Affine Transformations

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Affine Transformations Overview

Several of the Fractal Science Kit fractal generator built-in programs make use of affine transformations. Affine transformations are transformations in the form:

f(z) = Complex(A*z.X + B*z.Y + E, C*z.X + D*z.Y + F)

The numbers A, B, C, D, E, and F are real numbers. The input to the function (z) and the function result are both complex.

This is sometimes given in matrix form as:

A B E     z.X     A*z.X + B*z.Y + E
C D F  *  z.Y  =  C*z.X + D*z.Y + F
0 0 1     1       1

The built-in macros define an Affine Object and a collection of Affine Functions.

While the understanding of affine transformations is not a prerequisite to using the application, it is an interesting and rewarding area of mathematics that can be used to produce beautiful fractals and is highly recommended.

The following web sites provide an introduction to affine transformations:

Eric W. Weisstein. "Affine Transformation."

From MathWorld--A Wolfram Web Resource.

 

"Affine transformation."

From Wikipedia, the free encyclopedia.

The following books cover affine transformations in detail:

Fractals Everywhere

Michael F. Barnsley

 

Chaos and Fractals

Peitgen, Jurgens, Saupe

 

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