Volume 9, Issue 3

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The Modular Group

# Möbius Transformations

We first consider a class of functions known as Möbius transformations. These transformations, named after the same mathematician with whom we associate the one-sided, half-twisted Möbius band, are one-to-one transformations on the extended complex plane . Möbius transformations are of the following type:

Over the reals, a Möbius transformation with real coefficients falls into one of two categories: either and the graph is a line, or and the graph is a hyperbola. Here is a representation of this latter type of function.

Figure 3. Graph of shown with asymptotes.

Our purpose will be to investigate how Möbius transformations stretch and twist regions in the extended complex plane. The complex plane is the usual Euclidean plane with each point identified as a complex number, namely, The extended complex plane is formed from the complex plane by adding the point at infinity.

When a Möbius transformation, , acts on a complex number, , we may view the action as "moving" the point onto the point . We will be helped in what follows by one important fact: Möbius transformations map circles and lines in the extended plane onto circles and lines in the extended plane. A comprehensive proof of this fact may be found in most elementary texts on complex variables (e.g., Complex Variables by Levinson and Redheffer [2, 158]).

The extended complex plane is the domain in which the figures of our animations "live." Each point of the figure is acted on by the Möbius transformations as though the point were a complex number. These transformations spin hyperbolic 2-space about a fixed point or shift the space in one direction or another.