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. 

Inversion

Inversion in the complex plane

Inversion using stereographic projection

A Möbius transformation is a combination of dilatation, inversion, translation, and rotation.

Rotation, inversion and translation in the plane
Rotation, inversion, and translation using stereographic projection
The following applet shows the stereographic projection representing different Möbius transformations. Move the sliders to see what happens.



Made with GeoGebra, link here: http://tube.geogebra.org/student/m839839. 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 revealed. Notices of the AMS. 55, 10: pp. 1226-1231.