# Find the line that is closest to 4 skew lines

If I have 4 skew lines in $$\mathbb{R}^3$$, how can I find the line $$L_c$$, that is closest to all of them?

I know that with 3 skew lines, there is always a line that intersects all of them, in fact infinite many:

• Construct a plane with $$L_1$$ and and arbitrary point on $$L_2$$,
• intersect the plane with $$L_3$$ to get a second point,
• the line through both points intersects $$L_1$$,$$L_2$$ and $$L_3$$

With 4 skew lines, there is at most two intersecting lines (I think, though I have not found a way to construct that yet). Assuming there is none, how do I find the line that is closest to them all?

This is related to a previous question by me: I am trying to find the axis for a surface of rotation, given 4 random points on the surface and some measurement error.