Details

For a given set of vertices , the -complex is a simplicial subcomplex of the Delaunay triangulation parameterized by . For any simplex , we have that if the pairwise distances between vertices in that simplex are all less than the given . That is, for all . This Demonstration generates a random set of planar points; you can vary to see how the complex changes. The simplicial homology groups and their corresponding Betti numbers are topological invariants that characterize the -dimensional "holes" in the complex. For example, gives the number of connected components, is the number of "tunnels," and gives the number of closed-off spaces with volume (however, in this Demonstration, the complex is planar, so remains trivial).

For more information on homology (and algebraic topology in general), see the following.

References

[1] A. Hatcher, Algebraic Topology, New York: Cambridge University Press, 2002.

[2] J. R. Munkres, Elements of Algebraic Topology, Menlo Park, CA: Addison-Wesley, 1984.