By Abraham Albert Ungar
This ebook introduces for the 1st time the hyperbolic simplex as a tremendous notion in n-dimensional hyperbolic geometry. The extension of universal Euclidean geometry to N dimensions, with N being any optimistic integer, leads to better generality and succinctness in similar expressions. utilizing new mathematical instruments, the ebook demonstrates that this can be 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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Additional resources for Analytic Hyperbolic Geometry in N Dimensions : An Introduction
As such, gyroalgebra is used extensively in the book in the study of analytic hyperbolic geometry in n dimensions. 2. Part II: Mathematical Tools for Hyperbolic Geometry. Part II of the book, Chapters 5–7, presents the adaptation of classical tools that are commonly used in Euclidean geometry for use in hyperbolic geometry. Specifically, the classical tools are: a) Cartesian coordinates (in Euclidean geometry); b) Barycentric coordinates; c) trigonometry; and d) vector algebra, 16 Analytic Hyperbolic Geometry in N Dimensions and their respective hyperbolic counterparts are: a) Cartesian coordinates (in hyperbolic geometry); b) gyrobarycentric coordinates; c) gyrotrigonometry; and d) gyrovector gyroalgebra.
66) Proof. 22), pp. 53). 31), p. 29, coincide. Special attention to three dimensional gyrations, which are of interest in physical applications, is paid in Chapter 13 in the study of Thomas precession. 8 From Einstein Velocity Addition to Gyrogroups Guided by analogies with groups, the key features of Einstein groupoids (Rns, ⊕), n = 1, 2, 3, . , suggest the formal gyrogroup definition in which gyrogroups form a most natural generalization of groups. Accordingly, definitions related to groups and gyrogroups follow.
10) that signaled the emergence of the link between hyperbolic geometry and special relativity. It was first studied by Sommerfeld  and Varičak [139, 140] in terms of rapidities, a term coined by Robb . 14) s then the gamma factor γv of v ∈ Rns is related to the rapidity ϕv of v by cosh ϕv = γv ||v|| sinh ϕv = γv . 15) s The gamma identity plays in hyperbolic geometry a role analogous to the role that the law of cosines plays in Euclidean geometry, as we will see in Sect. 3, p. 218. 10) formed the first link between special relativity and the hyperbolic geometry of Lobachevsky and Bolyai.
Analytic Hyperbolic Geometry in N Dimensions : An Introduction by Abraham Albert Ungar