FractionalGeometry-Chap2

The size of is indicated on the left using

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Unformatted text preview: he Euclidean distance between points. For example, the ǫ-thickening 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 ǫ-thickening of a line segment is a rectangle with semi-disc caps. See Fig. 2.14. ε Figure 2.14: ǫ-thickenings of a point, circle, and line segment. The size of ǫ is indicated on the left. Using ǫ-thickenings, 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.

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