Hausdorff distance
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The Hausdorff distance, or Hausdorff metric, measures how far two compact non-empty subsets of a metric space are from each other. It is named after Felix Hausdorff.
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 the closed r-neighborhood of X contains Y and the closed r-neighborhood of Y contains X. In other words, if |xy| denotes the distance in M, then
- <math> d_{\mathrm H}(X,Y) = \max\{\sup_{x \in X} \inf_{y \in Y} |xy|, \sup_{y \in Y} \inf_{x \in X} |xy|\}\mbox{.} \! </math>
This distance function turns the set of all compact non-empty subsets of M into a metric space, say F(M). The topology of F(M) depends only on the topology of M. If M is compact, then so is F(M).
Hausdorff distance can be defined the same way for closed not-necessarily-compact subsets of M, but in this case the distance may take infinite values, and the topology of F(M) starts to depend on particular metric on M (not only on its topology). The Hausdorff distance between not-necessarily-closed subsets can be defined as the Hausdorff distance between their closures. It gives a pre-metric (or pseudometric) on the set of all subsets of M (the Hausdorff distance between any two sets with the same closure is zero).
In Euclidean geometry, one often uses an 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
es:Distancia de Hausdorff fr:Distance de Hausdorff ru:Метрика Хаусдорфа zh:豪斯多夫距离