Shortest distance between a point and a line segment

distancegeometrylanguage-agnosticline-segment

I need a basic function to find the shortest distance between a point and a line segment. Feel free to write the solution in any language you want; I can translate it into what I'm using (Javascript).

EDIT: My line segment is defined by two endpoints. So my line segment AB is defined by the two points A (x1,y1) and B (x2,y2). I'm trying to find the distance between this line segment and a point C (x3,y3). My geometry skills are rusty, so the examples I've seen are confusing, I'm sorry to admit.

Best Answer

Eli, the code you've settled on is incorrect. A point near the line on which the segment lies but far off one end of the segment would be incorrectly judged near the segment. Update: The incorrect answer mentioned is no longer the accepted one.

Here's some correct code, in C++. It presumes a class 2D-vector class vec2 {float x,y;}, essentially, with operators to add, subract, scale, etc, and a distance and dot product function (i.e. x1 x2 + y1 y2).

float minimum_distance(vec2 v, vec2 w, vec2 p) {
  // Return minimum distance between line segment vw and point p
  const float l2 = length_squared(v, w);  // i.e. |w-v|^2 -  avoid a sqrt
  if (l2 == 0.0) return distance(p, v);   // v == w case
  // Consider the line extending the segment, parameterized as v + t (w - v).
  // We find projection of point p onto the line. 
  // It falls where t = [(p-v) . (w-v)] / |w-v|^2
  // We clamp t from [0,1] to handle points outside the segment vw.
  const float t = max(0, min(1, dot(p - v, w - v) / l2));
  const vec2 projection = v + t * (w - v);  // Projection falls on the segment
  return distance(p, projection);
}

EDIT: I needed a Javascript implementation, so here it is, with no dependencies (or comments, but it's a direct port of the above). Points are represented as objects with x and y attributes.

function sqr(x) { return x * x }
function dist2(v, w) { return sqr(v.x - w.x) + sqr(v.y - w.y) }
function distToSegmentSquared(p, v, w) {
  var l2 = dist2(v, w);
  if (l2 == 0) return dist2(p, v);
  var t = ((p.x - v.x) * (w.x - v.x) + (p.y - v.y) * (w.y - v.y)) / l2;
  t = Math.max(0, Math.min(1, t));
  return dist2(p, { x: v.x + t * (w.x - v.x),
                    y: v.y + t * (w.y - v.y) });
}
function distToSegment(p, v, w) { return Math.sqrt(distToSegmentSquared(p, v, w)); }

EDIT 2: I needed a Java version, but more important, I needed it in 3d instead of 2d.

float dist_to_segment_squared(float px, float py, float pz, float lx1, float ly1, float lz1, float lx2, float ly2, float lz2) {
  float line_dist = dist_sq(lx1, ly1, lz1, lx2, ly2, lz2);
  if (line_dist == 0) return dist_sq(px, py, pz, lx1, ly1, lz1);
  float t = ((px - lx1) * (lx2 - lx1) + (py - ly1) * (ly2 - ly1) + (pz - lz1) * (lz2 - lz1)) / line_dist;
  t = constrain(t, 0, 1);
  return dist_sq(px, py, pz, lx1 + t * (lx2 - lx1), ly1 + t * (ly2 - ly1), lz1 + t * (lz2 - lz1));
}

Here, in the function parameters, <px,py,pz> is the point in question and the line segment has the endpoints <lx1,ly1,lz1> and <lx2,ly2,lz2>. The function dist_sq (which is assumed to exist) finds the square of the distance between two points.

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