onsegment

Onsegment is a geometric predicate used to determine whether a point lies on a closed line segment AB. A point P is on segment AB if P lies on the line through A and B and is between A and B, with endpoints included. Equivalently, there exists a parameter t in [0,1] such that P = A + t(B − A). This condition applies in both two- and three-dimensional space.

A practical way to test on-segment is to first check collinearity of A, B, and P, and then verify that P lies within the inclusive bounding box determined by A and B. In coordinates, this means min(ax, bx) ≤ px ≤ max(ax, bx) and min(ay, by) ≤ py ≤ max(ay, by) in 2D (and similarly for additional dimensions). A common numerical criterion uses the cross product (or the area of triangle ABP) to test collinearity and the dot product condition (P − A) · (P − B) ≤ 0 to ensure P lies between A and B. A degenerate segment where A = B contains only that single point; then P lies on the segment if and only if P = A.

In practice, on-segment tests appear in polygon processing, segment-intersection algorithms, computer graphics, and geographic information systems. Robust implementations may separate the collinearity check from range checks and often employ exact arithmetic or tolerance thresholds to handle floating-point coordinates and numerical errors. Related concepts include on_line (P lies on the infinite line AB) and between (P lies between A and B along a line, without necessarily requiring collinearity).