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A hyperbolic geodesic joining two punctures on a Riemann surface has infinite length. To obtain a useful distance-like quantity we define a finite pseudo-length of such a geodesic in terms of the hyperbolic length of its surrounding geodesic loop. There is a well defined angle between two geodesics meeting at a puncture, and our pseudotrigonometry connects these angles with pseudo-lengths. We state and prove a theorem resembling Ptolemy’s classical theorem on cyclic quadrilaterals and three general lemmas on intersections of shortest (in the sense of pseudo-length) geodesic joins. These ideas are then applied to the description of an optimal fundamental region for the covering Fuchsian group of a five-punctured sphere, effectively also giving a fundamental region for the modular group M(0, 5).

A pseudo-trigonometry related to Ptolemy's theorem and the hyperbolic geometry of punctured spheres