The Hyperbolic Ptolemy’s Theorem in the Poincaré Ball Model of Analytic Hyperbolic Geometry
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 ass...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-07-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/15/8/1487 |
| _version_ | 1797583192664834048 |
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| author | Abraham A. Ungar |
| author_facet | Abraham A. Ungar |
| author_sort | Abraham A. Ungar |
| collection | DOAJ |
| description | 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 <i>gyrotrigonometry</i>, to which the Poincaré ball model gives rise, and by means of the duality of trigonometry and gyrotrigonometry. |
| first_indexed | 2024-03-10T23:32:20Z |
| format | Article |
| id | doaj.art-1d7921e38e924b679fbfafd93d5f5ac5 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T23:32:20Z |
| publishDate | 2023-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-1d7921e38e924b679fbfafd93d5f5ac52023-11-19T03:10:28ZengMDPI AGSymmetry2073-89942023-07-01158148710.3390/sym15081487The Hyperbolic Ptolemy’s Theorem in the Poincaré Ball Model of Analytic Hyperbolic GeometryAbraham A. Ungar0Department of Mathematics, North Dakota State University, Fargo, ND 58105, USAPtolemy’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 <i>gyrotrigonometry</i>, to which the Poincaré ball model gives rise, and by means of the duality of trigonometry and gyrotrigonometry.https://www.mdpi.com/2073-8994/15/8/1487Ptolemy’s theoremPoincaré ball modelhyperbolic geometryMöbius additiongyrogroupsgyrovector spaces |
| spellingShingle | Abraham A. Ungar The Hyperbolic Ptolemy’s Theorem in the Poincaré Ball Model of Analytic Hyperbolic Geometry Symmetry Ptolemy’s theorem Poincaré ball model hyperbolic geometry Möbius addition gyrogroups gyrovector spaces |
| title | The Hyperbolic Ptolemy’s Theorem in the Poincaré Ball Model of Analytic Hyperbolic Geometry |
| title_full | The Hyperbolic Ptolemy’s Theorem in the Poincaré Ball Model of Analytic Hyperbolic Geometry |
| title_fullStr | The Hyperbolic Ptolemy’s Theorem in the Poincaré Ball Model of Analytic Hyperbolic Geometry |
| title_full_unstemmed | The Hyperbolic Ptolemy’s Theorem in the Poincaré Ball Model of Analytic Hyperbolic Geometry |
| title_short | The Hyperbolic Ptolemy’s Theorem in the Poincaré Ball Model of Analytic Hyperbolic Geometry |
| title_sort | hyperbolic ptolemy s theorem in the poincare ball model of analytic hyperbolic geometry |
| topic | Ptolemy’s theorem Poincaré ball model hyperbolic geometry Möbius addition gyrogroups gyrovector spaces |
| url | https://www.mdpi.com/2073-8994/15/8/1487 |
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