on a rectifiable curve
A parameter $s$ for a curve $\gamma$ with parametric representation $\mathbf r=\mathbf r(s)$ such that the arc length on the curve between two points $\mathbf r(s_1)$ and $\mathbf r(s_2)$ is equal to $|s_1-s_2|$. The parametrization of a curve by the natural parameter is known as its natural parametrization. The natural parametrization of a $k$-times differentiable (analytic) curve with no singular points is also $k$ times differentiable (analytic).
See also (the references to) Natural equation.
Natural parameter. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Natural_parameter&oldid=32588