How do you determine the function representing the surjective map say for example $f: C\to S$ where $S=\left\{(x,y) \in R^2 : x^2+y^2=r^2\right\}\,?$

Question

How do you determine the function representing the surjective map say for example $f: C\to S$ where $S=\left\{(x,y) \in\mathbb R^2 : x^2+y^2=r^2\right\}\;?$

$C$ in this question stands for the cylinder.

Hi everyone! I am really confused with how to know which $f(x)$ should I have so that I can map $C$ into $S$. I don’t have a single idea on how this works at all. I have seen a few examples but I always end up not getting anything from it.