By Abraham A. Ungar
This is often the 1st publication 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 e-book provides a singular gyrovector house 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 sessions of directed segments that upload in response to the parallelogram legislation. within the ensuing “gyrolanguage” of the e-book 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 is the mathematical abstraction of the relativistic influence referred to as Thomas precession. Gyrolanguage seems to be the language one must articulate novel analogies that the classical and the trendy during this publication proportion. The scope of analytic hyperbolic geometry that the ebook offers is cross-disciplinary, concerning nonassociative algebra, geometry and physics. As such, it really is evidently suitable with the designated conception of relativity and, rather, with the nonassociativity of Einstein pace addition legislations. in addition to analogies with classical effects that the publication emphasizes, there are extraordinary disanalogies in addition. hence, for example, not like Euclidean triangles, the perimeters of a hyperbolic triangle are uniquely decided by means of its hyperbolic angles. stylish formulation for calculating the hyperbolic side-lengths of a hyperbolic triangle by way of its hyperbolic angles are offered within the booklet. The booklet starts off with the definition of gyrogroups, that's absolutely analogous to the definition of teams. Gyrogroups, either gyrocommutative and nongyrocommutative, abound in crew concept. strangely, the possible structureless Einstein pace addition of targeted 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 environment for analytic Euclidean geometry. through hybrid innovations of differential geometry and gyrovector areas, it truly is proven that Einstein (Möbius) gyrovector areas shape the atmosphere for Beltrami–Klein (Poincaré) ball types of hyperbolic geometry. eventually, novel functions of Möbius gyrovector areas in quantum computation, and of Einstein gyrovector areas in unique relativity, are provided.
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Additional info for Analytic hyperbolic geometry : mathematical foundations and applications
We cannot view it as the “hyperbolic” cosine of a hyperbolic angle since the term “hyperbolic cosine” is already in use in a different sense. Abusing notation, we use the same notation for the trigonometric functions and their gyro-counterparts. Thus, for instance, the gyrocosine function in Fig. 2 is denoted by cos. This notation for the elementary gyrotrigonometric functions cos, sin, tan, etc. is justified since the gyrotrigonometric functions are interrelated by the same identities that interrelate the trigonometric functions.
Let (G, @) be a gyrogroup. 15, gyrotranslations are bijective. 19 For any two elements a, b of a gyrogroup (G, +) and any automorphism A of (G, +), A E Aut(G, +), Agyr[a, b] = gyr[Aa, Ab]A Proof. 48) +) and any automorphism +) we have by the left gyroassociative law, + (Aa Ab) + Agyr[a, b + + + + ] =~ A((a b) gyr[a, bJz) = A(. 48). 20 Let a, b be any two elements of a gyrogroup (G, +) and let A E Aut(G) be an automorphism of G. Then gyrb, bl = gyr[Aa, Abl if and only if the automorphisms A and gyr[a, b] commute.
1 (Binary Operations, Groupoids, and Groupoid in a set S is a function : Automorphisms). A binary operation S x S -+ S . W e use the notation a b to denote +(a, b) for any a , b E S . A groupoid ( S , +) is a nonempty set, S , with a binary operation, +. An automorphism 4 of a groupoid ( S , +) is a bijective (that is, one-to-one) self-map of S , 6 : S 4 S , which preserves its groupoid operation, that is, 4(a b ) = 4(a) 4 ( b ) f o r all a , b E S . + + + + + Groupoids may have identity elements.