I see that Möbius transformation is defined as $$f(z) = \dfrac{az + b}{cz + d}, \qquad ad -bc \neq 0$$ on the extended complex plane $\mathbb{C} \cup \{ \infty\}$.
But further in the text
Thus a Möbius transformation is always a bijective holomorphic function from the Riemann sphere to the Riemann sphere
I understand, that there are inverse stereographic projections which maps points in $\mathbb{C}$ to the Riemann sphere.
So, what is going on formally? Do we just name(associate) points on the "standard" unit sphere (like around the $(0,0,0)$ with unit radius) with the name $z$ of the pre-image under some "standard" inverse stereographic projection?
I.e., let
- $S^2$ be a Riemann Sphere,
- $\psi : \mathbb{C} \cup \{\infty\} \to S^2$ be an inverse stereographic projection to it,
- $z_s = \psi(z)$
Then, do we technically define $f_S : S^2 \to S^2$ as $$f_S = \psi \circ f \circ \psi^{-1}?$$
$^{1}$ please ignore the shading cone lines in the image