The Hyperbolic Ptolemy’s Theorem in the Poincaré Ball Model of Analytic Hyperbolic Geometry

Autor:	Abraham A. Ungar
Jazyk:	angličtina
Rok vydání:	2023
Předmět:	Ptolemy’s theorem Poincaré ball model hyperbolic geometry Möbius addition gyrogroups gyrovector spaces Mathematics QA1-939
Zdroj:	Symmetry, Vol 15, Iss 8, p 1487 (2023)
Druh dokumentu:	article
ISSN:	2073-8994
DOI:	10.3390/sym15081487
Popis:	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 Poincaré ball model of analytic hyperbolic geometry, which is based on the Möbius addition and its associated symmetry gyrogroup. The translation of Ptolemy’s theorem from Euclidean geometry into hyperbolic geometry is achieved by means of hyperbolic trigonometry, called gyrotrigonometry, to which the Poincaré ball model gives rise, and by means of the duality of trigonometry and gyrotrigonometry.
Databáze:	Directory of Open Access Journals
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