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From what I understand, a Möbius transformation is of the form f(z) = $\frac{Az+D}{Cz+B}$ where A,B,C, and D may be real or complex. What is f(z) doing to z exactly? And what are some of the significance of this?

I am also a little confused about the difference of real and complex numbers in a Möbius transformation. What are some differences if the variables are real or complex?

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  • $\begingroup$ I assume you wanted a $B$ somewhere in that defineition of $f(z)$ $\endgroup$ Jun 4, 2014 at 20:22
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    $\begingroup$ They do beautiful things! youtube.com/watch?v=JX3VmDgiFnY $\endgroup$
    – PPP
    Jun 4, 2014 at 20:23
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    $\begingroup$ Fundamentally, there is a relationship between $f(z)$ and the matrix $\begin{pmatrix}A&D\\C&B\end{pmatrix}$. Also, the complex projective line. $\endgroup$ Jun 4, 2014 at 20:25
  • $\begingroup$ Yes, I made a typo in the equation. @Lucas, I saw this video and it looks very cool! But this video doesn't exactly explain some of the differences between real and complex and what its doing to z. $\endgroup$
    – Instinct
    Jun 4, 2014 at 20:26
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    $\begingroup$ the difference between real and complex Möbius transformation, is that the real ones preserves the upper half semi-plane, search for Fuchsian groups (the complex analogue are the Kleinian groups), if you are interested in these stuffs you should take a look at Teichmüller theory and modular forms (I suggest Shimura's book and Hubbard's book) $\endgroup$
    – user40276
    Jun 4, 2014 at 20:46

1 Answer 1

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Any Möbius transformation is a composition of translation, rotation, reflection, homothety (scaling) and inversion, see here. The first three are straightforward to picture, and the last one is an analog of reflection, only across a circle rather than a line, a picture here gives the idea. Their properties of mapping lines and circles into lines and circles, and mapping "point at infinity" into a finite point are used extensively in complex analysis.

If $A,B,C,D$ are real then real numbers $x$ will be mapped into real numbers $f(x)$ or infinity, i.e. the transformation preserves the extended real line. Moreover, the (extended) upper half plane and lower half plane, that real line separates, get mapped into themselves. The upper half-plane (with a non-standard way to measure distances) is a model of hyperbolic geometry, the Poincaré half-plane model. In this model real Möbius transformations are the isometries, i.e. they preserve the hyperbolic distances, so they are 'translations, rotations and reflections' of the hyperbolic plane.

Complex Möbius transformations map the upper half plane into other domains that serve as different models of hyperbolic geometry, e.g. Poincaré disk model. Another group of complex Möbius transformations preserve that disk and serve as hyperbolic isometries of it. They produce spectacular tilings, used in art by Escher, that are created by starting with a seed pattern and spreading it around using hyperbolic isometries like you would use ordinary translations, rotations and reflections in the Euclidean plane.

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  • $\begingroup$ Thank you! This is a really good definition that definitely makes thing clearer for me. $\endgroup$
    – Instinct
    Jun 10, 2014 at 3:53

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