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.