Natural parameter

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).

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See also (the references to) Natural equation.