CONFIGURATIONS OF CYCLES
AND THE APOLLONIUS PROBLEM

BORUT JURCIC ZLOBEC AND
NEZA MRAMOR KOSTA

Abstract:

Given $n+1$ spheres and planes of dimension $n-1$ in ${\bf R}^n$, the Apollonius problem is to find a common tangent sphere or plane, and the generalized Apollonius problem is to find a sphere or plane intersecting them under prescribed angles. In Lie geometry, an Apollonius problem is given by an $(n+1)$-frame of points on the Lie quadric $\Omega\subset{\bf P}^n+2$. The solutions are described as the intersections of the projective line determined by the orthogonal complement to this frame with respect to the Lie product in ${\cal R}^{n+3}$ and the quadric. Two special points span this line, and the connection between the position of these two points and the existence and geometric properties of the solutions of the Apollonius problem are described.