The size of is indicated on the left using

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

Ask a homework question - tutors are online