Duality between Hyperbolic and de Sitter Geometry

TL;DR

This paper explores the relationship between hyperbolic and de Sitter geometries by developing a trigonometric framework on the de Sitter surface, utilizing polar triangles to connect and simplify hyperbolic and de Sitter triangle properties.

Contribution

It introduces a novel approach to de Sitter trigonometry using polar triangles, clarifying the hyperbolic law of cosines for angles and establishing a duality between hyperbolic and de Sitter geometries.

Findings

Characterization of geodesics on de Sitter surface

Development of triangle types and their properties

Simplified proof of hyperbolic law of cosines for angles

Abstract

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.