Möbius Transformations


TL; DR: Use the arrow keys to rotate the view. Click in a form field and use the q-a and w-s keys to transform the sphere.


Loosely, a Möbius transformation is a complex function of the form $$ T(z) = \frac{\alpha z + \beta}{\gamma z + \delta},\quad \text{$\alpha$, $\beta$, $\gamma$, $\delta$ complex and $\alpha\delta - \beta\gamma \neq 0$.} $$ More carefully, a Möbius transformation is a holomorphic bijection of the Riemann sphere, identified with the set of lines through the origin in $\mathbf{C}^{2}$.

If $z_{0}$, $z_{1}$ and $z_{\infty}$ are distinct points of the Riemann sphere, there exists a unique Möbius transformation $T$ satisfying $T(z_{0}) = 0$, $T(z_{1}) = 1$, and $T(z_{\infty}) = \infty$, given by the formula $$ T(z) = \frac{(z - z_{0})(z_{1} - z_{\infty})}{(z - z_{\infty})(z_{1} - z_{0})}. $$ The form fields allow values of these parameters to be entered manually, in the form a + bi for finite numbers, or Inf for infinity.

These parameters can also be adjusted continuously by clicking on a form field, then using q-a to increase or decrease the longitude of that parameter, and using w-s to increase or decrease the latitude. Holding down any of these keys therefore traces a circle in the space of Möbius transformations.