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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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