Journal of Differential Geometry

A variational principle for weighted Delaunay triangulations and hyperideal polyhedra

Boris A. Springborn

Full-text: Open access

Abstract

We use a variational principle to prove an existence and uniqueness theorem for planar weighted Delaunay triangulations (with non-intersecting site-circles) with prescribed combinatorial type and circle intersection angles. Such weighted Delaunay triangulations may be interpreted as images of hyperbolic polyhedra with one vertex on and the remaining vertices beyond the infinite boundary of hyperbolic space. Thus, the main theorem states necessary and sufficient conditions for the existence and uniqueness of such polyhedra with prescribed combinatorial type and dihedral angles. More generally, we consider weighted Delaunay triangulations in piecewise flat surfaces, allowing cone singularities with prescribed cone angles in the vertices. The material presented here extends work by Rivin on Delaunay triangulations and ideal polyhedra.

Article information

Source
J. Differential Geom. Volume 78, Number 2 (2008), 333-367.

Dates
First available in Project Euclid: 14 February 2008

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1203000270

Digital Object Identifier
doi:10.4310/jdg/1203000270

Mathematical Reviews number (MathSciNet)
MR2394026

Zentralblatt MATH identifier
1181.52018

Citation

Springborn, Boris A. A variational principle for weighted Delaunay triangulations and hyperideal polyhedra. J. Differential Geom. 78 (2008), no. 2, 333--367. doi:10.4310/jdg/1203000270. https://projecteuclid.org/euclid.jdg/1203000270


Export citation