Why image of a line in complex plane is a circle in the Riemann sphere?

Question

Let $ax+by+c=0$ be a line in complex plane. If $\pi$ be the stereographic progection, then since

$$\pi^{-1}(x_1,x_2,x_3)=\left(\frac{2x_1}{2-x_3},\frac{2x_2}{2-x_3}\right)$$

we have

$$a\left(\frac{2x_1}{2-x_3}\right)+b\left(\frac{2x_2}{2-x_3}\right)+c=0$$

or equivalently

$$2ax_1+2bx_2-cx_3=-2c$$

But this is clearly the equation of a plane in $\mathbb R^3$ and not a circle. So could someone explain for me what's the problem? Thanks!

Answer 1

Stereographic projection in the first place is a geometrically defined map. You are given a plane $C$ and a sphere $S\subset{\mathbb R}^3$. Then you choose a projection center $N\in S$ and match the points of $S\setminus\{N\}$ with those of $C$ by drawing rays through $N$. If $\ell\subset C$ is a line then the rays $N\vee Z$ for $Z\in\ell$ constitute a plane, and this plane intersects $S$ in a circle passing through $N$.