FractionalGeometry-Chap2

The decomposition a t1 a tn a suggests a method for

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Unformatted text preview: rem, we develop some experience with the inverse problem, finding an IFS to produce a given fractal. The decomposition A = T1 (A) ∪ · · · ∪ Tn (A) suggests a method for solving the inverse problem: given a self-similar or self-affine set A, find contractions Ti producing A. In principle, the method is relatively simple: 1. Decompose A into scaled copies: A = A1 ∪ · · · ∪ An . 2. For each Ai , find a transformation Ti with Ti (A) = Ai . Example 2.1.1 Three IFS examples. It is easy to see that fractal (a) consists of four copies of itself, three scaled by a factor of 1/2 and one by 1/4. The translations are straightforward. The parameters for the IFS generating this fractal can be presented in a table: r 1/2 1/2 1/2 1/4 s 1/2 1/2 1/2 1/4 θ 0 0 0 0 ϕ 0 0 0 0 e 0 1/2 0 3/4 f 0 0 1/2 3/4 28 CHAPTER 2. ITERATED FUNCTION SYSTEMS (b) (a) (c) Figure 2.4: Fractals produced by three IFS. Each is contained in the unit square with the origin at its lower left corner. Fractal (b) also consists of four copies of itself, but the lower right and...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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