An Extension of Poincare Model of Hyperbolic Geometry with Gyrovector Space Approach

Document Type : Special Issue: AIMC 51


1 Department of Mathematics, University of Kurdistan, P. O. Box 416 Sanandaj, I. R. Iran

2 Department of Mathematical Sciences, Isfahan University of Technology, 84156 Isfahan, I R Iran


‎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.


Main Subjects

  1. J. Chen, A. A. Ungar, From the group SL(2; C) to gyrogroups and gyrovector spaces and hyperbolic geometry, Found. Phys. 31 (11) (2001) 1611–1639.
  2. 2. H. Karzel, Recent developments on absolute geometries and algebraization by K-loops, Discrete Math. 208/209 (1999) 387–409.
  3. 3. ‎‎H‎. ‎Karzel‎, ‎M‎. ‎Marchi‎, ‎S‎. ‎Pianta‎, ‎The defect in‎ an invariant reflection structure‎, ‎J‎. ‎Geom.‎ 99 (1) (2010) 67-87‎.
  4. 4. S. -Gh. Taherian, On algebraic structures related to Beltrami-Klein model of hyperbolic geometry, Results Math. 57 (2010) 205–219.
  5. 5. M. Rostamzadeh, S. -Gh. Taherian, Defect and area in Beltrami-Klein model of hyperbolic geometry, Results Math. 63 (1-2) (2013) 229–239.
  6. 6. M. Rostamzadeh, S. -Gh. Taherian, On trigonometry in Beltrami-Klein model of hyperbolic geometry, Results Math. 65 (3-4) (2014) 361–369.
  7. 7. ‎A. A‎. ‎Ungar‎, ‎Thomas rotation and the parametrization of the Lorentz‎ transformation group‎, Found‎. ‎Phys‎. ‎Lett. 1 (1) (1988) 57-89‎.
  8. 8. A. A. Ungar, Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces, Kluwer Acad. Publ., Dordrecht, 2001.
  9. 9. A. A. Ungar, Analytic Hyperbolic Geometry: Mathematical Foundations and Applications, World Scientific, Singapore, 2005.
  10. 10. ‎A. A‎. ‎Ungar‎, Analytic Hyperbolic Geometry‎ and Albert Einstein's Special Theory of Relativity‎, World Scientific Publishing Co‎. ‎Pte‎. ‎Ltd.‎, ‎Hackensack‎, ‎NJ‎, ‎2008‎.
  11. 11. A. A. Ungar, A Gyrovector Space Approach to Hyperbolic Geometry, Morgan & Claypool Pub., San Rafael, California, 2009.
  12. 12. A. A. Ungar, Hyperbolic Triangle Centers: The Special Relativistic Approach, Springer-Verlag, New York, 2010.
  13. 13. A. A. Ungar, Barycentric Calculus in Euclidean and Hyperbolic Geometry: A Comparative Introduction, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010.
  14. 14. ‎A. A‎. ‎Ungar‎, Analytic Hyperbolic Geometry in n Dimensions‎: ‎An introduction‎, CRC Press‎, ‎Boca Raton‎, ‎FL‎, 2015‎.