Version 1
: Received: 13 March 2024 / Approved: 15 March 2024 / Online: 15 March 2024 (11:09:32 CET)
How to cite:
Ungar, A.A. When Four Cyclic Antipodal Points Are Ordered Counterclockwise in Euclidean and Hyperbolic Geometry. Preprints2024, 2024030910. https://doi.org/10.20944/preprints202403.0910.v1
Ungar, A.A. When Four Cyclic Antipodal Points Are Ordered Counterclockwise in Euclidean and Hyperbolic Geometry. Preprints 2024, 2024030910. https://doi.org/10.20944/preprints202403.0910.v1
Ungar, A.A. When Four Cyclic Antipodal Points Are Ordered Counterclockwise in Euclidean and Hyperbolic Geometry. Preprints2024, 2024030910. https://doi.org/10.20944/preprints202403.0910.v1
APA Style
Ungar, A.A. (2024). When Four Cyclic Antipodal Points Are Ordered Counterclockwise in Euclidean and Hyperbolic Geometry. Preprints. https://doi.org/10.20944/preprints202403.0910.v1
Chicago/Turabian Style
Ungar, A.A. 2024 "When Four Cyclic Antipodal Points Are Ordered Counterclockwise in Euclidean and Hyperbolic Geometry" Preprints. https://doi.org/10.20944/preprints202403.0910.v1
Abstract
A cyclic antipodal points of a circle is a pair of
points that are the intersection of the circle with a diameter
of the circle. A recent proof of Ptolemy’s Theorem, simultaneously
in both (i) Euclidean geometry; and (ii) the relativistic
model of hyperbolic geometry (which is identified with the
Klein model of hyperbolic geometry), motivates in this article
the study of four cyclic antipodal points 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. Formulas that extend
the Pythagorean formula in both Euclidean and hyperbolic
geometry are obtained as byproducts
Keywords
cyclic antipodal points; relativistic model of hyperbolic geometry; gyrovector space; 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.
