nLab natural parametrization

A smooth (or rectifiable?) curve s:IXs: I\to X, where II\subset \mathbb{R} is an interval and XX is a Riemannian manifold, is naturally parametrized, or parametrized by arclength, if the arclength of the arc from s(t)s(t) to s(t)s(t') is ttt' - t, for any t<tt\lt t', t,tIt,t'\in I. In other words, the parameter is the length of the curve counted from a given fixed point on the curve. Many classical formulas in the geometry of curves (where XX is typically n\mathbb{R}^n with its usual metric) assume natural parametrization; otherwise the analogues in terms of general parametrizations become far more complicated.

In particular, geodesics are generally taken to be naturally parametrized.

Last revised on February 9, 2010 at 15:55:55. See the history of this page for a list of all contributions to it.