Spherical arc-length as a global holomorphic parameter for analytic curves in the Riemann sphere

We prove that for every analytic curve in the complex plane C, Euclidean and spherical arc-lengths are global holomorphic parameters. We also prove that for any analytic curve in the hyperbolic plane, hyperbolic arc-length is also a global parameter. We generalize some of these results to the case of analytic curves in Rn and Cn and we discuss the situation of curves in the Riemann sphere C∪{∞}.