Möbius-invariant metrics and generalized angles in Ptolemeic spaces

We study Möbius and quasimobius mappings in spaces with a semimetric meeting the Ptolemy inequality. We construct a bimetrization of a Ptolemeic space which makes it possible to introduce a Möbius-invariant metric (angular distance) in the complement to each nonsingleton. This metric coincides with the hyperbolic metric in the canonical cases. We introduce the notion of generalized angle in a Ptolemeic space with vertices a pair of sets, determine its magnitude in terms of the angular distance and study distortion of generalized angles under quasimobius embeddings. As an application to noninjective mappings, we consider the behavior of the generalized angle under projections and obtain an estimate for the inverse distortion of generalized angles under quasimeromorphic mappings (mappings with bounded distortion).