 Chordal metric

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 $2\chi \le \rho \le \pi \chi$. These inequalities transfer uniform-continuity estimates between spherical and chordal distances, including at poles of meromorphic functions.