Finding IFS for Fractal Images

Background - IFS

IFS stands for Iterated Function System.

We are interested in the inverse problem:

Given a fractal, find transformations of an IFS that produces the fractal.

In general, we need a richer variety of transformations than those used to produce the gasket. Here are the fundamental geometric features of the transformations weshall use.

Scaling

Reflection

Rotation

Translation

Summary:

Scalings are given by r and s.

Negative r reflects across the y-axis.

Negative s reflects across the x-axis.

The most general rotation and distortion requires two angles, theta and phi. For rigid rotations, theta = phi. If a program uses only one angle, then all rotations are rigid. All exercises in this lab use rigid rotations.

Rotations are about the origin, counterclockwise angles are positive.

Horizontal translations are given by e.

Vertical translations are given by f.

Order matters Note that a reflection followed by rotation does not necessarily produce the same image as this rotation followed by this reflection. Our software performs transformations in this way:

first scalings,

then reflections,

then rotations,

and translations last.

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