Abraham A. Ungar's Analytic Hyperbolic Geometry: Mathematical Foundations and PDF

By Abraham A. Ungar

ISBN-10: 9812564578

ISBN-13: 9789812564573

ISBN-10: 9812703276

ISBN-13: 9789812703279

This can be the 1st e-book 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 provides a unique gyrovector house method of analytic hyperbolic geometry, absolutely analogous to the well known vector area method of Euclidean geometry. A gyrovector is a hyperbolic vector. Gyrovectors are equivalence sessions of directed gyrosegments that upload in accordance with the gyroparallelogram legislation simply as vectors are equivalence periods of directed segments that upload based on 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 impact referred to as Thomas precession. Gyrolanguage seems to be the language one must articulate novel analogies that the classical and the fashionable during this e-book share.The scope of analytic hyperbolic geometry that the booklet provides is cross-disciplinary, regarding nonassociative algebra, geometry and physics. As such, it truly is obviously suitable with the particular thought of relativity and, relatively, with the nonassociativity of Einstein pace addition legislations. besides analogies with classical effects that the e-book emphasizes, there are notable disanalogies to boot. hence, for example, in contrast to Euclidean triangles, the perimeters of a hyperbolic triangle are uniquely made up our minds through its hyperbolic angles. based 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 with the definition of gyrogroups, that's totally analogous to the definition of teams. Gyrogroups, either gyrocommutative and nongyrocommutative, abound in workforce thought. strangely, the probably structureless Einstein speed addition of specified 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 surroundings for analytic hyperbolic geometry simply as vector areas shape the environment for analytic Euclidean geometry. by means of hybrid concepts of differential geometry and gyrovector areas, it really is proven that Einstein (Möbius) gyrovector areas shape the surroundings for Beltrami-Klein (Poincaré) ball versions of hyperbolic geometry. ultimately, novel purposes of Möbius gyrovector areas in quantum computation, and of Einstein gyrovector areas in unique relativity, are awarded.

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Extra resources for Analytic Hyperbolic Geometry: Mathematical Foundations and Applications

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31 (The Cogyroautomorphic Inverse Property). 103) (-a) f o r any a , b g G . Proof. 103) we note that by Def. 104) a)} - gyr-'[a, -b]a) - gyr[b, -u]u) (-a)} Inverting both extreme sides we obtain the desired identity. 105) Proof. 105) follows from the first one by replacing b by 4. 106) an a gyrogroup (G,+) are equivalent for any a, X ,y E G. Proof. 106) are symmetric so that it is enough to show that the first equation implies the second. 93), and (v) by a left cancellation again, that as desired.

Following Def. 25) in a gyrogroup (G, @). 14 rogroup (G,+). Proof. (-a The gyropolygonal gyroaddition is associative in any gy- On the one hand + b) @ ((4+ c) @ (-c + d ) } = (-a + b) @ (4 + d ) = -a + d and on the other hand {(-a + b) @ (-b + c)} + (-c + d ) = (-a + C) @ ( -C + d ) = -a + d 0 The gyropolygonal gyrosubtraction is just the gyropolygonal gyroaddition in the reversed direction along the gyropolygonal path. 14 uncovers an associative addition, the gyropolygonal gyroaddition, defined under special circumstances in the nonassociative environment of the gyrogroup.

Thus, for instance, elements of a gyrovector space are called points rather than gyropoints, as they should be called in gyrolanguage. But, “vectors” of a gyrovector space are called gyrovectors since they do not exist classically. Furthermore, we use the terms gyrogeodesics and (hyperbolic) geodesics interchangeably since, for instance, the gyrogeodesics (also called gyrolines) of Mobius gyrovector spaces are nothing else but the familiar geodesics of the Poincark model of hyperbolic geometry.

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Analytic Hyperbolic Geometry: Mathematical Foundations and Applications by Abraham A. Ungar

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