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The Hyperbolic Ptolemy’s Theorem in the Poincare Ball Model of Analytic Hyperbolic Geometry
Version 1
: Received: 22 June 2023 / Approved: 23 June 2023 / Online: 23 June 2023 (12:31:10 CEST)
A peer-reviewed article of this Preprint also exists.
Ungar, A.A. The Hyperbolic Ptolemy’s Theorem in the Poincaré Ball Model of Analytic Hyperbolic Geometry. Symmetry 2023, 15, 1487. Ungar, A.A. The Hyperbolic Ptolemy’s Theorem in the Poincaré Ball Model of Analytic Hyperbolic Geometry. Symmetry 2023, 15, 1487.
Abstract
Ptolemy’s Theorem in Euclidean geometry, named after the Greek astronomer and mathematician Claudius Ptolemy, is well known. We translate Ptolemy’s Theorem from analytic Euclidean geometry into the Poincar´e ball model of analytic hyperbolic geometry, which is based on M¨obius addition. The translation of Ptolemy’s Theorem from Euclidean geometry into hyperbolic geometry is achieved by means of the hyperbolic trigonometry, called gyrotrigonometry, to which the Poincar´e ball model gives rise, and by means of the duality of trigonometry and gyrotrigonometry.
Keywords
Ptolemy’s Theorem; Poincar´e ball model; Hyperbolic geometry; M¨obius addition; Gyrogroups; Gyrovector spaces; Gyrotrigonometry
Subject
Computer Science and Mathematics, Geometry and Topology
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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