 Contact time of translating line segments

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.