# Metric of space of plane curve

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?

Thanks.

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If you're willing to have all your curves be parametrized over $[0,1]$. Then you can use $d(f,g)=\int_0^1 d(f(t), g(t))\, dt$. –  Avi Steiner Feb 25 '13 at 22:15
Also, your metric for directed lines segments does in fact work. –  Avi Steiner Feb 25 '13 at 22:20
Thanks for the reply (and the positive confirmation)! Can you also give some links for me to explore? –  Pui Feb 25 '13 at 22:27
what's your mathematical background? –  Avi Steiner Feb 25 '13 at 22:30
I took a course in differential geometry ~30 years ago. I need a metric for plane curves for my programming project. –  Pui Feb 25 '13 at 22:35