Canonical tessellations of decorated hyperbolic surfaces

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