I know the formulas for stereographic projection on $S^1$, we have two functions, $\varphi_N : S^1 \setminus \{(0,1)\}\to \mathbb{R}$, $\varphi_N(x^1, x^2)=\frac{x^1}{1-x^2}$ and $\varphi_S : S^1 \setminus \{(0,-1)\}\to \mathbb{R}$, $\varphi_S(x^1, x^2)=\frac{x^1}{1+x^2}$, and that's about it.
But now I am given a circle of radius $r>0$ and I want to find the formulas for stereographic projection here. I thought that maybe I should just take the projection of the circle to $S^1$, i.e. $p(x^1, x^2)=\left(\frac{x^1}{r}, \frac{x^2}{r}\right)$, and compose it with $\varphi_S$ and $\varphi_N$. Is this right?