Theory

Arc length parametrization

A parametric curve $\mathbf{r}(t)$ is said to have unit speed when $|{\mathbf{r}}^{\prime }(t)|=1$.

Another description with the same meaning is to say that $\mathbf{r}(t)$ has an arc length parametrization. This terminology is used because the distance-traveled function is $s(t)=t$ when $|{\mathbf{r}}^{\prime }(t)|=1$.

Typically a parametric curve with arc length parametrization is written using the letter ‘$s$’ for the parameter, as in $\mathbf{r}(s)$, to emphasize the fact that the parameter also measures the distance traveled along the curve.

The distance traveled $s$ is also the only natural parameter, in that it is determined by the image curve in ${\mathbb{R}}^3$, whereas any other parameter $t$, for $t\ne s$, suffers from an undesirable arbitrariness. (For example, what geometric reason could be given to choose $t$ instead of $2t+1$?)

A consequence of the fact that $s$ is geometrically natural is that arc length parametrizations are very useful in abstract mathematics, where arguments about geometrical structure are sought. Unfortunately it is often impracticable to create arc length parametrizations with explicit formulas, so they are less useful in practical applications. There are some exceptions.

$$\left|\frac{d}{ds}\mathbf{r}(g(s))\right|=1.$$