  In 1833 Gauss defined the linking number of two disjoint curves in 3-space.
For open curves this double integral over the parameterised curves is
real-valued and invariant modulo rigid motions or isometries that preserve
distances between points, and has been recently used in the elucidation of
molecular structures. In 1976 Banchoff geometrically interpreted the linking
number between two line segments. An explicit analytic formula based on this
interpretation was given in 2000 without proof in terms of 6 isometry
invariants: the distance and angle between the segments and 4 coordinates
specifying their relative positions. We give a detailed proof of this formula
and describe its asymptotic behaviour that wasn't previously studied.
