On algebraic structures related to Beltrami-Klein model of hyperbolic geometry
In [\textit{H. Karzel}, Discrete Math. 208--209, 387--409 (1999; Zbl 0941.51002)], the algebraization of a point reflection structure is exhibited; it yields the loop of derivation at the reference point \(o\). In the present paper Karzel's procedure is executed especially for the Beltrami--Klein model of the hyperbolic plane; this is done in the more general frame of an algebra \((\mathbb K,\mathbb R)\) where \(\mathbb R\) denotes the reals and \(\mathbb K\) stands beside the complex numbers \(\mathbb C\) also for the quaternions \(\mathbb H\) or the octonions \(\mathbb O\). For \({\mathbb D}:=\{z\in{\mathbb K} \mid 1>|z|\}\) the author determines the ``relativity addition \(\oplus\)'' constructively and by formula. The author shows that \(({\mathbb D},\oplus)\) is a K-loop which is neither commutative nor associative but satisfies the Bol-identity. The paper is accompanied by six helpful figures.
scientific article; zbMATH DE number 5725185