(Despite the title, this is the second post in the series. Part 0 covered the definition of uniform distributions in discrete and continuous space. If this last sentence makes no sense, go back and read Part 0.)
Instead of tackling the problem of picking a point on a sphere right away, let’s consider the easier problem of picking a point on a circle. We will then extend this method to the harder problem. (Mathematicians like to do this.)
(Note that this topic has been covered in detail by several well-known websites. While their material is sound, I feel that some of the interesting details in this problem are obscured by multivariate calculus. In this series of posts, I will attempt to explain this problem assuming limited prior mathematical knowledge. Readers with a good grasp of probability will want to skip past the preliminaries.)
In this series of posts we will answer the question: how do we choose a point uniformly at random on the surface of a sphere? (Think: how do we pick a random place on a globe without bias to any particular location(s)?)
Before we get to the solution, we need to address a few basics. First consider the phrase “uniformly at random.” What does this mean?