A smooth (or rectifiable?) curve , where is an interval and is a Riemannian manifold, is naturally parametrized, or parametrized by arclength, if the arclength? of the arc from to is , for any , . 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 is typically 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.