TY - JOUR
T1 - On polyhedra induced by point sets in space
AU - Agarwal, Pankaj K.
AU - Hurtado, Ferran
AU - Toussaint, Godfried T.
AU - Trias, Joan
N1 - Copyright:
Copyright 2008 Elsevier B.V., All rights reserved.
PY - 2008/1/1
Y1 - 2008/1/1
N2 - 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, Grünbaum showed that an analogous theorem holds in R3. More precisely, if S is a set of n ≥ 4 points in R3 (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. Grünbaum's constructive proof may yield Schönhardt 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 (n log n) time. Furthermore, we show that a tetrahedralized, xy-monotonic, polyhedronization of S may be computed in time O (n1 + ε{lunate}), for any ε{lunate} > 0.
AB - 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, Grünbaum showed that an analogous theorem holds in R3. More precisely, if S is a set of n ≥ 4 points in R3 (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. Grünbaum's constructive proof may yield Schönhardt 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 (n log n) time. Furthermore, we show that a tetrahedralized, xy-monotonic, polyhedronization of S may be computed in time O (n1 + ε{lunate}), for any ε{lunate} > 0.
KW - Computational geometry
KW - Polyhedra
KW - Triangulation
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U2 - 10.1016/j.dam.2007.08.033
DO - 10.1016/j.dam.2007.08.033
M3 - Article
AN - SCOPUS:35648997008
VL - 156
SP - 42
EP - 54
JO - Discrete Applied Mathematics
JF - Discrete Applied Mathematics
SN - 0166-218X
IS - 1
ER -