to work with stereographic projections I want to know how to solve this example first:
It is it about finding a map $f:\mathbb{R} \rightarrow S$ where $S=\{(x,y)\in \mathbb{R}^2|x^2+(y-1)^2=1\}$ (shifted unit circle) such that $f(0)=(0,0)$ and $f(\pm \infty)=(0,2)$.
How can I find $f(x)$ for $x \in \mathbb{R}$ using the parametrized line $(xs,2(1-s)), s \in [0,1]$?