Arc Length for Vector Functions

We have seen how a vector-valued function describes a curve in either two or three dimensions. Recall Alternative Formulas for Curvature, which states that the formula for the arc length of a curve defined by the parametric functions [latex]x=x\,(t),\ y=t\,(t),\ t_{1}\,\leq\,t\,\leq\,t_{2}[/latex] is given by

[latex]s=\displaystyle\int_{t_{1}}^{t^{2}}\ \sqrt{\big(x'\,(t)\big)^{2}+\big(y'\,(t)\big)^{2}}\,dt[/latex]

In a similar fashion, if we define a smooth curve using a vector-valued function [latex]{\bf{r}}\,(t)=f\,(t)\,{\bf{I}}+g\,(t)\,{\bf{j}}[/latex], where [latex]a\,\leq\,t\,\leq\,b[/latex], the arc length is given by the formula

[latex]s=\displaystyle\int_{a}^{b}\ \sqrt{\big(f'\,(t)\big)^{2}+\big(g'\,(t)\big)^{2}}\,dt[/latex]

In three dimensions, if the vector-valued function is described by [latex]{\bf{r}}\,(t)=f\,(t)\,{\bf{I}}+g\,(t)\,{\bf{j}}+h\,(t)\,{\bf{k}}[/latex] over the same interval [latex]a\,\leq\,t\,\leq\,b[/latex], the arc length is given by

[latex]s=\displaystyle\int_{a}^{b}\ \sqrt{\big(f'\,(t)\big)^{2}+\big(g'\,(t)\big)^{2}+\big(h'\,(t)\big)^{2}}\,dt[/latex]

The two formulas are very similar; they differ only in the fact that a space curve has three component functions instead of two. Note that the formulas are defined for smooth curves: curves where the vector-valued function [latex]{\bf{r}}\,(t)[/latex] is differentiable with a non-zero derivative. The smoothness condition guarantees that the curve has no cusps (or corners) that could make the formula problematic.

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[latex]{\bf{r}}\,(t)=R\cos{\Big(\frac{2\pi\,N\,t}{h}\Big)}\,{\bf{i}}+R\sin{\Big(\frac{2\pi\,N\,t}{h}\Big)}\,{\bf{j}}+t\,{\bf{k}},\ 0\,\leq\,t\,\leq\,h,[/latex]

where [latex]R[/latex] represents the radius of the helix, [latex]h[/latex] represents the height (distance between two consecutive turns), and the helix completes [latex]N[/latex] turns. Let’s derive a formula for the arc length of this helix using Arc-Length Formulas. First of all,

[latex]{\bf{r}}'\,(t)=-\frac{2\pi\,N\,R}{h}\sin{\Big(\frac{2\pi\,N\,t}{h}\Big)}\,{\bf{i}}+\frac{2\pi\,N\,R}{h}\cos{\Big(\frac{2\pi\,N\,t}{h}\Big)}\,{\bf{j}}+{\bf{k}}.[/latex]

Therefore,

[latex]\begin{array}{ccc}\hfill {s} &=\hfill&{\displaystyle\int_{a}^{b}\ \left\Vert{\bf{r}}'\,(t)\right\Vert\ dt}\hfill \\ \hfill & =\hfill & {\displaystyle\int_{0}^{h}\ \sqrt{\Bigg(-\frac{2\pi\,N\,R}{h}\sin{\Big(\frac{2\pi\,N\,t}{h}\Big)}\Bigg)^{2}+\Bigg(\frac{2\pi\,N\,R}{h}\cos{\Big(\frac{2\pi\,N\,t}{h}\Big)}\Bigg)^{2}+1^{2}}\,dt} \hfill \\ \hfill & =\hfill & {\displaystyle\int_{0}^{h}\ \sqrt{\frac{4{\pi}^{2}N^{2}R^{2}}{h^{2}}\,\Bigg(\sin{^{2}\Big(\frac{2\pi{N}t}{h}\Big)}+\cos{^{2}\Big(\frac{2\pi{N}t}{h}\Big)\Bigg)}+1}\,dt}\hfill \\ \hfill & =\hfill & {\displaystyle\int_{0}^{h}\ \sqrt{\frac{4{\pi}^{2}N^{2}R^{2}}{h^{2}}+1}\,dt}\hfill \\ \hfill & =\hfill & {\Big[t\,\sqrt{\frac{4{\pi}^{2}N^{2}R^{2}}{h^{2}}+1}\Big]^h_0}\hfill \\ \hfill & =\hfill & {h\,\sqrt{\frac{4{\pi}^{2}N^{2}R^{2}+h^{2}}{h^{2}}}}\hfill \\ \hfill & =\hfill & {\sqrt{4{\pi}^{2}N^{2}R^{2}+h^{2}}.}\hfill \\ \hfill \end{array}[/latex]

This gives a formula for the length of a wire needed to form a helix with [latex]N[/latex] turns that has radius [latex]R[/latex] and height [latex]h[/latex].

