FractionalGeometry-Chap2

Say enough that the line joining t1 1 0 and t1 12 32

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Unformatted text preview: diagonal r = s in the r-s plane. For the r and s values of the picture, the shaded regions are the θ and ϕ values that give a contraction transformation. Note for r = s = 0.7, all θ and ϕ give a contraction map. Perhaps surprisingly, even for large values of r and s, the transformation remains a contraction for some range of θ and ϕ values. Figure 2.54: Top, left to right: r = s = 0.7, 0.8, 0.9, 1. Bottom, left to right: r = s = 1.1, 1.6, 2.5, 3.5 So for an IFS with n transformations, the parameter space is n copies of the 6-dimensional space P = P1 × [−1, 1]2 . Consequently, to find an ntransformation IFS for a particular image, we must search through a 6n-dimensional space. A natural concern, then, is how delicately does the attractor of the IFS depend on the parameters? How close must we come to the true parameters to get a shape that’s close enough to the desired attractor? Before we can answer this question, we must understand what close enough means. For IFS attractors this was done in Sect. 2.2: attractors are elements of K(R2 ), compact s...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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