A variational principle for weighted Delaunay triangulations and hyperideal polyhedra

Details A variational principle for weighted Delaunay triangulations and hyperideal polyhedra A variational principle for weighted Delaunay triangulations and hyperideal polyhedra

2006-03-03

arxiv.org/abs/math/0603097v2

Mathematics

Geometric Topology

27 pages, many figures. v2: slightly restructured, typos corrected, references updated. Final version

Scientific paper

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.

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