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Schwarz Christoffel Mappings / Conformal Mappings for hyperbolic polygonspolygon

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 construction constructions in hyperbolic geometry?

Does anybody know of similiar construction constructions for any other domains?

Any idea , how to construct conformal mappings even for more general domains will also be very wellcomed! Perhaps I can spezialise them to my preferred domains. I am far from being an expert in conformal mappings and do only know some isolated examples!
show/hide this revision's text 2 I am searching for explicit construction for conformal mappings for other domains than euclidean polygons

Schwarz Christoffel Mappings / Conformal Mappings for hyperbolic polygons

The classical Schwarz Christoffel theorem gives an explicit construction

I am searching for a biholomorphic map conformal mapping from the upper halfplane to 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).

Is there

Does anybody know of a similar construction in hyperbolic geometry?

Does anybody know of similiar construction for hyperbolic polygonsany other domains?

Any idea, how to construct conformal mappings even for more general domains will also be very wellcomed! Perhaps I can spezialise them to my preferred domains. I am far from being an expert in conformal mappings and do only know some isolated examples!
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Schwarz Christoffel Mappings

The classical Schwarz Christoffel theorem gives an explicit construction for a biholomorphic map from the upper halfplane to a polygon (see e.g. http://en.wikipedia.org/wiki/Schwarz-Christoffel_mapping).

Is there a similiar construction for hyperbolic polygons?