*(This post continues where we left off in Parts 3a and 3b and provides an alternative solution to the problem that is more useful in practice.)*

We saw that using spherical coordinates to generate random points on the surface of a sphere required us to transform one of our angles to obtain a uniform distribution. In Part 3a we performed this transformation with the help of map projections, and in Part 3b we found it directly by using calculus. What would happen if we wanted to generalize this procedure to higher dimensions?