Real cross-ratio criterion for a generalized circle
= Real cross-ratio criterion for a generalized circle
Normalize three distinct points of the <Riemann sphere> by a <Möbius transformation> $f$ sending them to $\infty,0,1$. A fourth point lies on their <generalized circle> exactly when $f$ takes a real value there: the image of that <generalized circle> is the extended real line. Other standard <cross-ratio> ordering conventions differ by real fractional-linear transformations and preserve this reality criterion.