# Möbius Transformations are Isometries of a Sphere

We present a video on Möbius transformations and the geometry of the sphere. Anyone who has taken or will take complex analysis (that means you engineers!) should watch this. It shows not only the beautiful correspondence between the two, but it reveals the intuition behind a lot of complex analysis, when more often than not a student is left in the dust of rigorous formulas.

In short, this is a proof without words that the Möbius transformations are in correspondence with rigid motions of the unit sphere in $\mathbb{R}^3$. This 1-1 mapping is precisely Riemann’s stereographic projection. While we usually might see this in one context (showing $\mathbb{R}^2$ is the one-point compactification of the plane), it is not often connected to all of our interesting transformations in such a picturesque way.

Möbius transformations lay the foundation for much of hyperbolic geometry and other advanced topics in complex analysis. They’re quite fascinating.