Basic Primitive Tests - Real-Time Collision Detection - page 129

void ClosestPtPointSegment(Point c, Point a, Point b, float &t, Point &d)
{

Vector ab=b-a;
// Project c onto ab, but deferring divide by Dot(ab, ab)
t = Dot(c - a, ab);
if (t <= 0.0f) {
// c projects outside the [a,b] interval, on the a side; clamp to a
t = 0.0f;
d=a;
} else {
float denom = Dot(ab, ab); // Always nonnegative since denom = ||ab|| ∧ 2

if (t >= denom) {
// c projects outside the [a,b] interval, on the b side; clamp to b
t = 1.0f;
d=b;
} else {
// c projects inside the [a,b] interval; must do deferred divide now
t=t/denom;
d=a+t*ab;

}

}

}

The same basic method can be used for finding the closest point on a ray or the
closest point on a line. For a ray, it is necessary to clamp t only when it goes negative.
For a line, there is no need to clamp t at all.

5.1.2.1 Distance of Point to Segment

The squared distance between a point C and a segment AB can be directly computed
without explicitly computing the point D on AB closest to C . As in the preceding
section, there are three cases to consider. When AC · AB ≤ 0, A is closest to C and

the squared distance is given by AC · AC . When AC · AB ≥ AB · AB , B is closest to C

and the squared distance is BC · BC . In the remaining case, 0 < AC · AB < AB · AB ,

·

the squared distance is given by CD CD , where

AC · AB

D = A +

AB AB .

AB ·

However, because the expression CD · CD simplifies to

2

− (

AC · AB )

AC · AC ,

·

AB AB