Center of shortest connection between two lines in 3D

Question

I have two skew lines:

$g_l: \overrightarrow{x}=\overrightarrow{a}+r\overrightarrow{u}$

$g_r: \overrightarrow{x}=\overrightarrow{b}+s\overrightarrow{v}$

How can I calculate the center point of the shortest connection of these lines $g_l$ and $g_r$?

Answer 1

The method is as follows.

Write $$\overrightarrow{w}=\overrightarrow{a}-\overrightarrow{b}+r\overrightarrow{u}-s\overrightarrow{v}$$

This is the vector joining two points, one on each of the lines.

Form and solve a simultaneous equation for $r$ and $s$ by setting $$\overrightarrow{w}\cdot\overrightarrow{u}=0$$ and $$\overrightarrow{w}\cdot\overrightarrow{v}=0$$

This ensures that the line joining is perpendicular to both given lines.

Then the position vector of the centre of the shortest line, using the values of $r$ and $s$ obtained, is $$\frac 12(\overrightarrow{a}+\overrightarrow{b}+r\overrightarrow{u}+s\overrightarrow{v})$$