Coaxial Wireframe

thingiverse

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 defines the first family. To get the second family, you take the planes through each of the two generating points that are tangent to the sphere. When the generating points are not antipodal (on opposite poles) these two planes will intersect at some line L. Now take any plane that contains L and intersects the sphere. It will intersect the sphere at a circle. The set of intersections of all such planes through L with the sphere defines the second family. If the two points are antipodal (on opposite poles), then L is a line at infinity and the two planes are parallel. The set of planes through L is then defined to be all parallel planes. Notice that in this case, the two families of circles are the longitude and latitude lines of the sphere. Moreover, any Apollonian family generated as above can be found by applying a Mobius transformation to the longitude and latitude lines in the sphere which takes the North pole to the first generating point and the South pole to the second generating point.