FractionalGeometry-Chap2

A metric d on the set x is a function d x x 0

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: 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 (positive-definiteness), (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 ǫ-thickening. For any compact set A and any number ǫ ≥ 0, the ǫ-thickening of A is Aǫ = {(x, y ) ∈ R2 : d((x, y ), (x′ , y ′ )) ≤ ǫ for some point (x′ , y ′ ) ∈ A} where d is t...
View Full Document

Ask a homework question - tutors are online