 Shortest distance between skew lines

ID: shortest-distance-between-skew-lines

For lines

a_{1} + t m_{1}

and

a_{2} + u m_{2}

with nonparallel direction vectors, their shortest separation is the absolute orthogonal projection of

a_{2} - a_{1}

onto their common normal

m_{1} \times m_{2}

. Components in the span of

m_{1}, m_{2}

can be cancelled by choosing

t, u

, while the normal component cannot. The same formula gives zero for intersecting nonparallel lines. Parallel lines instead have separation

∣ (a_{2} - a_{1}) \times m_{1} ∣/∣ m_{1} ∣

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