Common lot

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The common perpendicular lies on the minimal transversal of two skewed straight lines in space . It is a segment whose endpoints lie on one of the two straight lines and which runs perpendicular to both straight lines. Their length indicates the shortest distance between these two straight lines.

The common perpendicular can (in the three-dimensional case) be determined using methods of analytical geometry as follows:

The straight lines and are given by the parametric equations ${\displaystyle g}$ ${\displaystyle h}$

${\displaystyle g\colon {\vec {X}}={\vec {A}}+\lambda {\vec {u}};\qquad h\colon {\vec {X}}={\vec {B}}+\mu {\vec {v}}}$.

If the direction vectors and , for example, the cross product of these vectors are a normal vector , then the approach corresponds ${\displaystyle {\vec {n}}}$ ${\displaystyle {\vec {u}}}$ ${\displaystyle {\vec {v}}}$

${\displaystyle {\vec {A}}+\lambda {\vec {u}}+\nu {\vec {n}}={\vec {B}}+\mu {\vec {v}}}$

a system of linear equations that can be solved for , and . Inserting these parameter values ​​into the equations of the straight lines and yields the position vectors of the two base points of the common perpendicular and thus its equation. ${\displaystyle \lambda }$ ${\displaystyle \mu }$ ${\displaystyle \nu }$ ${\displaystyle g}$ ${\displaystyle h}$