Coaxial Wireframe

This is a wireframe of a coaxial family of circles and its orthogonal family of circles on the sphere (also known as Apollonian circles). The circles are constructed as surfaces of revolution in such a way that each is the intersection of a sphere with a hollowed out cone.

What is remarkable about Apollonian circles like these is that every intersection of any two circle from one family with any circle from the other is at right angles. If you look closely you can see that the two families of circles are (1) a family which all intersect each other at two particular points on the sphere and (2) a family of circles that (informally) surround the two points that generate the other family.

More precisely, the two families can be generated using the following construction.

First, select two points on the sphere which we will call the generating points. Any plane that passes through those two points intersects the sphere at a circle. The set of all circles defined in this way defi