FractionalGeometry-Chap2

# 1 y r sin s cos y f that is any real 2 2 matrix can be

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Unformatted text preview: ten as x r cos(θ) −s sin(ϕ) x e T = + (2.1) y r sin(θ) s cos(ϕ) y f That is, any real 2 × 2 matrix can be expressed as the 2 × 2 in eq (2.1). (See Prob. 2.1.1.) Let d(, ) denote the Euclidean distance. A transformation T is a d-contraction with contraction factor t, 0 ≤ t < 1, if for all points (x1 , y1 ) and (x2 , y2 ), dT x x1 ,T 2 y2 y1 ≤t·d x x1 ,2 y2 y1 >s·d x x1 ,2 y2 y1 , (2.2) and for any number s < t, dT x x1 ,T 2 y2 y1 for at least one pair of points (x1 , y1 ) and (x2 , y2 ). For example, if T (x, y ) = (x/2, y/2), then dT x x1 ,T 2 y2 y1 = x2 x1 − 2 2 2 + y2 y1 − 2 2 2 = 1 d 2 x x1 ,2 y2 y1 for any pair of points. Consequently, this T is a contraction with contraction factor 1/2. A slightly more diﬃcult example is given in Prob. 2.1.2. Any ﬁnite collection T1 , . . . , Tn of contractions is called an iterated function system (IFS) and determines a collage map T deﬁned on compact subsets S of the plane by n T (S ) = {Ti (x, y ) : (x, y ) ∈ S } i=1 For example, deﬁne T1 , T2 , and T3 by Ti (x, y ) = (x/2, y/2) + (ai , bi ) where (a1 , b1 ) = (0, 0), (a2 , b2 ) = (1/2, 0), (a3 , b3...
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## This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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