I am looking for a metric $d$ for smooth 2D curves. Hence $d(x,y)$ is the distance between the curves x and y. For the moment, we may assume that $x$ and $y$ are just directed line segments. Do you think the sum of distances between the corresponding points would work (for the said restricted kind of curves). And in general, how to do it?
Thus, does the definition $d(AB,CD) = d'(A,C)+d'(B,D)$ works for directed line segments $AB$, $CD$ (where $d'$ is a metric of points in the plane)?
Is there any example? Where can I find out more details?