A Möbius transformation is a special kind of complex function defined by:

$$ f(z) = \frac{a z + b}{c z + d} $$

where $a, b, c, d \in \mathbb{C}$ and $ad - bc \neq 0$.

These transformations are the building blocks of complex geometry. They map lines and circles to other lines and circles, and they preserve angles (i.e., they’re conformal), but they can distort distances and shapes. They basically represent the composition of translations, similarities, orthogonal transformations, and inversions all in one formula.

In the Desmos graph above, each of the values $a, b, c, d$ is a complex number, adjustable with the point sliders (each point is a complex number). You can observe how changing them warps the complex grid.

Key Properties of Möbius Transformations: #

• They form a group under composition.
• They send circles and lines to other circles and lines.
• If $c \neq 0$, the function has a pole at $z = -\frac{d}{c}$.
• They can represent inversions, rotations, translations, and scaling — all in one expression.

For a deeper dive, check out the Wikipedia article on Möbius transformations and try composing a few of them using different coefficients in the Desmos graph!