If you want the points to be evenly distributed across the surface, you need to do this in two steps:
- Determine which face to put the point on. Each face needs to be picked with a relative probability of the area of that face.
- Once the face is determined, pick a random point on the face. This can easily be done by scaling barycentric 0…1 coordinates for “u” and “v” along the two face basis (edge) vectors.