Hausdorff distance

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Hausdorff distance measures how far two compact subsets of a metric space are from each other.

Definitions

Let X and Y be two compact subsets of a metric space M. Then Hausdorff distance dH(X,Y) is the minimal number r such that closed r-neighborhood of X contains Y and closed r-neighborhood of Y contains X. In other words, if |xy| denotes the distance in M then

<math>d_H(X,Y)=\max\{\sup_{x\in X}\inf_{y\in Y}|xy|,\sup_{y\in Y}\inf_{x\in X}|xy|\}.<math>

This distance function turns the set of all compact subsets of M into a metric space, say F(M). The topology of F(M) depends only on topology of M. If M is compact then so is F(M).

Hausdorff distance can be defined the same way for closed non-compact subsets of M, but in this case the distance may take infinite value and the topology of F(M) starts to depend on particular metric on M (not only on its topology). The Hausdorff distance between not closed subsets can be defined as the Hausdorff distance between its closures. It gives a pre-metric (or pseudometric) on the set of all subsets of M (Hausdorff distance between any two sets and with the same closures is zero).

In Euclidean geometry it is often used its analog, Hausdorff distance up to isometry. Namely let X and Y be two compact figures in a Euclidean space, then DH(X,Y) is the minimum of dH(I(X),Y) along all isometries I of Euclidean space . This distance measures how far X and Y are from being isometric.

See also:

ru:Метрика Хаусдорфа

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