I am given two metric spaces as two arrays of the same size. Each one is supposed to represent distance between vertices on a mesh in R^3. The meshes are assumed to have the same number of vertices and the correspondence betweeen the vertices is also given. Is there a way to find the a meaningful distance between these two matrices (other than the trivial ones)? I would like the distance be invariant under scaling : in other words if the first mesh's is just a scaling of the other mesh then I want the distance between the two corresponding metric spaces to be zero (even though the distances in the matrices have also been scaled).

I know that Hausdorff distance measures the distance between two sets in the same metric space but note here that my question is a little different so that notion is not exactly useful.

Thanks.

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