Hausdorff distance
From Wikipedia, the free encyclopedia
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.
[edit] 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 d(x, y) denotes the distance in M, then
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 in the same way for closed subsets of M which are not compact, but in this case the distance may take infinite values, and the topology of F(M) then depends on the metric on M and not only on its topology. The Hausdorff distance between 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.
