geometry - How do I find the points of intersection between a small circle and a great circle?

I only actually need to intersect a great circle arc i.e. the shortest path between two points, with a given small circle.

I found this tutorial for intersecting two great circle arcs, but unfortunately it doesn't cover small circles.

Answer

Here is a full algebraic solution for two arbitrary small circles, using Cartesian coordinates to simplify the maths.

It's assumed the earth is a spherical, but that's fine for my purposes.