)
in case of weighted alpha shapes.
Alpha shapes definition is based on an underlying
triangulation that may be a Delaunay triangulation
in case of basic alpha shapes or a regular triangulation
(cf. )
in case of weighted alpha shapes.
Let us consider the basic case with a Delaunay triangulation.
We first define the alpha complex of the set of points S.
The alpha complex is a subcomplex
of the Delaunay triangulation.
For a given value of 
, the alpha complex includes
all the simplices in the Delaunay triangulation which have
an empty circumsphere with squared radius equal or smaller than .
Here ``empty'' means that the open sphere
do not include any points of S.
The alpha shape is then simply the domain covered by the simplices
of the alpha complex (see [EM94]).
In general, an alpha complex is a non-connected and non-pure complex.
This means in particular that the alpha complex may have
singular faces. For 0 
k d-1,
a k-simplex of the alpha complex is said to be
singular if it is not a facet of a (k+1)-simplex of the complex
CGAL provides two versions of the alpha shapes. In the general mode,
the alpha shapes correspond strictly to the above definition.
The regularized mode provides a regularized version of the alpha shapes
corresponding to the domain covered by a regularized version
of the alpha complex where singular faces are removed.
The alpha shapes of a set of points
S form a discrete family, even though they
are defined for all real numbers .
The entire family of alpha shapes can be represented
through the underlying triangulation of S. In this representation
each k-simplex of the underlying triangulation is associated with an
interval that specifies for which values of the k-simplex
belongs to the alpha complex. Relying on this fact, the family of
alpha shapes can be computed efficiently and relatively
easily. Furthermore, we can select the optimal value
of to get an alpha shape including all data points
and having less than a given number of connected components.
The definition is analog in the case of weigthed alpha shapes.
The input set is now a set of weighted points (which can be regarded
as spheres) and the underlying triangulation
is the regular triangulation of this set.
Two spheres, or two weighted points , with centers C1, C2
and radii r1, r2 are said to be orthogonal iff
C1C2 2 = r12 + r22 and suborthogonal
iff C1C2 2 < r12 + r22.
For a given value of
the weighted alpha complex is formed with the simplices of the
regular triangulation triangulation
such that there is a sphere orthogonal to the weighted points associated
with the vertices of the simplex and suborthogonal to all the other
input weighted points. Once again the alpha shape is then defined as
the domain covered by a the alpha complex and arise in two versions
general or regularized.
AlphaShapeTraits_3
WeightedAlphaShapeTraits_3
AlphaShapeCell_3
AlphaShapeVertex_3