Ptolemy’s Theorem in the Relativistic Model of Analytic Hyperbolic Geometry

Abstract

Ptolemy’s Theorem in Euclidean geometry, named after the Greek astronomer and mathematician Ptolemy, is well-known. By means of the relativistic model of hyperbolic geometry, we translate Ptolemy’s Theorem from Euclidean geometry into the hyperbolic geometry of Lobachevsky and Bolyai. The relativistic model of hyperbolic geometry is based on the Einstein addition of relativistically admissible velocities and, as such, it coincides with the well-known Beltrami–Klein ball model of hyperbolic geometry. The translation of Ptolemy’s Theorem from Euclidean geometry into hyperbolic geometry is achieved by means of hyperbolic trigonometry, called gyrotrigonometry, to which the relativistic model of analytic hyperbolic geometry gives rise.