Abstract
Barycentric coordinates are commonly used in Euclidean geometry. Following the adaptation of barycentric coordinates for use in hyperbolic geometry in recently published books on analytic hyperbolic geometry, known and novel results concerning triangles and circles in the hyperbolic geometry of Lobachevsky and Bolyai are discovered. Among the novel results are the hyperbolic counterparts of important theorems in Euclidean geometry. These are: (i) the Inscribed Gyroangle Theorem, (ii) the Gyrotangent–Gyrosecant Theorem, (iii) the Intersecting Gyrosecants Theorem, and (iv) the Intersecting Gyrochord Theorem. Here in gyrolanguage, the language of analytic hyperbolic geometry, we prefix a gyro to any term that describes a concept in Euclidean geometry and in associative algebra to mean the analogous concept in hyperbolic geometry and nonassociative algebra. Outstanding examples are gyrogroups and gyrovector spaces, and Einstein addition being both gyrocommutative and gyroassociative. The prefix “gyro” stems from “gyration,” which is the mathematical abstraction of the special relativistic effect known as “Thomas precession.”
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References
Crowe, M.J.: A History of Vector Analysis. Dover Publications Inc., New York (1994). (The Evolution of the Idea of a Vectorial System, Corrected reprint of the 1985 edition)
Einstein, A.: Zur Elektrodynamik Bewegter Körper [on the electrodynamics of moving bodies]. Ann. Physik. (Leipzig) 17, 891–921 (1905). (We use the English translation in [3] or in [8], or in http://www.fourmilab.ch/etexts/einstein/specrel/www/)
Einstein, A.: Einstein’s Miraculous Years: Five Papers that Changed the Face of Physics. Princeton University Press, Princeton (1998). Edited and introduced by John Stachel. Includes bibliographical references. Einstein’s dissertation on the determination of molecular dimensions, Einstein on Brownian motion, Einstein on the theory of relativity, Einstein’s early work on the quantum hypothesis. A new English translation of Einstein’s 1905 paper on pp. 123–160
Fock, V.: The Theory of Space, Time and Gravitation. The Macmillan Co., New York (1964). (Second revised edition. Translated from the Russian by Kemmer, N. A Pergamon Press Book)
Gray, J.: Möbius’s geometrical mechanics. In: Fauvel, J., Flood, R., Wilson, R. (eds.) Möbius and His Band, Mathematics and Astronomy in Nineteenth-Century Germany, 78–103. The Clarendon Press Oxford University Press, New York (1993)
Kelly, P.J., Matthews, G.: The Non-Euclidean, Hyperbolic Plane. Its Structure and Consistency. Springer-Verlag, New York (1981). (Universitext)
Kimberling, C.: Clark Kimberling’s Encyclopedia of Triangle Centers—ETC (2012). http://faculty.evansville.edu/ck6/encyclopedia/ETC.html
Lorentz, H.A., Einstein, A., Minkowski, H., Weyl, H.: The Principle of Relativity. Dover Publications Inc., New York (1923) (undated). A collection of original memoirs on the special and general theory of relativity, with notes by A. Sommerfeld, Translated by W. Perrett and G.B. Jeffery
Maor, E.: Trigonometric Delights. Princeton University Press, Princeton (1998)
Millman, R.S., Parker, G.D.: Geometry: A Metric Approach with Models, 2nd edn. Springer-Verlag, New York (1991)
Møller, C.: The Theory of Relativity. Clarendon Press, Oxford (1952)
Rassias, T.M.: Book review: analytic hyperbolic geometry and Albert Einstein’s special theory of relativity, by Abraham A. Ungar. Nonlinear Funct. Anal. Appl. 13(1), 167–177 (2008)
Rassias, T.M.: Book review: A gyrovector space approach to hyperbolic geometry, by Abraham A. Ungar. J. Geom. Symm. Phys. 18, 93–106 (2010)
