Contact time of translating line segments ($D+sU-rT+t(V-W)=0$)

For segments with initial direction vectors $U,T$, initial endpoint displacement $D$, and translation velocities $V,W$, contact is feasibility of the displayed linear equation with $s,r\in [0,1]$ and $t\ge 0$. Minimizing $t$ is a small linear programming problem. A nonsingular three-column coefficient matrix gives at most one candidate; singular, parallel and degenerate cases require the constrained feasibility problem rather than division by a vanishing determinant.

The earliest approach within distance $d$ is a convex optimization problem with a norm constraint and the same segment and time bounds as contact. An elementary alternative enumerates whether each segment parameter is free or at an endpoint. In each nonsingular active case the free closest-point parameters are affine functions of time, leaving a quadratic distance inequality on a valid time interval. Rank-deficient cases need their null-space constraints retained. Unlike contact, the two segment parameters and time are no longer determined by a vector equality.