Arc Length

Arc length is the distance between two points along a section of a curve.

General

The arc length of a smooth curve in ${\mathbb{R}}^3$ where $\vec{r}(t)=⟨x(t),y(t),z(t)⟩$ from $t=a$ to $t=b$, that is traced out exactly once is

$$L={\int }_{t=a}^{t=b}\Vert {\vec{r}}^{\prime }(t)\Vert dt={\int }_{t=a}^{t=b}\sqrt{x^{\prime }(t{)}^2+y^{\prime }(t{)}^2+z^{\prime }(t{)}^2}dt$$

Note that the first form of the arc length is generally applicable for all ${\mathbb{R}}^n$

Parameterization

The same as General, except only our starting time is fixed.

$$s(t)={\int }_{\tau =t_0}^{\tau =t}\Vert {\vec{r}}^{\prime }(\tau )\Vert d\tau$$