Motivation
The definition of the Hausdorff distance can be derived by a series of natural extensions of the distance function d(x, y) in the underlying metric space M, as follows:
- Define a distance function between any point x of M and any non-empty set Y of M by:
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- .
- For example, d(1, ) = 2 and d(7, ) = 1.
- Define a distance function between any two non-empty sets X and Y of M by:
-
- .
- For example, d(, ) = d(1, ) = 2.
- If X and Y are compact then d(X,Y) will be finite; d(X,X)=0; and d inherits the triangle inequality property from the distance function in M. As it stands, d(X,Y) is not a metric because d(X,Y) is not always symmetric, and d(X,Y) = 0 does not imply that X = Y (It does imply that ). For example, d(, ) = 2, but d(, ) = 0. However, we can create a metric by defining the Hausdorff distance to be:
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