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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

Citations:

Zbl 0271.55012

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