Combinatorial Structure of Delaunay Triangulations of the Plane (Extended
Abstract)
M.B. Dillencourt, University of California;I. Rivin, NECI;W.D. Smith,
NECI
11/06/91
ABSTRACT
Delaunay triangulations are closely related to inscribable polyhedra. A
polyhedron is {\it inscribable} if it can be realized as the convex hull of
a set of points on the surface of the sphere. The question of characterizing
inscribable polyhedra dates back to Descartes and was posed as an open
problem by Steiner in 1832. This problem has been solved by Rivin and
Smith in 1991 using methods from hyperbolic geometry.
This extended abstract
sketches the following extensions and applications of the Rivin-Smith
characterization.
\begin{enumerate}
\item An algorithm for producing an
inscription of a polyhedron.
\item A proof that ``almost any'' completion of a
degenerate Delaunay triangulation can be realized as a nondegenerate
Delaunay triangulation of an arbitrarily small perturbation of the original
point set. This implies, in particular, that general-purpose schemes for
resolving degeneracies in geometric data are unnecessary when computing
planar Delaunay triangulations.
\item A proof that any 4-connected
simplicial polyhedron is both inscribable and circumscribable. This result
implies that any plane triangulation without chords or nonfacial triangles is
realizable as a Delaunay triangulation. Moreover, any ``medial'' polyhedron
is both inscribable and circumscribable.
\item Upper and lower bounds for
the number of inscribable and circumscribable simplicial polyhedra.
\end{enumerate}
We conclude by mentioning some related results and open
problems.
KEYWORDS:
INSCRIBABLE GRAPHS, POLYHEDRA, DELAUNAY
TRIANGULATIONS, DEGENERACIES, CONVEX PROGRAMMING