I understand how given a point in $\mathbb{C}$ you can map to a point on the unit sphere using the inverse of stereographic projection with the following formula:
$\pi^{-1}(x+iy)=\frac{2x,2y,x^2+y^2+1}{1+x^2+y^2}$
I'm struggling to understand how you can map a circle, say $(x-2)^2+y^2=2$ to its image using the formula.
My first intuition is to do it point by point--that is, select four easy to understand points from the circle, apply the inverse projection, and then try to formulate a cline; however, I feel like there's likely a more generalized approach than this.