  We study contact interactions for long world-lines on a curved surface,
focusing on the average number of times two world-lines intersect as a function
of their end-points. The result can be used to extend the concept of
path-ordering, as employed in the Wilson loop, from a closed curve into the
interior of a surface spanning the curve. Taking this surface as a string
world-sheet yields a generalisation of the string contact interaction
previously used to represent the Abelian Wilson loop as a tensionless string.
We also describe a supersymmetric generalisation.
