- Determine the length of a particle’s path in space by using the arc-length function
- Calculate and understand what curvature means for curves in space
- Understand what normal and binormal vectors tell us about a curve’s behavior
Arc Length for Vector Functions
You’ve already seen that a vector-valued function describes a curve in either two or three dimensions. Recall from earlier courses that the arc length of a curve defined by parametric functions [latex]x=x(t)[/latex], [latex]y=y(t)[/latex], [latex]t_{1}\leq t\leq t_{2}[/latex] is:
[latex]s=\int_{t_{1}}^{t_{2}} \sqrt{(x'(t))^{2}+(y'(t))^{2}}dt[/latex]
Similarly, 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:
[latex]s=\int_{a}^{b} \sqrt{(f'(t))^{2}+(g'(t))^{2}}dt[/latex]
In three dimensions, if [latex]{\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}+h(t){\bf{k}}[/latex] over the interval [latex]a\leq t\leq b[/latex], the arc length is:
[latex]s=\int_{a}^{b} \sqrt{(f'(t))^{2}+(g'(t))^{2}+(h'(t))^{2}}dt[/latex]
arc-length formulas theorem
Let [latex]C[/latex] be a smooth curve defined by [latex]{\bf{r}}(t)[/latex], where [latex]t[/latex] lies within the interval [latex][a, b][/latex].
For plane curves [latex]{\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}[/latex]:
[latex]s=\int_{a}^{b} \sqrt{[f'(t)]^{2}+[g'(t)]^{2}}dt=\int_{a}^{b} \left\Vert{\bf{r}}'(t)\right\Vert dt[/latex]
For space curves [latex]{\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}+h(t){\bf{k}}[/latex]:
[latex]s=\int_{a}^{b} \sqrt{[f'(t)]^{2}+[g'(t)]^{2}+[h'(t)]^{2}}dt=\int_{a}^{b} \left\Vert{\bf{r}}'(t)\right\Vert dt[/latex]
The two formulas are nearly identical—they differ only in that a space curve has three component functions instead of two. Notice that arc length is simply the integral of the magnitude of the derivative vector [latex]\left\Vert{\bf{r}}'(t)\right\Vert[/latex].
These formulas are defined for smooth curves: curves where [latex]{\bf{r}}(t)[/latex] is differentiable with a non-zero derivative. This smoothness condition guarantees that the curve has no cusps or corners that could make the formula problematic.
- [latex]{\bf{r}}\,(t)=(3t-2)\,{\bf{i}}+(4t+5)\,{\bf{j}},\ 1\,\leq\,t\,\leq\,5[/latex]
- [latex]{\bf{r}}\,(t)=\langle{t}\cos{t}, \ t\sin{t},\ 2t\rangle,\ 0\,\leq\,t\,\leq\,2\pi[/latex]
Arc Length of a Helix
Let’s return to the helix introduced earlier in this chapter. A vector-valued function describing a helix can be written as:
[latex]{\bf{r}}(t)=R\cos\Big(\frac{2\pi Nt}{h}\Big){\bf{i}}+R\sin\Big(\frac{2\pi Nt}{h}\Big){\bf{j}}+t{\bf{k}}[/latex], [latex]0\leq t\leq h[/latex]
where [latex]R[/latex] is the radius of the helix, [latex]h[/latex] is the height (distance between two consecutive turns), and the helix completes [latex]N[/latex] turns. Let’s derive a formula for the arc length using our formula. First:
[latex]{\bf{r}}'(t)=-\frac{2\pi NR}{h}\sin\Big(\frac{2\pi Nt}{h}\Big){\bf{i}}+\frac{2\pi NR}{h}\cos\Big(\frac{2\pi Nt}{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 formula gives the length of wire needed to form a helix with [latex]N[/latex] turns, radius [latex]R[/latex], and height [latex]h[/latex]:
[latex]s=\sqrt{4{\pi}^{2}N^{2}R^{2}+h^{2}}[/latex]