= Cross-ratio construction of generalized-circle reflection
{title2=$J=f^{-1}\circ(z\mapsto\overline z)\circ f$}
Normalize three distinct points on a <generalized circle> by a <Möbius transformation> $f$ taking them to $1,0,\infty$. The inverse image of the extended real line is their generalized circle, and $J=f^{-1}\circ c\circ f$, with $c(z)=\overline z$, fixes that circle pointwise and squares to the identity. Changing the three normalizing points replaces $f$ by $M\circ f$ with a real-coefficient Möbius map $M$. Since $M$ commutes with conjugation, $J$ is unchanged. Thus this construction depends only on the circle. It is <inversion in a circle> for an ordinary circle and <Euclidean reflection> for a straight line, each extended to the <Riemann sphere>.
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