I am searching for a conformal mapping from the upper halfplane onto a hyperbolic polygon, i.e. the sides of the polygon have to be geodesics.

The classical Schwarz Christoffel theorem does the job for euclidean polygons (see e.g. http://en.wikipedia.org/wiki/Schwarz-Christoffel_mapping).

Does anybody know of a similar constructions in hyperbolic geometry?

Does anybody know of similiar constructions for any other domains?

Any idea will be very wellcomed! I am far from being an expert in conformal mappings and do only know some isolated examples!