Distance from a point to a polygon

Question

I am trying to determine the distance from a point to a polygon in 2D space. The point can be inside or outside the polygon; The polygon can be convex or concave.

If

Answer 1

Your best bet is to iterate over all the lines and find the minimum distance from a point to a line segment.

To find the distance from a point to a line segment, you first find the distance from a point to a line by picking arbitrary points P1 and P2 on the line (it might be wise to use your endpoints). Then take the vector from P1 to your point P0 and find (P2-P1) . (P0 - P1) where . is the dot product. Divide this value by ||P2-P1||^2 and get a value r.

Now if you picked P1 and P2 as your points, you can simply check if r is between 0 and 1. If r is greater than 1, then P2 is the closest point, so your distance is ||P0-P2||. If r is less than 0, then P1 is the closest point, so your distance is ||P0-P1||.

If 0, then your distance is sqrt(||P0-P1||^2 - (r * ||P2-P1||)^2)

The pseudocode is as follows:

for p1, p2 in vertices:

  var r = dotProduct(vector(p2 - p1), vector(x - p1))
  //x is the point you're looking for

  r /= (magnitude(vector(p2 - p1)) ** 2)

  if r < 0:
    var dist = magnitude(vector(x - p1))
  else if r > 1:
    dist = magnitude(vector(p2 - x))
  else:
    dist = sqrt(magnitude(vector(x - p1)) ^ 2 - (r * magnitude(vector(p2-p1))) ^ 2)

  minDist = min(dist,minDist)