(This post continues where we left off in Part 2.)
I have split the solution into two posts. First I show how we could use results from cartography to solve our problem. Then I show how to derive those same results directly using some calculus. While I try to explain the calculus, I realize that I may not be able to reach everyone who is unfamiliar with the concepts. I hope that the first part at least can give all readers a sense of the solution in what I think is a more relatable way than the standard treatment.
(This post continues where we left off in Part 1.)
In Part 1 we devised a method for choosing a point uniformly at random on a circle. When we tried to extend this to a sphere we saw that the chosen points were clumped together instead of spread out evenly upon the surface (see Figure 1).