Chordal metric ($\chi (z,w)$)

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

A rational map is smooth as a map between compact round spheres, including at its poles in reciprocal coordinates. Its spherical differential norm therefore has a finite supremum $L$. Integrating along a shortest spherical geodesic gives $\rho (Rz,Rw)\le L\rho (z,w)$. Since $2\chi \le \rho \le \pi \chi$, we obtain $\chi (Rz,Rw)\le (\pi L/2)\chi (z,w)$. In finite coordinates the differential norm is $R^{\#}(z)=|R^{\prime }(z)|(1+|z{|}^2)/(1+|R(z){|}^2)$, extending continuously to the whole sphere. This uniform bound is useful for passing equicontinuity through compositions.