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Displaying similar documents to “Planar Geodesic Immersions in Pseudo-Euclidean Space.”

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

Joachim A. Hempel (2004)

Annales Polonici Mathematici

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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 pseudo-trigonometry connects these angles with pseudo-lengths. We state and prove a theorem resembling Ptolemy's classical theorem on cyclic quadrilaterals and...