  We introduce decorated piecewise hyperbolic and spherical surfaces and
discuss their discrete conformal equivalence. A decoration is a choice of
circle about each vertex of the surface. Our decorated surfaces are closely
related to inversive distance circle packings, canonical tessellations of
hyperbolic surfaces, and hyperbolic polyhedra.
  We prove the corresponding uniformization theorem. Furthermore, we show that
on can deform continuously between decorated piecewise hyperbolic, Euclidean,
and spherical surfaces sharing the same fundamental discrete conformal
invariant. Therefore, there is one master theory of discrete conformal
equivalence in different background geometries. Our approach is based on a
variational principle, which also provides a way to compute the discrete
uniformization and geometric transitions.
