A variational principle for weighted Delaunay triangulations and hyperideal polyhedra

TL;DR

This paper establishes a variational principle to prove existence and uniqueness of weighted Delaunay triangulations with prescribed angles, linking them to hyperbolic polyhedra and extending Rivin's work to piecewise flat surfaces.

Contribution

It introduces a variational approach to characterize weighted Delaunay triangulations and hyperideal polyhedra with prescribed geometric data, generalizing previous results.

Findings

Proves existence and uniqueness of weighted Delaunay triangulations with given angles.

Characterizes hyperbolic polyhedra with prescribed dihedral angles.

Extends Rivin's work to include piecewise flat surfaces with cone singularities.

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.