nLab natural parametrization

A smooth (or rectifiable?) curve $s: I\to X$, where $I\subset \mathbb{R}$ is an interval and $X$ is a Riemannian manifold, is naturally parametrized, or parametrized by arclength, if the arclength of the arc from $s(t)$ to $s(t')$ is $t' - t$, for any $t\lt t'$, $t,t'\in I$. 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 $X$ is typically $\mathbb{R}^n$ 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.