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Showing posts from March, 2015

Möbius transformations with stereographic projections

A Möbius transformation of the plane is a rational function of the form $$f(z) = \frac{a z + b}{c z + d}$$ of one complex variable $z$. Here the coefficients $a, b, c, d$ are complex numbers satisfying $ad - bc\neq 0.$
Geometrically, a Möbius transformation can be obtained by stereographic projection of the complex plane onto an admissible sphere in $\mathbb R^3$, followed by a rigid motion of the sphere in $\mathbb R^3$ which maps it to another admissible sphere, followed by stereographic projection back to the plane. 

A Möbius transformation is a combination of dilatation, inversion, translation, and rotation.
The following applet shows the stereographic projection representing different Möbius transformations. Move the sliders to see what happens.

Made with GeoGebra, link here: This applet was made based on the work of D. N. Arnold and J. Rogness.
Further reading:
Arnold, D. N. & Rogness, J. (2008).  Möbius transformations revea…