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We now return to the helix introduced earlier in this chapter. A vector-valued function that describes a helix can be written in the form
where R represents the radius of the helix, h represents the height (distance between two consecutive turns), and the helix completes N turns. Let’s derive a formula for the arc length of this helix using [link] . First of all,
Therefore,
This gives a formula for the length of a wire needed to form a helix with N turns that has radius R and height h.
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:
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 u is to distinguish between time and the variable of integration. Since $s\left(t\right)$ measures distance traveled as a function of time, ${s}^{\prime}\left(t\right)$ measures the speed of the particle at any given time. Since we have a formula for $s\left(t\right)$ in [link] , we can differentiate both sides of the equation:
If we assume that $\text{r}\left(t\right)$ defines a smooth curve, then the arc length is always increasing, so ${s}^{\prime}\left(t\right)>0$ for $t>a.$ Last, if $\text{r}\left(t\right)$ is a curve on which $\Vert {r}^{\prime}\left(t\right)\Vert =1$ for all t , then
which means that t represents the arc length as long as $a=0.$
Let $\text{r}\left(t\right)$ describe a smooth curve for $t\ge a.$ Then the arc-length function is given by
Furthermore, $\frac{ds}{dt}=\Vert {r}^{\prime}\left(t\right)\Vert >0.$ If $\Vert {r}^{\prime}\left(t\right)\Vert =1$ for all $t\ge a,$ then the parameter t represents the arc length from the starting point at $t=a.$
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 $\text{r}\left(t\right)=\u27e83\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}t,3\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}t\u27e9,0\le t\le 2\pi $ that parameterizes a circle of radius 3, we can change the parameter from t to $4t,$ obtaining a new parameterization $\text{r}\left(t\right)=\u27e83\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}4t,3\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}4t\u27e9.$ The new parameterization still defines a circle of radius 3, but now we need only use the values $0\le t\le \pi \text{/}2$ to traverse the circle once.
Suppose that we find the arc-length function $s\left(t\right)$ and are able to solve this function for t as a function of s. We can then reparameterize the original function $\text{r}\left(t\right)$ by substituting the expression for t back into $\text{r}\left(t\right).$ The vector-valued function is now written in terms of the parameter s. Since the variable s represents the arc length, we call this an arc-length parameterization of the original function $\text{r}\left(t\right).$ One advantage of finding the arc-length parameterization is that the distance traveled along the curve starting from $s=0$ is now equal to the parameter s. 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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