Signatures

 

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),

combinatorial properties:

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

and topological properties:

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].