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

A cyclic antipodal pair of a circle is a pair of points that are the intersection of the circle with the diameter of the circle. In this article, a recent proof of Ptolemy’s Theorem—simultaneously in both (i) Euclidean geometry and (ii) the relativistic model of hyperbolic geometry (also known as th...

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Bibliographic Details
Published inSymmetry (Basel) Vol. 16; no. 6; p. 729
Main Author Ungar, Abraham A
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.06.2024
Subjects
Circles (geometry)
cyclic antipodal pairs
Euclidean geometry
Geometry
gyrotrigonometry
gyrovector spaces
hyperbolic Pythagorean identity
relativistic model of hyperbolic geometry
Trigonometry
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Summary:A cyclic antipodal pair of a circle is a pair of points that are the intersection of the circle with the diameter of the circle. In this article, a recent proof of Ptolemy’s Theorem—simultaneously in both (i) Euclidean geometry and (ii) the relativistic model of hyperbolic geometry (also known as the Klein model)—motivates the study of four cyclic antipodal pairs of a circle, ordered arbitrarily counterclockwise. The translation of results from Euclidean geometry into hyperbolic geometry is obtained by means of hyperbolic trigonometry, called gyrotrigonometry, to which Einstein addition gives rise. Identities that extend the Pythagorean identity in both Euclidean and hyperbolic geometry are obtained.
ISSN:2073-8994
2073-8994
DOI:10.3390/sym16060729