An Extension of Poincare Model of Hyperbolic Geometry with Gyrovector Space Approach
The aim of this paper is to show the importance of analytic hyperbolic geometry introduced in [9]. In [1], Ungar and Chen showed that the algebra of the group SL(2, C) naturally leads to the notion of gyrogroups and gyrovector spaces for dealing with the Lorentz group and its underlying hyper...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
University of Kashan
2016-01-01
|
| Series: | Mathematics Interdisciplinary Research |
| Subjects: | |
| Online Access: | https://mir.kashanu.ac.ir/article_13923_45adafb5f2e2797a3bb789d969d94705.pdf |
| _version_ | 1797630967748231168 |
|---|---|
| author | Mahfouz Rostamzadeh Sayed-Ghahreman Taherian |
| author_facet | Mahfouz Rostamzadeh Sayed-Ghahreman Taherian |
| author_sort | Mahfouz Rostamzadeh |
| collection | DOAJ |
| description | The aim of this paper is to show the importance of analytic hyperbolic geometry introduced in [9]. In [1], Ungar and Chen showed that the algebra of the group SL(2, C) naturally leads to the notion of gyrogroups and gyrovector spaces for dealing with the Lorentz group and its underlying hyperbolic geometry. They defined the Chen addition and then Chen model of hyperbolic geometry. In this paper, we directly use the isomorphism properties of gyrovector spaces to recover the Chen's addition and then Chen model of hyperbolic geometry. We show that this model is an extension of the Poincaré model of hyperbolic geometry. For our purpose we consider the Poincaré plane model of hyperbolic geometry inside the complex open unit disc D. Also we prove that this model is isomorphic to the Poincaré model and then to other models of hyperbolic geometry. Finally, by gyrovector space approach we verify some properties of this model in details in full analogue with Euclidean geometry. |
| first_indexed | 2024-03-11T11:15:24Z |
| format | Article |
| id | doaj.art-4fdc9969954e44a6876877f20afd9c6b |
| institution | Directory Open Access Journal |
| issn | 2476-4965 |
| language | English |
| last_indexed | 2024-03-11T11:15:24Z |
| publishDate | 2016-01-01 |
| publisher | University of Kashan |
| record_format | Article |
| series | Mathematics Interdisciplinary Research |
| spelling | doaj.art-4fdc9969954e44a6876877f20afd9c6b2023-11-11T06:27:31ZengUniversity of KashanMathematics Interdisciplinary Research2476-49652016-01-011118719810.22052/mir.2016.1392313923An Extension of Poincare Model of Hyperbolic Geometry with Gyrovector Space ApproachMahfouz Rostamzadeh0Sayed-Ghahreman Taherian1Department of Mathematics, University of Kurdistan, P. O. Box 416 Sanandaj, I. R. IranDepartment of Mathematical Sciences, Isfahan University of Technology, 84156 Isfahan, I R IranThe aim of this paper is to show the importance of analytic hyperbolic geometry introduced in [9]. In [1], Ungar and Chen showed that the algebra of the group SL(2, C) naturally leads to the notion of gyrogroups and gyrovector spaces for dealing with the Lorentz group and its underlying hyperbolic geometry. They defined the Chen addition and then Chen model of hyperbolic geometry. In this paper, we directly use the isomorphism properties of gyrovector spaces to recover the Chen's addition and then Chen model of hyperbolic geometry. We show that this model is an extension of the Poincaré model of hyperbolic geometry. For our purpose we consider the Poincaré plane model of hyperbolic geometry inside the complex open unit disc D. Also we prove that this model is isomorphic to the Poincaré model and then to other models of hyperbolic geometry. Finally, by gyrovector space approach we verify some properties of this model in details in full analogue with Euclidean geometry.https://mir.kashanu.ac.ir/article_13923_45adafb5f2e2797a3bb789d969d94705.pdfhyperbolic geometrygyrogroupgyrovector spacepoincar'e modelanalytic hyperbolic geometry |
| spellingShingle | Mahfouz Rostamzadeh Sayed-Ghahreman Taherian An Extension of Poincare Model of Hyperbolic Geometry with Gyrovector Space Approach Mathematics Interdisciplinary Research hyperbolic geometry gyrogroup gyrovector space poincar'e model analytic hyperbolic geometry |
| title | An Extension of Poincare Model of Hyperbolic Geometry with Gyrovector Space Approach |
| title_full | An Extension of Poincare Model of Hyperbolic Geometry with Gyrovector Space Approach |
| title_fullStr | An Extension of Poincare Model of Hyperbolic Geometry with Gyrovector Space Approach |
| title_full_unstemmed | An Extension of Poincare Model of Hyperbolic Geometry with Gyrovector Space Approach |
| title_short | An Extension of Poincare Model of Hyperbolic Geometry with Gyrovector Space Approach |
| title_sort | extension of poincare model of hyperbolic geometry with gyrovector space approach |
| topic | hyperbolic geometry gyrogroup gyrovector space poincar'e model analytic hyperbolic geometry |
| url | https://mir.kashanu.ac.ir/article_13923_45adafb5f2e2797a3bb789d969d94705.pdf |
| work_keys_str_mv | AT mahfouzrostamzadeh anextensionofpoincaremodelofhyperbolicgeometrywithgyrovectorspaceapproach AT sayedghahremantaherian anextensionofpoincaremodelofhyperbolicgeometrywithgyrovectorspaceapproach AT mahfouzrostamzadeh extensionofpoincaremodelofhyperbolicgeometrywithgyrovectorspaceapproach AT sayedghahremantaherian extensionofpoincaremodelofhyperbolicgeometrywithgyrovectorspaceapproach |