Novel Tools to Determine Hyperbolic Triangle Centers

Abstract

Recently discovered tools to study analytic hyperbolic geometry in terms of analogies with analytic Euclidean geometry are presented and employed. Special attention is paid to the study of two novel hyperbolic triangle centers that we call hyperbolic Cabrera points of a hyperbolic triangle and to the way they descend to their novel Euclidean counterparts. The two novel hyperbolic Cabrera points are the (1) Cabrera gyrotriangle ingyrocircle gyropoint and the (2) Cabrera gyrotriangle exgyrocircle gyropoint. Accordingly, their Euclidean counterparts to which they descend are the two novel Euclidean Cabrera points, which are the (1) Cabrera triangle incircle point and the (2) Cabrera triangle excircle point.

Parametric Realization of the Lorentz Transformation Group in Pseudo-Euclidean Spaces

“…The concept of the gyrogroup emerged from the 1988 study of the parametrization of the Lorentz group in [18]. Presently, the gyrogroup concept plays a universal computational role, which extends far beyond the domain of special relativity, as noted by Chatelin in [1, p. 523] and in references therein and as evidenced, for instance, from [2,4,5,6,7,11,14,15,16] and [12,21,26,28]. In a similar way, the concept of the bi-gyrogroup emerges in this paper from the study of the parametrization of the Lorentz group SO(m, n), m, n ∈ N. Hence, like gyrogroups, bi-gyrogroups are capable of playing a universal computational role that extends far beyond the domain of Lorentz transformations in pseudo-Euclidean spaces.…”