Topological modeling with simplicial complexes

Simplicial complexes are useful for modeling shape of a discrete geometric domain and for discretizing continuous domains. A geometric triangulation of a point set $S$ is a simplicial complex whose vertex set is contained in $S$ and whose underlying space is the convex hull of $S$. In this thesis we study different approaches for constructing subcomplexes of a geometric triangulation to obtain a good model of a given domain. The work described in this thesis is about regular triangulations, weighted $\alpha$-shapes and homeomorphic triangulations. We develop the notion of a regular triangulation of a set on $n$ weighted points in general position in $\real^d$. Regular triangulations generalise Delaunay triangulations, and are related to convex hulls in $\real^{d+1}$. We present an efficient randomized incremental algorithm for computing the regular triangulation of a finite weighted point set in $\real^d$. The expected running time for the worst set of $n$ points in $\real^d$ is O$(n\log n + n^{\lceil d/2 \rceil})$. We also discuss some implementation issues related to degenerate point sets. For $\alpha\in\real$, a weighted $\alpha$-shape of a finite set of weighted points in $\real^d$ is obtained from a subcomplex of the regular triangulation of the point set. Weighted $\alpha$-shapes are useful for molecular modeling and surface reconstruction. We present a definition for weighted $\alpha$-shapes that applies to any input, including degenerate data. We also give a straightforward algorithm to compute them. Finally, we introduce the Delaunay simplicial complex of a point set $S$ restricted by a given topological space, a subset of $\real^d$. This concept is useful in discretizing continuous domains, especially when the dimension of the domain and the imbedding dimension are different. The restricted Delaunay simplicial complex is a subcomplex of the Delaunay triangulation of $S$. We present sufficient conditions for the underlying space of the restricted Delaunay simplicial complex to be homeomorphic to the given topological space.