> [!Definition] Definition (Unit speed parametrisation) > > A [[Regular Parametrization|regular parametrisation]] $\underline{r}(t)$ has unit speed if $\left\lvert \left\lvert \frac{d\underline{r}}{dt} \right\rvert \right\rvert =1 .$ # Properties Let $r(s)$ and $r(u)$ be two unit-speed parameterisations of a curve, then $u=\pm s+c$ for some $c\in\mathbb{R}$ >*Proof*. We have $\left\lvert \left\lvert \frac{d\underline{r}}{dt} \right\rvert \right\rvert= \left\lvert \left\lvert \frac{d\underline{r}}{du} \right\rvert \right\rvert =1$By chain rule, $\frac{d\underline{r}}{ds}=\frac{d\underline{r}}{du} \frac{du}{ds}$Take length: $\left\lvert \left\lvert \frac{d\underline{r}}{ds} \right\rvert \right\rvert = \left\lvert \frac{du}{ds} \right\rvert \left\lvert \left\lvert \frac{d\underline{r}}{du} \right\rvert \right\rvert \implies \left\lvert \frac{du}{ds} \right\rvert = 1 \implies \frac{du}{ds} = \pm 1 $so $u=\pm s+c$for some $c\in \mathbb{R}$. # Examples - [[Arc length of regular parametrized curve|Arc length parametrisation]]. - [[Trigonometric Functions]].