# Frechet distance between surfaces

1. Aug 2, 2008

### Dedalus

I am wondering whether or not anybody has any ideas of how to visualize and calculate the Frechet distance between two surfaces, or the sets that they encompass.

Let M be an m-dimensional finitely triangulated manifold (with or without boundary). Let f1 and f2 be continuous maps M---->R^n, n>m≥0. The Frechet Distance between maps is defined as
σ_F (f1,f2)=〖inf〗_(α,β) 〖max〗_xϵM∥[f1(α(x)) -f2(β(x))∥(Euclidean norm) where α,β are all the possible homeomorphisms (injective, bi-continuous) from M to M.

I'm particularly interested in calculating the Frechet Distance between two convex sets, one contained in the other. I'm wondering if ther'es a way to consider instead the Hausdorff distance between the surfaces, in which case the maximum epsilon of th epsilon-neighborhood of the outer surface would be the maximum distance from the surface to the center of mass of the set it bounds. I know there's a bound on the Frechet distance in two-space using the hausdorff distance, but it hasn't yet been extended to 3 space.

Thanks for any ideas or direction.