Arc-length parametrization

Re-timing a curve so its own parameter measures distance traveled instead of time.

Reparametrizing $\mathbf{r}(t)$ by arc length means replacing $t$ with $s(t)=\int_a^t|\mathbf{r}'(u)|\,du$ , the distance traveled by time $t$ , then solving for $t$ in terms of $s$ and substituting back. The payoff is $\left|\dfrac{d\mathbf{r}}{ds}\right|=1$ for every $s$ — moving at unit speed through arc length — which is what makes facts like curvature's definition depend only on the curve's shape, not on how fast a particular $\mathbf{r}(t)$ happens to move along it.

Examples

Two cars driving the same road at different speeds have different $\mathbf{r}(t)$ 's but share exactly one arc-length parametrization $\mathbf{r}(s)$ , since the odometer only cares about the road.