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.