Transversals to Line Segments in Three-Dimensional Space

We completely describe the structure of the connected components of transversals to a collection of n line segments in R. Generically, the set of transversal to four segments consist of zero or two lines. We catalog the non-generic cases and show that n > 3 arbitrary line segments in R admit at most n connected components of line transversals, and that this bound can be achieved in certain configurations when the segments are coplanar, or they all lie on a hyperboloid of one sheet. This implies a tight upper bound of n on the number of geometric permutations of line segments in R.