Sexl, R.U., Urbantke, H.K.: Relativity, Groups, Particles. Special Relativity and Relativistic Symmetry in Field and Particle Physics. Springer-Verlag, Vienna (2001). (Revised and translated from the third German (1992) edition by Urbantke)
Sönmez, N., Ungar, A.A.: The Einstein relativistic velocity model of hyperbolic geometry and its plane separation axiom. Adv. Appl. Clifford Alg. 23, 209–236 (2013)
Ungar, A.A.: Quasidirect product groups and the Lorentz transformation group. In: Rassias, T.M. (ed.) Constantin Carathéodory: An international Tribute, vol. I, II, pp. 1378–1392. World Sci. Publishing, Teaneck (1991)
Ungar, A.A.: Gyrovector spaces in the service of hyperbolic geometry. In: Rassias, T.M. (ed.) Mathematical Analysis and Applications, pp. 305–360. Hadronic Press, Palm Harbor (2000)
Ungar, A.A.: Möbius transformations of the ball, Ahlfors` rotation and gyrovector spaces. In: Rassias, T.M. (ed.) Nonlinear Analysis in Geometry and Topology, pp. 241–287. Hadronic Press, Palm Harbor (2000)
Ungar, A.A.: Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces, vol. 117 of Fundamental Theories of Physics. Kluwer Academic, Dordrecht (2001)
Ungar, A.A.: Analytic Hyperbolic Geometry: Mathematical Foundations and Applications. World Scientific Publishing Co. Pte. Ltd., Hackensack (2005)
Ungar, A.A.: The hyperbolic square and Möbius transformations. Banach J. Math. Anal. 1(1), 101–116 (2007)
Ungar, A.A.: Analytic Hyperbolic Geometry and Albert Einstein’s Special Theory of Relativity. World Scientific Publishing Co, Pte. Ltd., Hackensack (2008)
Ungar, A.A.: A Gyrovector Space Approach to Hyperbolic Geometry. Morgan & Claypool, San Rafael (2009)
Ungar, A.A.: The hyperbolic triangle incenter. Dedicated to the 30th anniversary of Themistocles M. Rassias` stability theorem. Nonlinear Funct. Anal. Appl. 14(5), 817–841 (2009)
Ungar, A.A.: Barycentric Calculus in Euclidean and Hyperbolic Geometry: A Comparative Introduction. World Scientific Publishing Co, Pte. Ltd., Hackensack (2010)
Ungar, A.A.: Hyperbolic Triangle Centers: The Special Relativistic Approach. Springer-Verlag, New York (2010)
Ungar, A.A.: Gyrations: the missing link between classical mechanics with its underlying Euclidean geometry and relativistic mechanics with its underlying hyperbolic geometry. In: Pardalos, P.M., Rassias, T.M. (eds.) Essays in Mathematics and its Applications in Honor of Stephen Smale’s 80th Birthday, pp. 463–504. Springer, Heidelberg (2012). arXiv 1302.5678 (math-ph)
Ungar, A.A.: Möbius transformation and Einstein velocity addition in the hyperbolic geometry of Bolyai and Lobachevsky. In: Pardalos, P.M., Georgiev, P.G., Srivastava, H.M. (eds.) Nonlinear Analysis, vol. 68 of Springer Optim. Appl., pp. 721–770. Springer, New York (2012). Stability, Approximation, and Inequalities. In honor of Themistocles M. Rassias on the occasion of his 60th birthday
Ungar, A.A.: An introduduction to hyperbolic barycentric coordinates and their applications. In: Pardalos, P., Rassias, T.M. (eds) Mathematics Without Boundaries: Surveys in Interdisciplinary Research. Springer Optim. Appl. Springer, New York (2013) (in print)
Walter, S.: Book review: beyond the Einstein addition law and its gyroscopic Thomas precession: the theory of gyrogroups and gyrovector spaces, by Abraham A. Ungar. Found. Phys. 32(2), 327–330 (2002)
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Ungar, A. (2014). On the Study of Hyperbolic Triangles and Circles by Hyperbolic Barycentric Coordinates in Relativistic Hyperbolic Geometry. In: Rassias, T., Pardalos, P. (eds) Mathematics Without Boundaries. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1106-6_22
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