Canonical tessellations of decorated hyperbolic surfaces

A decoration of a hyperbolic surface of finite type is a choice of circle, horocycle or hypercycle about each cone-point, cusp or flare of the surface, respectively. In this article we show that a decoration induces a unique canonical tessellation and dual decomposition of the underlying surface. They are analogues of the weighted Delaunay tessellation and Voronoi decomposition in the Euclidean plane. We develop a characterisation in terms of the hyperbolic geometric equivalents of Delaunay’s empty-discs and Laguerre’s tangent-distance, also known as power-distance. Furthermore, the relation between the tessellations and convex hulls in Minkowski space is presented, generalising the Epstein–Penner convex hull construction. This relation allows us to extend Weeks’ flip algorithm to the case of decorated finite type hyperbolic surfaces. Finally, we give a simple description of the configuration space of decorations and show that any fixed hyperbolic surface only admits a finite number of combinatorially different canonical tessellations.