Generalized Rabl mappings and Apollonius-type problems. (English) Zbl 1165.51012

Bácsó, Sándor; Szilasi, Zoltán

J. Geom. Graph. 11, No. 1, 27-38 (2007).

The cyclographic mapping, an indispensable tool in sphere geometry, assigns to each point \((x_1,\dots,x_{n-1}, r) \in {\mathbb R}^n\) the oriented \((n-2)\)-sphere with center \((x_1,\dots,x_{n-1})\) and signed radius \(r\). A variant of this mapping, proposed by K. Rabl [Elem. Math. 29, 6–12 (1974; Zbl 0271.55012)] for the case of \(n=3\), maps \((x_1,\dots,x_{n-1},r)\) to the non-oriented \((n-2)\)-sphere with center \((x_1,\dots,x_{n-1})\) and radius \(\sqrt{r}\).
The authors use this generalized Rabl mapping for solving Apollonius-type problems in space: (a) Find a paraboloid of revolution tangent to two spheres in \({\mathbb R}^3\), (b) find a quadric surface tangent to three spheres in \({\mathbb R}^3\) along circles.
The analogous problems in \({\mathbb R}^2\) are solved as well. Constructive solutions are based on methods from descriptive geometry of four-dimensional space or elementary projective geometry. For the spatial case, the authors provide some formulas.

Reviewer: Hans-Peter Schröcker (Innsbruck)

MSC:

51N15	Projective analytic geometry
51N05	Descriptive geometry

Keywords:

cyclography; Rabl mapping; Apollonius-type problem

Citations:

Zbl 0271.55012

Software:

Maple