Question

I am looking for an algorithm to generate equally distributed points inside a polygon.

Here is the scenario:

I have a polygon specified by the coordinates of the points at the corners (x, y) for each point. And I have the number of points to generate inside the polygon.

For example lets say I have a polygon containing 5 points: (1, 1) ; (1, 2) ; (2, 3) ; (3, 2) ; and (3, 1)

And I need to generate 20 equally distanced points inside that polygon.

Note: Some polygons may not support equally distributed points, but I’m looking to distribute the points in a way to cover all the region of the polygon with as much consistency as possible. (what i mean is I don’t want a part with a lot more points than another)

Is there an algorithm to do so? or maybe a library

I am working on a C# application, but any language is ok, since I only need the algorithm and I can translate it.

Thanks a lot for any help

Answer 1

The simple approach I use is:

1. Triangulate the polygon. Ear clipping is entirely adequate, as all you need is a dissection of the polygon into a set of non-overlapping triangles.

2. Compute the area of each triangle. Sample from each triangle proportionally to the area of that triangle relative to the whole. This costs only a single uniform random number per sample.

3. Once a point is determined to have come from a given triangle, sample uniformly over the triangle. This is itself easier than you might think.

So really it all comes down to how do you sample within a triangle. This is easily enough done. A triangle is defined by 3 vertices. I’ll call them P1, P2, P3.

1. Pick ANY edge of the triangle. Generate a point (P4) that lies uniformly along that edge. Thus if P1 and P2 are the coordinates of the corresponding end points, then P will be a uniformly sampled point along that edge, if r has uniform distribution on the interval [0,1].

   P4 = (1-r)*P1 + r*P2

2. Next, sample along the line segment between P3 and P4, but do so non-uniformly. If s is a uniform random number on the interval [0,1], then

   P5 = (1-sqrt(s))*P3 + sqrt(s)*P4

r and s are independent pseudo-random numbers of course. Then P5 will be randomly sampled, uniform over the triangle.

The nice thing is it needs no rejection scheme to implement, so long, thin polygons are not a problem. And for each sample, the cost is only in the need to generate three random numbers per event. Since ear clipping is rather simply done and an efficient task, the sampling will be efficient, even for nasty looking polygons or non-convex polygons.