FractionalGeometry-Chap2

To get some understanding of how ifs parameters

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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...
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