Random points inside a parallelogram

Question

I have a 4 side convex Polygon defined by 4 points in 2D, and I want to be able to generate random points inside it.

If it really simplifies the problem, I can limit

Answer 1

This works for general, convex quadrilaterals:

You can borrow some concepts from the Finite Element Method, specifically for quadrilateral (4-sided) elements (refer to section 16.5 here). Basically, there is a bilinear parameterization that maps a square in u-v space (for u, v \in [-1, 1] in this case) to your quadrilateral that consists of points p_i (for i = 1,2,3,4). Note that In the provided reference, the parameters are called \eta and \xi.

Basic recipe:

1. Choose a suitable random number generator to generate well-distributed points in a square 2D domain
2. Generate random u-v pairs in the range [-1, 1]
3. For each u-v pair, the corresponding random point in your quad = 1/4 * ((1-u)(1-v) * p_1 + (1+u)(1-v) * p_2 + (1+u)(1+v) * p_3 + (1-u)(1+v) * p_4)

The only problem is that uniformly distributed points in the u-v space won't produce uniformly distributed points in your quad (in the Euclidean sense). If that is important, you can work directly in 2D within the bounding box of the quad and write a point-in-quad (maybe by splitting the problem into two point in tris) test to cull random points that are outside.