Examples

Arc length parametrization of a line

Arc length parametrization of a line.

Calculate an arc length parametrization of the line in the plane given by $y=mx$.

Solution: We create a natural parametrization of the line using $\mathbf{r}(t)=(t,mt)$. Observe that ${\mathbf{r}}^{\prime }(t)=(1,m)$ and $|{\mathbf{r}}^{\prime }(t)|=\sqrt{1+m^2}$. Using the definition of $s(t)$, we then find:

$$s(t)={\int }_0^t\sqrt{1+m^2}du=t\sqrt{1+m^2}.$$

Solving for $t$, we find $t=s\frac{1}{\sqrt{1+m^2}}$. Plugging this into the original parametrization, obtain: $\mathbf{r}(s)=\left(s\frac{1}{\sqrt{1+m^2}},s\frac{m}{\sqrt{1+m^2}}\right).$