Decompositions of polyhedra in three dimensions

Tamal Krishna Dey, Purdue University

Abstract

This thesis deals with new theoretical and practical results on convex and CSG decompositions, and triangulations of polyhedra in three dimensions. Convex and CSG decompositions of polyhedra find applications in simpler algorithms in motion planning, computer graphics, and solid modeling. Triangulations of polyhedra are fundamental nontrivial steps in finite element simulations and CAD/CAM applications. To reduce ill conditioning as well as discretization error in finite element simulations, near regular shaped elements are desired. This motivates triangulation algorithms for polyhedra that produce well shaped tetrahedra. We present efficient algorithms for convex and CSG decompositions of polyhedra with arbitrary genus. A modification of this decomposition method gives an efficient algorithm for triangulations of polyhedra. The efficiency of these algorithms is mainly derived from the use of "zone" theorem on hyperplane arrangements, studied in combinatorial geometry. A triangulation algorithm that triangulates a convex polyhedron and a three dimensional point set, in general, with guaranteed quality tetrahedra is also presented. In particular, this algorithm guarantees that four out of five possible bad tetrahedra are never generated. Geometric algorithms, when implemented under finite precision arithmetic often crash or fail to produce valid output because of numerical errors. We have investigated this problem of output inconsistency under imprecise arithmetic computations in order to provide topologically robust implementations of the decomposition algorithms. Implementations are carried out as part of SHILP, a solid modeling and display toolkit that runs on Unix workstations under the X Window System.

Degree

Ph.D.

Advisors

Bajaj, Purdue University.

Subject Area

Computer science

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