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Unformatted text preview: he Euclidean distance between points. For example, the ǫthickening
of A = (x0 , y0 ) is the closed disc with center (x0 , y0 ) and radius ǫ. The ǫthickening of the unit circle is the annulus with inner radius 1 − ǫ and outer
radius 1 + ǫ, provided ǫ < 1, and is the disc of radius 1 + ǫ if ǫ > 1. The
ǫthickening of a line segment is a rectangle with semidisc caps. See Fig. 2.14. ε
Figure 2.14: ǫthickenings of a point, circle, and line segment. The size of ǫ is
indicated on the left.
Using ǫthickenings, we can deﬁne a notion of distance between compact
sets. Given compact sets A and B , the Hausdorﬀ distance between A and B is
h(A, B ) = inf {ǫ : A ⊆ Bǫ and B ⊆ Aǫ } (2.3) Before using the Hausdorﬀ distance to illustrate IFS convergence, we note a
common mistake in computing Hausdorﬀ distaces. Take A = {(0, y ) : 0 ≤ y ≤
1/2} and B = {(x, 0) : 0 ≤ x ≤ 1}. Certainly, B ⊆ A1 and A ⊆ B1/2 . So
because 1/2 < 1, h(A, B ) = 1/2. No, of course not. The Hausdorﬀ distance
bet...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.
 Fall '14
 AmandaFolsom
 Geometry, Fractal Geometry, Limits, The Land

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