Duality between Hyperbolic and de Sitter Geometry

Details Duality between Hyperbolic and de Sitter Geometry Duality between Hyperbolic and de Sitter Geometry

2008-10-29

arxiv.org/abs/0810.5303v2

Mathematics

Differential Geometry

32 pages, 4 figures

Scientific paper

In this paper we describe trigonometry on the de Sitter surface. For that a characterization of geodesics is given, leading to various types of triangles. We define lengths and angles of these. Then, transferring the concept of polar triangles from spherical geometry into the Minkowski space, we relate hyperbolic with de Sitter triangles such that the proof of the hyperbolic law of cosines for angles becomes much clearer and easier than it is traditionally. Furthermore, polar triangles turn out to be a powerful tool for describing de Sitter trigonometry.

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