Zur Möbiusschen Geometrie und Kinematik in \(H^ 3\). (Möbius geometry and kinematics in \(H^ 3)\). (German) Zbl 0695.51006

Jankovský, Zdeněk

Apl. Mat. 34, No. 6, 453-465 (1989).

Let the half-space \(H^ 3=\{(x_ 1,x_ 2,x_ 3)\in {\mathbb{R}}^ 3\); \(x_ 3>0\}\) be represented by quaternions of the type \(q=u+vj\), where \(v\in {\mathbb{R}}\) and \(v>0\) [cf. H. Karzel and G. Kist, NATO ASI Ser., Ser. C 160, 437-509 (1985; Zbl 0598.51012)].
The author defines a group \(M(H^ 3)\) by means of the group SL(2,C) acting transitively on \(H^ 3\) by \[ m(\xi)=(\alpha \xi +\beta)(\gamma \xi +\delta)^{-1},\quad where\quad \left( \begin{matrix} \alpha \\ \gamma \end{matrix} \begin{matrix} \beta \\ \delta \end{matrix} \right)\in SL(2,C)\quad and\quad \xi \in H^ 3. \] A one-parameter family of such transformations with the real parameter \(t\in I\subset {\mathbb{R}}\) represents a Möbius motion in \(H^ 3\). Then some algebraic (e.g. cross-ratios) and kinematic (e.g. instantaneous centres) invariants are investigated.

Reviewer: M.Uscki

MSC:

51B10	Möbius geometries
53A17	Differential geometric aspects in kinematics

Keywords:

Möbius geometry; kinematics in the half-space \(H^ 3\); i-pole; i- velocity

Citations:

Zbl 0598.51012

References:

[1]	L. V. Ahlfors: Möbius transformations in several dimensions. University of Minnesota 1981 · Zbl 0517.30001
[2]	W. Benz: Vorlesungen über Geometrie der Algebren. Springer-Verlag, Berlin, Heidelberg, New York, 1973. · Zbl 0258.50024
[3]	Z. Jankovský: Zu einigen Fragen der ebene kinematische Geometrie auf der \(\mathcal M\)-Gruppe. Acta polytechnica-Práce ČVUT v Praze, 7 (IV, 3), 1978, 43-51
[4]	Z. Jankovský: Zu den Möbiusschen Feldern der i-Geschrindigkeiten. Acta polytechnica-Práce ČVUT v Praze, 17 (IV, 2), 1980, 91-105