Random IFS

Probability and the Random IFS Algorithm

In the Random IFS Algorithm the transformations T_i are applied in random order, but they need not be applied equally often.

Associated with each T_i is a probability p_i, 0 < p_i < 1, representing how often each transformation is applied. That is,

when N points are generated, each T_i is applied about N*p_i times.

To illustrate the effect of changing the probabilities, we use the IFS

T3(x, y) = (x/2, y/2) + (0, 1/2)	T4(x, y) = (x/2, y/2) + (1/2, 1/2)
T1(x, y) = (x/2, y/2)	T2(x, y) = (x/2, y/2) + (1/2, 0)

We take

p₄ to range from 0 to 1 in steps of .05,

and

p₁ = p₂ = p₃ = (1 - p₄)/3.

Starting with p₄ = 0, the first picture is the gasket. Do you see why?

Click the picture to start the animation.

Here is a way to find the probabilities that give approximately uniform fill of the attractor.

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