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 pseudo-trigonometry 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).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap84-2-5,
author = {Joachim A. Hempel},
title = {A pseudo-trigonometry related to Ptolemy's theorem and the hyperbolic geometry of punctured spheres},
journal = {Annales Polonici Mathematici},
volume = {83},
year = {2004},
pages = {147-167},
zbl = {1102.30040},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap84-2-5}
}
Joachim A. Hempel. A pseudo-trigonometry related to Ptolemy's theorem and the hyperbolic geometry of punctured spheres. Annales Polonici Mathematici, Tome 83 (2004) pp. 147-167. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap84-2-5/