Take the 2-minute tour ×  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!  share|improve
this question I'm no expert on the subject, but presumably you
could take normal polygons and and map them into the unit disc
in the usual way. –  Adam Hughes Nov 15 '10 at 7:00 @Adam: a
hyperbolic polygon means one whose sides are geodesics for the
hyperbolic metric, so they are not usually straight lines. If
you take a polygon with straight edges in the upper half plane
then the edges are not geodesic for the hyperbolic m