Mobius Transformations

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Mobius Transformations Overview

Several of the Fractal Science Kit fractal generator built-in programs make use of Mobius transformations, also called linear fractional transformations. Mobius transformations are complex transformations in the form:

f(z) = (A*z + B) / (C*z + D)

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

Mobius transformation can be classified as 1 of 4 types: elliptic, hyperbolic, loxodromic, or parabolic.

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

While the understanding of Mobius 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 Mobius transformations:

Eric W. Weisstein. "Linear Fractional Transformation."

From MathWorld--A Wolfram Web Resource.

 

"Mobius transformation."

From Wikipedia, the free encyclopedia.

The following books cover Mobius transformations in detail:

Indra's Pearls - The Vision of Felix Klein

David Mumford, Caroline Series, David Wright

 

Visual Complex Analysis

Tristan Needham

 

Introduction to Geometry

H. S. M. Coxeter, F. R. S.

 

Geometry

David A. Brannan, Matthew F. Esplen, Jeremy J. Gray

 

Geometry - A Comprehensive Course

Dan Pedoe

 

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