Shortest distance between a point and a line segment

Given a line segment from point \(\mathbf{A}\) to point \(\mathbf{B}\), what is the shortest distance to a point \(\mathbf{P}\)?

Above the line itself, the shortest distance is the length from point \(\mathbf{P}\) to the point orthogonal on the line; everywhere else the shortest distance is between point \(\mathbf{P}\) and \(\mathbf{A}\) or \(\mathbf{B}\) respectively. Now the scalar projection \(t\) of vector \(\mathbf{a} = \mathbf{AP}\) onto \(\mathbf{b} = \mathbf{AB}\) is

\[t = |\mathbf{a}|\cos\theta = |\mathbf{a}|\underbrace{\frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|\cdot|\mathbf{b}|}}_{\cos\theta}=\frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|}\]

Scaling the projected length \(t\) by the length \(|\mathbf{b}|\) gives a ratio \(\hat{t}\) between 0 and 1 of the projection on the line \(\mathbf{AB}\):

\[\hat{t} = \frac{t}{|\mathbf{b}|} = \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|\cdot |\mathbf{b}|} =\frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|^2}=\frac{\mathbf{a}\cdot\mathbf{b}}{\mathbf{b}\cdot\mathbf{b}}\]

Now \(\mathbf{R} = \mathbf{A} + \hat{t}(\mathbf{B} - \mathbf{A}) = \mathbf{A} + \hat{t}\mathbf{b}\) is the point on the line, and by clamping \(\hat{t}\) strictly into the interval \([0, 1]\) it is guaranteed to land on the segment \(\overline{AB}\) rather than its infinite extension. The final result is then simply \(|\mathbf{P} - \mathbf{R}|\).

This clamped \(\hat{t}\) is exactly the same quantity used to test whether a point lies between two other points: a value below 0 means the closest point is \(\mathbf{A}\), a value above 1 means it is \(\mathbf{B}\), and anything in between is the orthogonal foot on the segment itself. Since the whole derivation only relies on dot products, it carries over unchanged to 3D or any higher dimension — only the vector components change, not the formula.

Implementation

Implemented in JavaScript this gives

function minLineDist(A, B, P) {
    const a = {
        x: P.x - A.x,
        y: P.y - A.y
    };
    const b = {
        x: B.x - A.x,
        y: B.y - A.y
    };
    const bb = b.x * b.x + b.y * b.y;

    if (0 === bb) {
        return Math.hypot(a.x, a.y);
    }

    const ab = a.x * b.x + a.y * b.y;
    const t = Math.max(0, Math.min(1, ab / bb));

    const R = {
        x: A.x + t * b.x,
        y: A.y + t * b.y
    };
    return Math.hypot(P.x - R.x, P.y - R.y);
}