Stereographic projection preserves circles

Statement

Setup

Further information: Stereographic projection

Consider $\mathbb {R} ^{3}$, three-dimensional Euclidean space, with coordinates $x,y,z$. Denote:

$S^{2}:=\{(x,y,z)\in \mathbb {R} ^{3}\mid x^{2}+y^{2}+z^{2}=1\}$

Identify $\mathbb {C}$ with the $xy$-plane under the map:

$(x,y,0)\mapsto x+iy$

Let $N=(0,0,1)$ denote the north pole in $S^{2}$, and define the stereographic projection as a bijective map:

$S^{2}\setminus \{N\}\to \mathbb {C}$

which sends a point $P\in S^{2}\setminus \{N\}$ to the unique point in $\mathbb {C}$ that is collinear with $N$ and $P$.

Actual statement

The following two facts are true:

• Circles in $S^{2}\setminus \{N\}$ get mapped to circles in $\mathbb {C}$
• A circle in $S^{2}$ that passes through $N$, minus the point $N$, gets mapped to a straight line in $\mathbb {C}$