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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.

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