  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.
