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

By Abraham Albert Ungar

This publication introduces for the 1st time the hyperbolic simplex as an immense inspiration in n-dimensional hyperbolic geometry. The extension of universal Euclidean geometry to N dimensions, with N being any confident integer, ends up in higher generality and succinctness in comparable expressions. utilizing new mathematical instruments, the publication 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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Z = 0. 53) with w = z, gyr[u, v]z = z. 63) we say that the gyration axis in Rn of the gyration gyr[u, v] : Rn → Rn, generated by u, v ∈ Rns, 38 Analytic Hyperbolic Geometry in N Dimensions is parallel to the vector z. 65) x  0, for any coefficients cu, cv ∈ R, excluding cu = cv = 0. 65). Moreover, we have the following result. 7 (Gyration–Thomas Precession Angle). Let u, v, x ∈ Rns be relativistically admissible velocities such that u  −v (so that u⊕v  0). 66) Proof. 22), pp. 53). 31), p. 29, coincide.

87. 45) x2 + y 2 < s of 2-dimensional relativistically admissible velocities, equipped with the Cartesian coordinate system Σ = (x, y). 46) x b , so that 0 x ⊕ b b = γb |x|. 6 Einstein Addition vs. 48) Rns is neither commutative for all u, v, w ∈ R . In contrast, Einstein addition, ⊕, in nor associative. 49) 34 Analytic Hyperbolic Geometry in N Dimensions for all u, v, w ∈ Rs3. 49) presents the application to w of the gyration gyr[u, v] generated by u and v. Gyrations turn out to be automorphisms of the Einstein groupoid (Rs3, ⊕).

Part III: Hyperbolic Triangles and Circles. Part III of the book, Chapters 8–9, employs the tools developed in Part II for the discovery of properties of hyperbolic triangles (gyrotriangles) and hyperbolic circles (gyrocircles). Several important, well-known results in Euclidean geometry are translated into corresponding results in hyperbolic geometry. Thus, for instance, a) the Inscribed Angle Theorem; b) the Tangent–Secant Theorem, p. 319; c) the Intersecting Secants Theorem, p. 320; and d) the Intersecting Chords Theorem, p.