  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.
