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

Autor:	Abraham A. Ungar
Jazyk:	angličtina
Rok vydání:	2023
Předmět:	Ptolemy’s theorem analytic hyperbolic geometry einstein addition gyrogroups gyrovector spaces gyrotrigonometry Mathematics QA1-939
Zdroj:	Symmetry, Vol 15, Iss 3, p 649 (2023)
Druh dokumentu:	article
ISSN:	2073-8994
DOI:	10.3390/sym15030649
Popis:	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.
Databáze:	Directory of Open Access Journals
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