Posted 26 October 2013 - 07:09 AM
If you want a point in the frustum, pick a number v between 0 and the volume of the frustum (Pi*(R2^2*+R1*R2+R1^2)*h/3, I believe). You then need to find the height z at which the volume below z is v; you can do that by using the formula for the volume above, where z plays the role of h and one of the radii is linked to it by a simple relationship, and then solving for z. You then pick a random point in the disk at height z (this should be easy to find).
If you want a point on the frustum, I assume you need to generate a random point on the curvy side of the truncated cone. For that, you reason similarly but picking a random breakpoint for area, not volume. You then pick a random point on the circle at the desired height, which is trivial. If you need to generate points on the limiting disks as well, roll a die at the beginning to pick which of the three pieces the point will come from (of course the probability of each piece should be proportional to its area).
In both cases, I would pick the height first and then pick a point in the slice at that height.