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Unformatted text preview: ircles, the distance
should be positive for r > 0 and should go to 0 as r → 0. How can we do this? 35 2.2. THE HAUSDORFF METRIC The general notion of distance between elements of a set X – think of X as
the set of points in the plane, the set of curves in the plane, the set of compact
sets in the plane or in space – is captured by the general deﬁnition of a metric.
A metric d on the set X is a function d : X × X → [0, ∞) possessing three
properties. For all x, y, z ∈ X ,
(i) d(x, y ) ≥ 0, and d(x, y ) = 0 if and only if x = y (positivedeﬁniteness),
(ii) d(x, y ) = d(y, x) (symmetry), and
(iii) d(x, y ) ≥ d(x, z ) + d(z, y ) (triangle inequality).
The Hausdorﬀ metric is a metric on X = K(R2 ), the set of compact subsets of the plane. Deﬁning the Hausdorﬀ metric requires using the noton of
ǫthickening.
For any compact set A and any number ǫ ≥ 0, the ǫthickening of A is
Aǫ = {(x, y ) ∈ R2 : d((x, y ), (x′ , y ′ )) ≤ ǫ for some point (x′ , y ′ ) ∈ A} where d is t...
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 Fall '14
 AmandaFolsom
 Geometry, Fractal Geometry, Limits, The Land

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