The filter of implicitly represents all alpha
complexes defined by **B** as prefixes.
The availability of this simplex sequence favors
incremental algorithms for computing properties of alpha shapes.
The software considers metric properties:

- volume, area, and length (defined below),

- number of tetrahedra, triangles, edges, and vertices,
distinguishing between simplices on the boundary
and in the interior,

- number of components, independent tunnels, and voids,
as expressed by the three betti numbers,
,
,
.

As shown in [11],
every additive and continuous map from
the set of convex bodies to invariant under rigid motion
is a linear combination of quermassintegrals,
see also [14, chapter 4,].
In , the quermassintegrals are basically
volume, area, mean width, and the Euler number.
Length is defined as an extension of the mean width
to non-convex bodies.
Specifically, *length* is the sum of edge lengths, each
weighted by the (possibly negative) complementary angle.
The Euler number is .

Each property defines a *signature*
,
where .
Signatures are useful in studying shapes and convenient in
quickly identifying the ``interesting'' ones in the typically huge
family of alpha shapes.

The signatures expressing the above metric and combinatorial
properties are straightforward to compute:
scan the filter from 0 through **n** and increment or decrement the
current value depending on the next simplex.
Such a strategy also works for the three betti numbers, but is
less obvious [2].

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