Generalized circle - Knowlpedia

Definition

A generalized circle in the Riemann sphere is either an ordinary Euclidean circle in $\mathbb C$ or a set

L\cup\{\infty\},

where $L\subset\mathbb C$ is a Euclidean straight line.

Spherical picture

Under stereographic projection, generalized circles correspond exactly to the circles cut out on the round sphere by affine planes that meet it transversely. A spherical circle through the projection point becomes a line together with $\infty$ ; all other spherical circles become ordinary plane circles.

Möbius invariance

Every Möbius transformation maps generalized circles to generalized circles. Inversion $z\mapsto1/z$ explains why both types are needed: a circle through $0$ is sent to a line, while a circle avoiding $0$ is sent to another circle.

Cross-ratio criterion

Four distinct points of the sphere lie on a common generalized circle exactly when their cross-ratio is real. The property is therefore projectively invariant.

References

Alan F. Beardon, The Geometry of Discrete Groups, Springer, 1983. Publisher record. Relevant: Chapter 3.