= Chordal metric
{title2=$\chi(z,w)$}
The chordal metric on the <Riemann sphere> is $\chi(z,w)=|z-w|/\sqrt{(1+|z|^2)(1+|w|^2)}$ for finite points, with $\chi(z,\infty)=(1+|z|^2)^{-1/2}$. It is half the Euclidean chord distance between stereographic images on the unit sphere, so the triangle inequality follows from the Euclidean one. If $\rho$ is round spherical geodesic distance, then $\rho=2\arcsin\chi$ and $2\chi\leq\rho\leq\pi\chi$. These inequalities transfer uniform-continuity estimates between spherical and chordal distances, including at poles of meromorphic functions.
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