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?
(This post continues where we left off in Part 2 and gives a more direct solution than Part 3a. However this post uses multivariable/vector calculus, and I do not provide many explanations for readers unfamiliar with that material.)
As we saw in Part 2, the problem with picking our spherical angles and uniformly was that the area on our two-dimensional grid changed when we transformed that surface into the sphere in three dimensions. In particular if we wanted to calculate the probability mass in a region of our spherical angles, we have the formula from Part 0:
where is our probability density function. However when we move to the sphere our area elements are no longer little rectangles with area ; they are little patches on the surface of a sphere. What area do these corresponding patches have?
(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).
(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?