The Experts below are selected from a list of 54 Experts worldwide ranked by ideXlab platform
Keiichi Watanabe - One of the best experts on this subject based on the ideXlab platform.
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Finitely generated Gyrovector subSpaces and orthogonal gyrodecomposition in the Möbius Gyrovector Space
2017Co-Authors: Toshikazu Abe, Keiichi WatanabeAbstract:In this study, we show that any finitely generated Gyrovector subSpace in the Möbius Gyrovector Space coincides with the intersection of the vector subSpace generated by the same generators and the Möbius ball. As an application, we present a notion of orthogonal gyrodecomposition and clarify the relationship with the orthogonal decomposition.
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Orthogonal Gyroexpansion in Möbius Gyrovector Spaces
2017Co-Authors: Keiichi WatanabeAbstract:We investigate the Möbius Gyrovector Spaces which are open balls centered at the origin in a real Hilbert Space with the Möbius addition, the Möbius scalar multiplication, and the Poincaré metric introduced by Ungar. In particular, for an arbitrary point, we can easily obtain the unique closest point in any closed Gyrovector subSpace, by using the ordinary orthogonal decomposition. Further, we show that each element has the orthogonal gyroexpansion with respect to any orthogonal basis in a Möbius Gyrovector Space, which is similar to each element in a Hilbert Space having the orthogonal expansion with respect to any orthonormal basis. Moreover, we present a concrete procedure to calculate the gyrocoefficients of the orthogonal gyroexpansion.
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a confirmation by hand calculation that the mobius ball is a Gyrovector Space
2016Co-Authors: Keiichi WatanabeAbstract:We give a confirmation that the Mobius ball of any real inner product Space is a Gyrovector Space by using only elementary hand calculation. Some remarks to [2] will also be made.
Abraham A. Ungar - One of the best experts on this subject based on the ideXlab platform.
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A Gyrovector Space Approach to Hyperbolic Geometry
2009Co-Authors: Abraham A. UngarAbstract:The mere mention of hyperbolic geometry is enough to strike fear in the heart of the undergraduate mathematics and physics student. Some regard themselves as excluded from the profound insights of hyperbolic geometry so that this enormous portion of human achievement is a closed door to them. The mission of this book is to open that door by making the hyperbolic geometry of Bolyai and Lobachevsky, as well as the special relativity theory of Einstein that it regulates, accessible to a wider audience in terms of novel analogies that the modern and unknown share with the classical and familiar. These novel analogies that this book captures stem from Thomas gyration, which is the mathematical abstraction of the relativistic effect known as Thomas precession. Remarkably, the mere introduction of Thomas gyration turns Euclidean geometry into hyperbolic geometry, and reveals mystique analogies that the two geometries share. Accordingly, Thomas gyration gives rise to the prefix "gyro" that is extensively used in the gyrolanguage of this book, giving rise to terms like gyrocommutative and gyroassociative binary operations in gyrogroups, and Gyrovectors in Gyrovector Spaces. Of particular importance is the introduction of Gyrovectors into hyperbolic geometry, where they are equivalence classes that add according to the gyroparallelogram law in full analogy with vectors, which are equivalence classes that add according to the parallelogram law. A gyroparallelogram, in turn, is a gyroquadrilateral the two gyrodiagonals of which intersect at their gyromidpoints in full analogy with a parallelogram, which is a quadrilateral the two diagonals of which intersect at their midpoints. Table of Contents: Gyrogroups / Gyrocommutative Gyrogroups / Gyrovector Spaces / Gyrotrigonometry
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From the Group SL(2, C) to Gyrogroups and Gyrovector Spaces and Huperbolic Geometry
2007Co-Authors: Jingling Chen, Abraham A. UngarAbstract:this paper is to present a natural way in which the algebra of the SL(2; C) group leads to gyrogroups and Gyrovector Spaces. This natural way convincingly demonstrates that the theory of gyrogroups and Gyrovector Spaces provides a most powerful formalism for dealing with the Lorentz group and hyperbolic geometry, the geometry that governs the special theory of relativity as well as other areas of physics (see, for instance, [9] and [10]). It is therefore hoped that, following this article, gyrogroup and Gyrovector Space theoretic techniques will provide standard tools in the study of relativity physics and, as such, will become part of the lore learned by all explorers who are interested in relativity physics. Links between gyrogroups and other mathematical objects are presented in [11] [12] [13] and [14]. Furthermore, our approach to gyrogroups and scalar multiplication in a gyrogroup of Gyrovectors can be used as a preparation for the study of a related, but more abstract study of Sabinin's odules in [15]. A related study of quasigroups in differential geometry is presented by Sabinin and Miheev on pp. 357 -- 430 of [16]. 6 2 THE ALGEBRA OF THE SL(2; C) GROUP Let R 3 c be the set of all relativistically admissible velocities, R 3 c = fv 2 R 3 : kvk < cg It is the ball of radius c, c > 0, of the Euclidean 3-Space R 3 , c being the vacuum speed of light. A boost L(v) is a pure Lorentz transformation, that is, a Lorentz transformation without rotation, parametrized by a velocity parameter v 2 R 3 c . The boost L(v) is a linear transformation of Spacetime coordinates which has the matrix representation Lm (v), v = (v 1 ; v 2 ; v 3 ) t , Lm (v) = 0 B B B B B B @ v c 2 v v 1 c 2 v v 2 c 2 v v 3 v v 1 1 + c 2 2 v v +1 v 2 1 c ..
