Unit Tangent Vector

Definition

$T(t)=\frac{{\mathbf{r}}^{\prime }(t)}{||{\mathbf{r}}^{\prime }(t)||}$

Facts

$\frac{d\mathbf{r}}{ds}=\frac{d\mathbf{r}}{dt}\frac{dt}{ds}={\mathbf{r}}^{\prime }\frac{1}{||r^{\prime }(t)||}=T$ where $s$ is arc length parametrization