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.
Recently, Schlenker has treated weighted Delaunay triangulations in piecewise flat and piecewise hyperbolic surfaces using a deformation method @cite_12 . He obtains an existence and uniqueness theorem [Theorem 1.4] schlenker05b with the same scope as Theorem , but the conditions for existence are in terms of angle sums over paths like in Theorem . This seems to be the first time that this type of conditions was obtained for circle patterns with cone singularities. It would be interesting to show directly that the conditions of his theorem are equivalent to the conditions of Theorem .
@cite_12
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We consider hyperideal'' circle patterns, i.e. patterns of disks appearing in the definition of the Delaunay decomposition associated to a set of disjoint disks, possibly with cone singularities at the center of those disks. Hyperideal circle patterns are associated to hyperideal hyperbolic polyhedra. We describe the possible intersection angles and singular curvatures of those circle patterns, on Euclidean or hyperbolic surfaces with conical singularities. This is related to results on the dihedral angles of ideal or hyperideal hyperbolic polyhedra.