An interview with me on the Wolfram blog: http://blog.wolfram.com/2017/12/14/creating-mathematical-gems-in-the-wolfram-language/
Mapping a rotating circle on the sphere to the plane. Here's the map: first, send (almost every) point on the sphere to the point on the \(z=1\) plane contained in the same line through the origin. Then, invert the plane in the unit circle.
Take the square grid, inverse stereographic project to the sphere, then orthogonally project to the unit disk. Now, before mapping, translate the whole grid by \(-t(1,2)\) as \(t\) varies from \(-1\) to \(1\). This is the result.