An analog of the Apollonian circle on the Lobachevski plane and on the sphere (Q5941862)

For points \(A,B\in E^2\) and any \(k>1\), the set \(K=\{M\subset E^2:\frac{|MB|}{|MA|}=k\}\) is known to be the Apollonian circle of \(A\) and \(B\). The authors describe the analogue of \(K\) in the Lobachevski plane (which is a closed analytic curve bounding a strictly convex set containing \(A\)), and they continue with a similar construction on the unit sphere \(S^2\). In this spherical case the respective curve can be nonconvex and may contain a singular point. On the other hand a condition is presented under which the convexity of this ``Apollonian curve'' is assured. From these considerations it follows that there are two compacta \(A,B\subset S^n\), \(n\geq 2\), such that \(A\) contains more than one point, \(B\) is nonempty, and there exist more than one extremal spherical belts separating \(A\) from \(B\) and having the maximum possible ratio of exterior and interior radii.