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

Joachim A. Hempel. "A pseudo-trigonometry related to Ptolemy's theorem and the hyperbolic geometry of punctured spheres." Annales Polonici Mathematici 84.2 (2004): 147-167. <http://eudml.org/doc/280429>.

@article{JoachimA2004,
abstract = {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).},
author = {Joachim A. Hempel},
journal = {Annales Polonici Mathematici},
keywords = {uniformization; hyperbolic geometry; Fuchsian groups; moduli spaces; punctured spheres},
language = {eng},
number = {2},
pages = {147-167},
title = {A pseudo-trigonometry related to Ptolemy's theorem and the hyperbolic geometry of punctured spheres},
url = {http://eudml.org/doc/280429},
volume = {84},
year = {2004},
}

TY - JOUR
AU - Joachim A. Hempel
TI - A pseudo-trigonometry related to Ptolemy's theorem and the hyperbolic geometry of punctured spheres
JO - Annales Polonici Mathematici
PY - 2004
VL - 84
IS - 2
SP - 147
EP - 167
AB - 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).
LA - eng
KW - uniformization; hyperbolic geometry; Fuchsian groups; moduli spaces; punctured spheres
UR - http://eudml.org/doc/280429
ER -