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beyond the einstein addition law and its gyroscopic thomas precession the theory of gyrogroups and Gyrovector Spaces
2001Co-Authors: Abraham A. UngarAbstract:List of Figures. List of Tables. Preface. Acknowledgments. Introduction A.A. Ungar. 1. Thomas Precession: The Missing Link. 2. Gyrogroups: Modeled on Einstein's Addition. 3. The Einstein Gyrovector Space. 4. Hyperbolic Geometry of Gyrovector Spaces. 5. The Ungar Gyrovector Space. 6. The Mobius Gyrovector Space. 7. Gyrogeometry. 8. Gyrooperations -- The SL(2,C) Approach. 9. The Cocycle Form. 10. The Lorentz Group and its Abstraction. 11. The Lorentz Transformation Link. 12. Other Lorentz Groups. 13. References. About the Author. Topic Index. Author Index.
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Hyperbolic trigonometry and its application in the Poincare ball model of hyperbolic geometry
2001Co-Authors: Abraham A. UngarAbstract:Hyperbolic trigonometry is developed and illustrated in this article along lines parallel to Euclidean trigonometry by exposing the hyperbolic trigonometric law of cosines and of sines in the Poincare ball model of n-dimensional hyperbolic geometry, as well as their application. The Poincare ball model of three-dimensional hyperbolic geometry is becoming increasingly important in the construction of hyperbolic browsers in computer graphics. These allow in computer graphics the exploitation of hyperbolic geometry in the development of visualization techniques. It is, therefore, clear that hyperbolic trigonometry in the Poincare ball model of hyperbolic geometry, as presented here, will prove useful in the development of efficient hyperbolic browsers in computer graphics. Hyperbolic trigonometry is governed by Gyrovector Spaces in the same way that Euclidean trigonometry is governed by vector Spaces. The capability of Gyrovector Space theory to capture analogies and its powerful elegance is thus demonstrated once more.
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The Einstein Gyrovector Space
2001Co-Authors: Abraham A. UngarAbstract:In this chapter we introduce scalar multiplication ⊗E in the Einstein gyrogroup (Vc,⊕E), obtaining the Einstein Gyrovector Space (Vc, ⊕E, ⊗E). This, in turn, results in the emergence of the hyperbolic analytic geometry of the Einstein Gyrovector Space, which turns out to be the familiar Beltrami ball model of hyperbolic geometry. The ball Vc is equipped with the coordinates it inherits from its real inner product Space V, relative to which Gyrovectors are represented. We close the chapter with the observation that the unique hyperbolic ‘straight line’ called a geodesic, passing through two given points a, b ∈ Vc is the set of all points $$a{{\oplus }_{E}}\left( {{\ominus }_{E}}a{{\oplus }_{E}}b \right){{\otimes }_{E}}t $$ of Vc, t ∈ ℝ, ⊖Ea = −a, which is analogous to its counterpart in Euclidean analytic geometry.
Toshikazu Abe - One of the best experts on this subject based on the ideXlab platform.
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Finitely generated Gyrovector subSpaces and orthogonal gyrodecomposition in the Möbius Gyrovector Space
2017Co-Authors: Toshikazu Abe, Keiichi WatanabeAbstract:In this study, we show that any finitely generated Gyrovector subSpace in the Möbius Gyrovector Space coincides with the intersection of the vector subSpace generated by the same generators and the Möbius ball. As an application, we present a notion of orthogonal gyrodecomposition and clarify the relationship with the orthogonal decomposition.
Ungar A.a. - One of the best experts on this subject based on the ideXlab platform.
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Hyperbolic trigonometry and its application in the Poincaré ball model of hyperbolic geometry
2001Co-Authors: Ungar A.a.Abstract:AbstractHyperbolic trigonometry is developed and illustrated in this article along lines parallel to Euclidean trigonometry by exposing the hyperbolic trigonometric law of cosines and of sines in the Poincaré ball model of n-dimensional hyperbolic geometry, as well as their application. The Poincaré ball model of three-dimensional hyperbolic geometry is becoming increasingly important in the construction of hyperbolic browsers in computer graphics. These allow in computer graphics the exploitation of hyperbolic geometry in the development of visualization techniques. It is, therefore, clear that hyperbolic trigonometry in the Poincaré ball model of hyperbolic geometry, as presented here, will prove useful in the development of efficient hyperbolic browsers in computer graphics. Hyperbolic trigonometry is governed by Gyrovector Spaces in the same way that Euclidean trigonometry is governed by vector Spaces. The capability of Gyrovector Space theory to capture analogies and its powerful elegance is thus demonstrated once more
Sayedghahreman Taherian - One of the best experts on this subject based on the ideXlab platform.
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an extension of poincare model of hyperbolic geometry with Gyrovector Space approach
2016Co-Authors: Mahfouz Rostamzadeh, Sayedghahreman TaherianAbstract: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 $SL(2,mathbb C)$ 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 Poincar'e model of hyperbolic geometry. For our purpose we consider the Poincar'e plane model of hyperbolic geometry inside the complex open unit disc $mathbb{D}$. Also we prove that this model is isomorphic to the Poincar'e 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.