Spherical arc-length as a global conformal parameter for analytic curves   in the Riemann sphere
  We prove that for every analytic curve in the complex plane, Euclidean and spherical arc-lengths are global conformal 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 Euclidean n-space and complex n-space and we discuss the situation of curves in the Riemann sphere. 
Paul Gauthier, Vassili Nestoridis, Athanase Papadopoulos (IRMA)
math.CV math.GT