If I have define a Möbius transformation as "a map on the extended complex plane, $\bar{\mathbb{C}} \rightarrow \bar{\mathbb{C}}$, given by $\omega = \frac{az + b}{cz + d}$ where $a,b,c,d \in \mathbb{C}$ and $ad - bc \neq 0$", then can I define an affine Möbius transormation as
"A map in the extended complex plane, $\bar{\mathbb{C}} \rightarrow \bar{\mathbb{C}}$, given by $\omega = az + d$, i.e
$\left\{ \infty \rightarrow \infty, a \neq 0 : \omega = \frac{1z + b}{cz + 1}\right\}$
Is this correct?