Arc Length

Definition

Let $\mathbf{r}(t)=x(t)\mathbf{i}+y(t)\mathbf{j}+z(t)\mathbf{k},a\le t\le b$ be a Parametric Curve. The length of the curve is defined as $L={\int }_a^b||{\mathbf{r}}^{\prime }(t)||dt={\int }_a^b\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2+{\left(\frac{dz}{dt}\right)}^2}dt$

Arc Length Parametrization

Let $\mathbf{r}(t)$ be a curve for $t\ge a$. Then the arc-length function is given by $s(t)={\int }_a^t||{\mathbf{r}}^{\prime }(\tau )||d\tau$ Furthermore, by Fundamental Theorem of Calculus, $s^{\prime }(t)=\frac{d}{dt}{\int }_a^t|{\mathbf{r}}^{\prime }(\tau )|d\tau =||r^{\prime }(t)||>0$