# nLab natural parametrization

A smooth (or rectifiable?) curve $s:I\to X$, where $I\subset ℝ$ is an interval and $X$ is a Riemannian manifold, is naturally parametrized, or parametrized by arclength, if the arclength? of the arc from $s\left(t\right)$ to $s\left(t\prime \right)$ is $t\prime -t$, for any $t, $t,t\prime \in I$. 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 $X$ is typically ${ℝ}^{n}$ 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.

Revised on February 9, 2010 15:55:55 by Toby Bartels (12.53.235.42)