How to generate random points on the surface of a quadrilateral

Question

Alright, I'm sure there has to be a simple way to do this but it eludes me at the moment.

I want to be able to generate random points on the surface of a quadrilateral in 3D space. (Defined simply as four points) What is the best way to go about doing this?

If the quad is a rectangle, this is trivial to do with random interpolation between points. However, this won't work for me since my quads aren't guaranteed to have rectangular properties.

Answer 1

Cut the quad into two trianles, get their area size, then first random-pick one of the triangles (based on their areas) and finally pick a random point in the triangle using your favourite standard algorithm for triangles.

Answer 2

You can easily do this with a variant of bilinear interpolation. For example

Point randomInQuad(Point a, b, c, d) {
  double s = random(0.0,1.0);  // uniform in [0,1]
  Point e = s*a + (1-s)*b;
  Point f = s*c + (1-s)*d;
  double t = random(0.0,1.0);
  return t*e + (1-t)*f;
}

You can interpret this as putting each of the vertices of your quad at a corner of a unit square. Then, you randomly pick a point in the unit square. Finally, you use bilinear interpolation (which is a series of affine combinations) to interpolate a vertex at that point.