On polyhedra induced by point sets in space

  • Authors:

  • Pankaj K. Agarwal;Ferran Hurtado;Godfried T. Toussaint;Joan Trias

  • Affiliations:

  • Department of Computer Science, Box 90129, Duke University, Durham, NC 27708-0129, USA;Departament de Matemática Aplicada II, Universitat Politécnica de Catalunya (UPC), Jordi Girona, 1-3, 08034 Barcelona, Spain;School of Computer Science, 3480 University Street, McGill University, Montreal, Que., Canada H3A 2A7;Departament de Matemática Aplicada II, Universitat Politécnica de Catalunya (UPC), Jordi Girona, 1-3, 08034 Barcelona, Spain

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2008

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Abstract

Given a set S of n=3 points in the plane (not all on a line) it is well known that it is always possible to polygonize S, i.e., construct a simple polygon P such that the vertices of P are precisely the given points in S. For example, the shortest circuit through S is a simple polygon. In 1994, Grunbaum showed that an analogous theorem holds in R^3. More precisely, if S is a set of n=4 points in R^3 (not all of which are coplanar) then it is always possible to polyhedronize S, i.e., construct a simple (sphere-like) polyhedron P such that the vertices of P are precisely the given points in S. Grunbaum's constructive proof may yield Schonhardt polyhedra that cannot be triangulated. In this paper several alternative algorithms are proposed for constructing such polyhedra induced by a set of points, which may always be triangulated, and which enjoy several other useful properties as well. Such properties include polyhedra that are star-shaped, have Hamiltonian skeletons, and admit efficient point-location queries. We show that polyhedronizations with a variety of such useful properties can be computed efficiently in O(nlogn) time. Furthermore, we show that a tetrahedralized, xy-monotonic, polyhedronization of S may be computed in time O(n^1^+^@e), for any @e0.