By Abraham A. Ungar

This can be the 1st booklet on analytic hyperbolic geometry, absolutely analogous to analytic Euclidean geometry. Analytic hyperbolic geometry regulates relativistic mechanics simply as analytic Euclidean geometry regulates classical mechanics. The booklet offers a unique gyrovector area method of analytic hyperbolic geometry, totally analogous to the well known vector house method of Euclidean geometry. A gyrovector is a hyperbolic vector. Gyrovectors are equivalence periods of directed gyrosegments that upload based on the gyroparallelogram legislation simply as vectors are equivalence periods of directed segments that upload in accordance with the parallelogram legislation. within the ensuing "gyrolanguage" of the booklet one attaches the prefix "gyro" to a classical time period to intend the analogous time period in hyperbolic geometry. The prefix stems from Thomas gyration, that's the mathematical abstraction of the relativistic impression referred to as Thomas precession. Gyrolanguage seems to be the language one must articulate novel analogies that the classical and the trendy during this ebook share.The scope of analytic hyperbolic geometry that the booklet offers is cross-disciplinary, related to nonassociative algebra, geometry and physics. As such, it's evidently suitable with the unique idea of relativity and, fairly, with the nonassociativity of Einstein pace addition legislation. besides analogies with classical effects that the publication emphasizes, there are outstanding disanalogies besides. therefore, for example, in contrast to Euclidean triangles, the edges of a hyperbolic triangle are uniquely decided by way of its hyperbolic angles. dependent formulation for calculating the hyperbolic side-lengths of a hyperbolic triangle when it comes to its hyperbolic angles are provided within the book.The ebook starts off with the definition of gyrogroups, that is totally analogous to the definition of teams. Gyrogroups, either gyrocommutative and nongyrocommutative, abound in crew concept. unusually, the doubtless structureless Einstein pace addition of detailed relativity seems to be a gyrocommutative gyrogroup operation. Introducing scalar multiplication, a few gyrocommutative gyrogroups of gyrovectors develop into gyrovector areas. The latter, in flip, shape the environment for analytic hyperbolic geometry simply as vector areas shape the atmosphere for analytic Euclidean geometry. by means of hybrid recommendations of differential geometry and gyrovector areas, it's proven that Einstein (Möbius) gyrovector areas shape the surroundings for Beltrami-Klein (Poincaré) ball types of hyperbolic geometry. ultimately, novel functions of Möbius gyrovector areas in quantum computation, and of Einstein gyrovector areas in particular relativity, are offered.

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**Example text**

10. (111) The PV space model of hyperbolic geometry (also called the Ungar model, a term coined by Jing-Ling Chen in 2001 [Chen and Ungar (2001)l) is governed by PV gyrovector spaces where PV addition plays a role analogous to the role that vector addition plays in vector spaces. The geodesics of this model (gyrolines) are Euclidean hyperbolas with asymptotes that intersect 18 Analytic Hyperbolic Geometry at the space origin, Fig. 12. PV addition turns out to be the “proper velocity” addition of proper velocities in special relativity.

Surprisingly, the map gyr[a, b] turns out to be an automorphism of the Mobius groupoid. We recall that a map 4 : R: 4 R: of the groupoid (R:, @) is an automorphism if it is bijective (that is, one-to-one) and preserves the groupoid binary operation, that is, $(a@b)= q5(a)@$(b). Analytic Hyperbolic Geometry 12 Fig. 6 A gyrotriangle uvw in the Poincark disc model of hyperbolic geometry (that is, in the Mobius gyrovector plane (iR:,@,@)) is shown with the gyromidpoints muv, mu, and mvw of its sides, its gyromedians urnvw,vm,, and wm,,, and its gyrocen,, .

Hence, by (l),a 0 = a for all a E G so that 0 is a right identity. (6) Suppose 0 and O* are two left identities, one of which, say 0, is also a right identity. Then 0 = O* 0 = O*. (7) Let z be a left inverse of a. Then z ( a z) = (z a ) gyr[z, a]z =0 z = z = z 0, by left gyroassociativity, (G2), (3), (5), and (6) above. By (1) we have a z = 0 so that z is a right inverse of + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + a. (8) Suppose z and y are left inverses of a. By (7) above, they are also right inverses, so a z = 0 = a y.