Define a fish-bowl as a sphere comprised between two horizontal disks. That is, a sphere where we have replaced the top and bottom sectors by horizontal disks. See picture below.
How to sample uniformly from this surface?
My approach is to:
- Choose either the diskless bowl or one of the disk with probability proportional to its surface.
- Then, sample uniformly from this surface.
- For the disks, we know how to do it in close form.
- For the sphere with truncated sectors, I do rejection sampling by sampling from the sphere and rejecting until the sampled point is not in one of the removed sectors.
If my computation is correct, a surface area wholly comprised within a disk or within the truncated sphere has same probability. But what about the boundaries? My intuition is that the points neighbors to the circle of contact between a disk and the sphere won't be uniformly distributed.