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https://hdl.handle.net/2142/20105
Description
Title
Topological modeling with simplicial complexes
Author(s)
Shah, Nimish Rameshbhai
Issue Date
1994
Doctoral Committee Chair(s)
Edelsbrunner, Herbert
Department of Study
Computer Science
Discipline
Computer Science
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Computer Science
Language
eng
Abstract
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 $\IR\sp{d}$. Regular triangulations generalise Delaunay triangulations, and are related to convex hulls in $\IR\sp{d+1}$. We present an efficient randomized incremental algorithm for computing the regular triangulation of a finite weighted point set in $\IR\sp{d}$. The expected running time for the worst set of n points in $\IR\sp{d}$ is O($n\log n$ + $n\sp{\lceil d/2\rceil}$). We also discuss some implementation issues related to degenerate point sets.
For $\alpha$ $\in$ $\IR$, a weighted $\alpha$-shape of a finite set of weighted points in $\IR\sp{d}$ 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 $\IR\sp{d}$. 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.
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