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

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.

### MSC:

 51N15 Projective analytic geometry 51N05 Descriptive geometry

### Keywords:

cyclography; Rabl mapping; Apollonius-type problem

Zbl 0271.55012

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