FractionalGeometry-Chap2

# To get some understanding of how ifs parameters

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

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: scale of the attractor by the same factor w. If we are interested only in the shape, not the size, of the attractor of an IFS {T1 , . . . Tn }, then we can restrict each ei and fi to lie in the square {(ei , fi ) : −1 ≤ ei ≤ 1, −1 ≤ fi ≤ 1}. Simply replace each ei and fi by ei /m and fi /m, where m = max{|e1 |, . . . , |en |, |f1 |, . . . , |fn |}. Certainly, θ and ϕ can take on any value in [0, 2π ], but we need to think a bit diﬀerently to understand the eﬀect of a small change of parameters. These variables represent angles, so θ = 1.9π is close to θ = 0.1π . Consequently, we shall think of both θ and ϕ as representing points on a circle, S 1 . Together, θ and ϕ range over the product of two circles, that is, a torus T 2 . For the r and s parameters, a ﬁrst thought is that a transformation is a contraction if max{|r|, |s|} < 1, and so the parameter space of values giving a contraction map is ((−1, 1) × (−1, 1)) × T 2 × R2 or if we are concerned only with the shape of the attractor P0 = ((−1, 1) × (−1, 1)) × T 2...
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