Spiral Fractals from IFS

Background - Probabilities

The Random IFS algorithm generates an image by applying the IFS transformations one at a time in random order, with frequency determined by the probability assigned to each transformation.

For example, if we apply the transformation for the small right piece and the transformation for everything else with equal probability, the small right piece fills in quickly, while everything else is filled slowly indeed.

equal probabilities

To achieve uniform fill, the probability of applying each transformation T_i should equal the fraction of the total shape S occupied by T_i(S). This fraction can be estimated in two ways.

Visual Approach Visually inspect the whole picture and estimate what percentage of the whole is occupied by each region of the decomposition. By trial and error adjustments, this initial estimate can be modified to give a satisfactory picture.

Computational Approach If all rotations are rigid (theta = phi for each transformation), then the contraction factor for T_i is r_is_i. If the pieces of the decomposition do not have significant overlaps, then the total contraction factor of the IFS is

|r₁s₁| + |r₂s₂| + ... + |r_Ns_N|,

and p_i, the probability of applying T_i is

p_i = |r_is_i|/(|r₁s₁| + |r₂s₂| + ... + |r_Ns_N|)

Adjust any rounding errors so that the probabilities sum to 1.

What probabilities should we use? What percentage of the area of the whole spiral do the two copies each occupy?

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