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

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


51B10 Möbius geometries
53A17 Differential geometric aspects in kinematics


Zbl 0598.51012
Full Text: EuDML


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