How to find a random point in a quadrangle?

I believe there are two suitable ways to solve this problem.

The first mentioned by other posters is to find the smallest bounding box that encloses the rectangle, then generate points in that box until you find a point which lies inside the rectangle.

  Find Bounding box (x,y,width, height)
  Pick Random Point x1,y1 with ranges [x to x+width] and [y to y+height]
  while (x1 or y1 is no inside the quadrangle){
       Select new x1,y1
  }

Assuming your quadrangle area is Q and the bounding box is A, the probability that you would need to generate N pairs of points is 1-(Q/A)^N, which approaches 0 inverse exponentially.

I would reccommend the above approach, espesially in two dimensions. It is very fast to generate the points and test.

If you wanted a gaurentee of termination, then you can create an algorithm to only generate points within the quadrangle (easy) but you must ensure the probablity distribution of the points are even thoughout the quadrangle.

http://mathworld.wolfram.com/TrianglePointPicking.html

Gives a very good explination

Split your quadrangle into two triangles and then use this excellent SO answer to quickly find a random point in one of them.

Update:

Borrowing this great link from Akusete on picking a random point in a triangle.

_{(from MathWorld - A Wolfram Web Resource: wolfram.com)}

Given a triangle with one vertex at the origin and the others at positions v₁ and v₂, pick
_{(from MathWorld - A Wolfram Web Resource: wolfram.com)}
where A₁ and A₂ are uniform variates in the interval [0,1] , which gives points uniformly distributed in a quadrilateral (left figure). The points not in the triangle interior can then either be discarded, or transformed into the corresponding point inside the triangle (right figure).