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 d_H(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 d(x, y) denotes the distance in M, then

[ d_(X,Y) = \max\ \inf_ d(x,y),\, \sup_ \inf_ d(x,y)\,\}\mbox \! ]

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 D_H(X,Y) is the minimum of d_H(I(X),Y) along all isometries I of Euclidean space. This distance measures how far X and Y are from being isometric.

Hausdorff distance

Definitions

See also