Analytic Hyperbolic Geometry in N Dimensions : An by Abraham Albert Ungar PDF

By Abraham Albert Ungar

ISBN-10: 1482236680

ISBN-13: 9781482236682

This ebook introduces for the 1st time the hyperbolic simplex as an incredible proposal in n-dimensional hyperbolic geometry. The extension of universal Euclidean geometry to N dimensions, with N being any confident integer, leads to better generality and succinctness in similar expressions. utilizing new mathematical instruments, the booklet demonstrates that this is often additionally the case with analytic hyperbolic geometry. for instance, the writer analytically determines the hyperbolic circumcenter and circumradius of any hyperbolic simplex.

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This e-book introduces for the 1st time the hyperbolic simplex as a huge proposal in n-dimensional hyperbolic geometry. The extension of universal Euclidean geometry to N dimensions, with N being any optimistic integer, ends up in higher generality and succinctness in comparable expressions. utilizing new mathematical instruments, the ebook demonstrates that this is often additionally the case with analytic hyperbolic geometry.

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1. The barycentric coordinate representation of the tetrahedron circumcenter is not determined directly. Rather, it is extracted from the gyrobarycentric coordinate representation of the gyrotetrahedron circumgyrocenter. 5 Parts of the Book The book is self-contained. The required background in the theory of gyrogroups and gyrovector spaces and in gyrotrigonometry is presented in Parts I and II. More about these topics is found in [122, 129, 133, 134]. The book is divided into six parts: 1. Part I: Einstein Gyrogroups and Gyrovector Spaces.

49), p. 69) for all u, v ∈ Rns where, by definition, 2⊗v = v⊕v. 23), p. 84), p. 194. 10 First Gyrogroup Properties While it is clear how to define a right identity and a right inverse in a gyrogroup, the existence of such elements is not presumed. Indeed, the existence of a unique identity and a unique inverse, both left and right, is a consequence of the gyrogroup axioms, as the following theorem shows, along with other immediate results about gyrogroups. 17 (First Gyrogroup Properties). Let (G, ⊕) be a gyrogroup.

8, and a gyrovector space, as we will see in Sect. 3. In the formal approach to gyrogroups in Def. 14, p. 39, the left reduction property is elevated to the Reduction Axiom. The gyration left reduction axiom is also known as the left loop property. The more revealing term, reduction axiom, was coined by F. 107), p. 47. 49) in terms of Einstein addition. 53) 1 γv {γ (γ + 1)(u·w) + (γu − 1)γv (v·w)} s2 γv + 1 u v D = γu γv (1 + u·v ) + 1 = γu⊕v + 1 ≥ 2 s2 for all u, v, w ∈ Rns. 5 (Gyration Domain Extension).

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Analytic Hyperbolic Geometry in N Dimensions : An Introduction by Abraham Albert Ungar


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