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!



Conformal Mappings for hyperbolic polygon
See Harmer and Martin's work on [Conformal Mappings from the Upper Half Plane to Fundamental Domains on the Hyperbolic Plane](http://www.math.auckland.ac.nz/Research/Reports/view.php?id=499).


Some of the ideas developed by [Christopher Bishop](http://www.math.sunysb.edu/~bishop/) in the context of computational geometry may also be of interest. 
See his [talks](http://www.math.sunysb.edu/~bishop/lectures/lec.html) and [papers](http://www.math.sunysb.edu/~bishop/papers/papers.html) on conformal maps.


