeFace_2`
- `AlphaShapeVertex_2`

\cgalCRPSection{Classes}
- `CGAL::Alpha_shape_2<Dt>`
- `CGAL::Weighted_alpha_shape_euclidean_traits_2<K>`
- `CGAL::Alpha_shape_vertex_base_2<AlphaShapeTraits_2>`
- `CGAL::Alpha_shape_face_base_2<AlphaShapeTraits_2, TriangulationFaceBase_2>`

*/

