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.

When Four Cyclic Antipodal Points Are Ordered Counterclockwise in Euclidean and Hyperbolic Geometry

“…Accordingly, if viewed in hyperbolic geometry, (1) is said to be a half-gyroangled gyrotrigonometric identity. Furthermore, we note the following observations in [1] and [2]:…”

“…Einstein (Möbius) gyrovector spaces form the algebraic setting for Klein (Poincaré) ball model of hyperbolic geometry with spectacular gain in clarity and simplicity, just as vector spaces form the algebraic setting for Euclidean geometry. In our work, the Poincaré ball model of hyperbolic geometry stems from Möbius addition [2] and the Klein model of hyperbolic geometry stems from Einstein addition [1]. Hence, when studied analytically by means of Einstein addition, we refer the Klein model to as the relativistic model.…”

“…The study of Ptolemy's Theorem (i) in Euclidean geometry and (ii) in the relativistic model of hyperbolic geometry in [1] and (iii) in the Poincaré ball model of hyperbolic geometry in [2] is solely based on the half-angled trigonometric identity…”

“…2 of 12 3. It is shown in [2] that if viewed in the Poincaré ball model of hyperbolic geometry, the halfgyroangled gyrotrigonometric identity (1) gives rise to the novel hyperbolic Ptolemy's Theorem in the Poincaré ball model of hyperbolic geometry.…”

“…The study of Ptolemy's Theorem in [1] and [2] is based on the half-angled trigonometric identity (1). Similarly, the study of cyclic antipodal points in this article is based on the half-angled trigonometric identity (4).…”