By Abraham Albert Ungar
This publication introduces for the 1st time the hyperbolic simplex as a big proposal 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 booklet 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 info for Analytic Hyperbolic Geometry in N Dimensions : An Introduction
A2N ⎟ ⎟ ⎟ .. ⎟ . ⎠ ... 23) along with its Cayley–Menger determinant, Det MN, where aij2 = || − Ai + Aj||2. Here we use the notation illustrated in Fig. 3. Analogously, in the study of higher dimensional gyrosimplices it proves useful to assign to each (N − 1)-gyrosimplex A1 . . 40), p. 378, ⎞ ⎛ 1 γ12 γ13 . . γ1N ⎟ ⎜ ⎜ γ12 1 γ23 . . γ2N ⎟ ⎟ ⎜ ΓN = ⎜ . 24) ⎟ ⎜ .. ⎝ ⎠ γ1N γ2N γ3N . . 1 along with its gamma determinant, Det ΓN, where γij = γaij = γ || Ai⊕Aj||. Here we use the notation illustrated in Fig.
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. Special attention to three dimensional gyrations, which are of interest in physical applications, is paid in Chapter 13 in the study of Thomas precession.
The Einstein gyroparallelogram law of gyrovector addition. Let A, B, C ∈ Rns be any three points of an Einstein gyrovector space (Rns, ⊕, ⊗), giving rise to the two gyrovectors u = A⊕B and v = A⊕C. Furthermore, let D be a point of the gyrovector space such that ABDC is a gyroparallelogram, that is, D = (B ⊞ C) A by Def. 2, p. 174, of the gyroparallelogram. Then, Einstein coaddition of gyrovectors u and v, u ⊞ v = w, expresses the gyroparallelogram law, where w = A⊕D. Einstein coaddition, ⊞, thus gives rise to the gyroparallelogram addition law of Einsteinian velocities, which is commutative and fully analogous to the parallelogram addition law of Newtonian velocities.
Analytic Hyperbolic Geometry in N Dimensions : An Introduction by Abraham Albert Ungar