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A measure for the dissimilarity of two shapes is given by Hausdorff distance up to isometry, denoted DH. Namely, let X and Y be two compact figures in a metric space M (usually a Euclidean space); then DH(X,Y) is the infimum of dH(I(X),Y) along all isometries I of the metric space M to itself. This distance measures how far the shapes X and Y are from being isometric.
The Gromov–Hausdorff convergence is a related idea: we measure the distance of two metric spaces M and N by taking the infimum of dH(I(M),J(N)) along all isometric embeddings I:M→L and J:N→L into some common metric space L.
Read more about this topic: Hausdorff Distance
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