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),

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].