Browse Reports

Simplicial complexes, or triangulated meshes, are widely used for computer graphics and for representing data in computational science and engineering. Often it is necessary to determine some topological characteristics of a complex, either for scientific study or to maintain properties while the data set is undergoing some transformation. In this paper we define the wave traversal algorithm and study its use for topological characterization of a complex. Wave traversal is a natural extension of breadth-first search of a graph to n-dimensional simplicial complexes. As defined, each wave is a subcomplex whose simplices have a certain distance from an arbitrary start vertex. We prove that if waves on a surface split and rejoin then there is a topological hole called a tunnel in the surface. We then prove stronger results about wave traversal and topology. We show that wave traversal can be extended to a discrete Morse function on triangulated manifolds. Critical points of a Morse function can be used to characterize the homotopy type of an n-manifold, and they give bounds on the betti numbers of the manifold. The Morse function we derive is straightforward to compute, but expensive computationally both in terms of space and time. So we show that we can achieve similar and even stronger results about the critical points using combinatorial methods, without first computing the function itself. We supply algorithms for doing this in the case of 2-manifolds. We also prove that barycentric subdivision eliminates degeneracies in wave traversal. Results from problems in other areas of research that have used the wave traversal paradigm could be re-examined in light of our findings. Finally, we demonstrate the use of wave traversal for creating an auditory display of a simplicial complex. This has application particularly when the complex is large or higher-dimensional, but audio has been shown to reinforce visual display in all cases. The auditory display is intended to convey information about the complex to the user in a interactive exploratory manner. We detail two of the auditory displays that we have developed using wave traversal.