Arc-Length Parameterization

We now have a formula for the arc length of a curve defined by a vector-valued function. Let’s take this one step further and examine what an arc-length function is.

If a vector-valued function represents the position of a particle in space as a function of time, then the arc-length function measures how far that particle travels as a function of time. The formula for the arc-length function follows directly from the formula for arc length:

[latex]s\,(t)=\displaystyle\int_{a}^{t}\ \sqrt{\big(f'\,(u)\big)^{2}+\big(g'\,(u)\big)^{2}+\big(h'\,(u)\big)^{2}}\,du[/latex]

If the curve is in two dimensions, then only two terms appear under the square root inside the integral. The reason for using the independent variable [latex]u[/latex] is to distinguish between time and the variable of integration. Since [latex]s(t)[/latex] measures distance traveled as a function of time, [latex]s'(t)[/latex] measures the speed of the particle at any given time. Since we have a formula for [latex]s(t)[/latex], we can differentiate both sides of the equation:

[latex]\begin{array}{ccc}\hfill {s'\,(t)} &=\hfill&{\frac{d}{dt}\bigg[\displaystyle\int_{a}^{t}\ \sqrt{\big(f'\,(u)\big)^{2}+\big(g'\,(u)\big)^{2}+\big(h'\,(u)\big)^{2}}\,du\bigg]}\hfill \\ \hfill & =\hfill & {\frac{d}{dt}\bigg[\displaystyle\int_{a}^{t}\ \left\Vert{\bf{r}}'\,(u)\right\Vert\ du\bigg]} \hfill \\ \hfill & =\hfill & {\left\Vert{\bf{r}}'\,(t)\right\Vert\,.} \hfill \\ \hfill \end{array}[/latex]

If we assume that [latex]{\bf{r}}\,(t)[/latex] defines a smooth curve, then the arc length is always increasing, so [latex]s'\,(t)\,>\,0[/latex] for [latex]t\,>\,a[/latex]. Last, if [latex]{\bf{r}}\,(t)[/latex] is a curve on which [latex]\left\Vert{\bf{r}}'\,(t)\right\Vert=1[/latex] for all [latex]t[/latex], then

[latex]s\,(t)=\displaystyle\int_{a}^{t}\ \left\Vert{\bf{r}}'\,(u)\right\Vert\ du=\displaystyle\int_{a}^{t}\ 1\,du=t-a,[/latex]

which means that [latex]t[/latex] represents the arc length as long as [latex]a=0[/latex].

A useful application of this theorem is to find an alternative parameterization of a given curve, called an arc-length parameterization. Recall that any vector-valued function can be reparameterized via a change of variables. For example, if we have a function [latex]{\bf{r}}\,(t)=\langle{3}\cos{t},\ 3\sin{t}\rangle,\ 0\,\leq\,t\,\leq\,2\pi[/latex] that parameterizes a circle of radius 3, we can change the parameter from [latex]t[/latex] to [latex]4t[/latex], obtaining a new parameterization [latex]{\bf{r}}\,(t)=\langle{3}\cos{4t},\ 3\sin{4t}\rangle[/latex]. The new parameterization still defines a circle of radius [latex]3[/latex], but now we need only use the values [latex]0\,\leq\,t\,\leq\,\frac{\pi}{2}[/latex] to traverse the circle once.

Suppose that we find the arc-length function [latex]s(t)[/latex] and are able to solve this function for [latex]t[/latex] as a function of [latex]s[/latex]. We can then reparameterize the original function [latex]{\bf{r}}(t)[/latex] by substituting the expression for [latex]t[/latex] back into [latex]{\bf{r}}(t)[/latex]. The vector-valued function is now written in terms of the parameter [latex]s[/latex]. Since the variable [latex]s[/latex] represents the arc length, we call this an arc-length parameterization of the original function [latex]{\bf{r}}(t)[/latex]. One advantage of finding the arc-length parameterization is that the distance traveled along the curve starting from [latex]s=0[/latex] is now equal to the parameter [latex]s[/latex]. The arc-length parameterization also appears in the context of curvature (which we examine later in this section) and line integrals, which we study in the Introduction to Vector Calculus.

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