How to find a random point in a quadrangle?

Question

I have to be able to set a random location for a waypoint for a flight sim. The maths challenge is straightforward:

\"To find a single random location within a quadrangl

Answer 1

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

Answer 2

This is an interesting problem and there's probably as really interesting answer, but in case you just want it to work, let me offer you something simple.

Here's the algorithm:

1. Pick a random point that is within the rectangle that bounds the quadrangle.
2. If it is not within the quadrangle (or whatever shape), repeat.
3. Profit!

edit

I updated the first step to mention the bounding box, per Bart K.'s suggestion.

Answer 3

So, this time tackling how to figure out if a point is within the quad:

The four edges can be expressed as lines in y = mx + b form. Check if the point is above or below each of the four lines, and taken together you can figure out if it's inside or outside.

Answer 4

The "brute force" approach is simply to loop through until you have a valid coordinate. In pseudocode:

left   = min(pa.x, pb.x, pc.x, pd.x)
right  = max(pa.x, pb.x, pc.x, pd.x)
bottom = min(pa.y, pb.y, pc.y, pd.y)
top    = max(pa.y, pb.y, pc.y, pd.y)
do {
    x = left   + fmod(rand, right-left)
    y = bottom + fmod(rand, top-bottom)
} while (!isin(x, y, pa, pb, pc, pd));

You can use a stock function pulled from the net for "isin". I realize that this isn't the fastest-executing thing in the world, but I think it'll work.

Answer 5

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).

Answer 6

Are you allowed to just repeatedly try anywhere within the rectangle which bounds the quadrangle, until you get something within the quad? Might this even be faster than some fancy algorithm to ensure that you pick something within the quad?

Incidentally, in that problem statement, I think the use of the word "find" is confusing. You can't really find a random value that satisfies a condition; the randomizer just gives it to you. What you're trying to do is set parameters on the randomizer to give you values matching certain criteria.