Suppose I have the following scenario:

[![enter image description here][1]][1]

And I am aware of an algorithm to draw uniformly from (in this case) the 2-sphere. Does this same algorithm readily extend to the situation where I randomly take a "cut/slice" (shown in red above) and what points uniformly over this disk?

Further, suppose now that I have a uniform random sampling algorithm over an arbitrary Reimannian manifold. Would the same principles necessarily apply?

  [1]: https://i.sstatic.net/iyK0B.png