  Let S be a compact surface with boundary and F be the set of the orbits of a
traversing flow on S. If the flow is generic, its orbit space is a spine G of
S, namely G is a graph embedded in S and S is a regular neighbourhood of G.
Moreover an extra structure on G turns it into a flow-spine, from which one can
reconstruct S and F. In this paper we study properly immersed curves C in S. We
do this by considering generic C's and their apparent contour relative to F,
namely the set of points of G corresponding to orbits that either are tangent
to C, or go through a self-intersection of C, or meet the boundary of C. We
translate this apparent contour into a decoration of G that allows one to
reconstruct C, and then we allow C to vary up to homotopy within a fixed
generic F, and next also F to vary up to homotopy, and we identify a finite set
of local moves on decorated graphs that translate these homotopies.
