  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.
