Arc Length

he material in this section requires students to understand the graph of a vector-valued function of one variable (from ) and tools from .

In the previous sections, we introduced vector-valued functions, which provided us with a useful tool for thinking about curves in three-dimensional space. We have also seen how the calculus operations of limits, derivatives, and integrals apply to these functions. In single-variable calculus, you likely learned that displacement (net change in position) and total distance traveled for an object moving in one-dimension will not be the same if the object changes direction. For motion along a curve in two or more dimensions, we have a similar distinction. For instance, consider the curve in .

Remember that the distance between two points in space is given by \[\begin{aligned}\end{aligned}\]. However, if this curve is a portion of a road on which we drive a car, the distance traveled as measured by the change in the car's odometer value from point \(P\) to point \(Q\) will be different than the straight-line distance \(\vert PQ\vert\) calculated by the distance formula. You likely saw how to calculate the length of a portion of a curve \(y=f(x)\) in your single-variable calculus class. In this section, we will use the idea of vector-valued functions to find the lengths of a larger family of curves.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

The central question we want to answer in this section is Given a curve \(C\) from \(P\) to \(Q\), what is the length of \(C\)? We will answer this question for well-behaved curves by applying the classic calculus approach to the ideas you investigated in the Preview Activity. For step 1 of the CCA, we approximate the length of the curve \(C\) by computing the length of line segments between points on \(C\). We will call these intermediate points \(\vr_0,\vr_1,...,\vr_n\). Remember that we are specifying location using vectors graphed in standard position. Using vectors to specify position means that the corresponding line segments between successive positions can be represented by the vector difference \(\Delta\vr_i = \vr_{i}-\vr_{i-1}\).

This leads to an approximation for the length of \(C\) given by the sum of the lengths of the vectors \(\Delta\vr_i = \vr_i-\vr_{i-1}\). Using \(\Delta x_i\), \(\Delta y_i\), and \(\Delta z_i\) to denote the change in each component, the magnitude of \(\Delta \vr_i\) is determined by the distance formula as we saw in the previous chapter: \[\begin{aligned}\end{aligned}\]. The length of the curve can now be approximated by the sum of these lengths, which gives \[\begin{aligned}L \approx\amp \sum_{i=1}^{n} \vecmag{\Delta\vr_i} \\ \approx\amp \sum_{i=1}^{n} \sqrt{(\Delta x_i)^2 + (\Delta y_i)^2 + (\Delta z_i)^2}\end{aligned}\].

Use the slider on to change the number of segments used in the approximation of the arc length of \(C\). Notice that as the number of segments increases, the difference between the actual length of the curve and the line segments gets smaller. You can also see how the sum of the lengths of the blue segments will approach the true length of the curve, which is approximately \(6.3286\).

Conceptually, you can use to see how more intermediate points will give rise to a better estimate for the true length of the curve, but we have not quantified how the approximation changes on a finer scale (step 2 of the CCA). In order to complete step 2 of the CCA, we will need to incorporate steps in our parameter \(t\) into this calculation.

This process leads to a specific approximation used in the classic calculus approach. Since this approximation approaches the actual length as we use smaller and smaller steps in the parameter, we define the arc length as follows:

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

In , we considered parameterizations of a circle and a helix. The computation of the arc length in these problems was made easier by the fact that the speed of an object with motion described by the parameterizations used did not depend on the time \(t\). That is, the speed of an object moving according to the parameterization was constant. Moving with constant speed is always feasible. For example, we can set our self-driving car to move at a constant speed of 60 miles per hour, and the car then moves with constant speed. (Of course, this is probably only recommended on a road where no obstacles or other vehicles will appear.) However, finding a parameterization with constant speed may involve some complex algebra, as we will see while working through this section.

We will focus on finding a parameterization with unit speed. That is, one for which \(\vecmag{\vv(t)} = \vecmag{\vr\, '(t)} = 1\) for all values of \(t\). We call such a parameterization a unit speed parameterization of the curve. The key idea involves recognizing that, in general, distance along the curve is not the same as the parameter being used. If an object moves with constant speed one, then the numerical value of the distance it travels is the numerical value of the time elapsed. For example, if you walk at exactly 1 meter per second for 47 seconds, how far have you gone? If you walk at exactly 1 meter per second, how long will it take you to travel 47 meters? The answer to both is 47 but the units are different. Remember that we do not have to walk in a straight line (i.e., with constant direction) to keep speed constant.

As a more concrete illustration of this idea, consider the curve defined by the parabola \(y = x^2/2\) with \(x\in[0,2]\). We can parameterize this curve by \(\vr(t) = \langle t, t^2/2\rangle\) for \(t\in[0,2]\). shows a plot of this parabola with equally-spaced points in the parameter \(t\) on the left. You can see in the plot on the left that equally-spaced points based on parameter values are not separated by equal distances along the curve. On the right, however, the points are equally-spaced by arc length. By this we mean that the distance along the curve between the point labeled \(s=0.5\) and the point labeled \(s=1.0\) is \(0.5\). Similarly, the distance along the curve between the point labeled \(s=0.5\) and the point labeled \(s=2.5\) is \(2\). For this reason, a unit speed parameterization is also often called a parameterization by arc length.

In the next example, we will work through how to transform a parameterization with variable speed into one that has unit speed.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

In , we will look at several measurements as we consider different drivers completing one lap around the track seen in .

All of the drivers are going the same way around the track, all start at the same location at \(t=0\), all complete one lap of the track, and all of the cars have perfect grip of the road (the race cars are never sliding). The picture of the racetrack above is given so you have an example to help you think about the tasks in this problem, not because any particular feature of the track needs to be considered for the following activity.

When we are using the analogy of the racetrack for a curve in space, each different driver corresponds to a different parameterization of the curve. If a measurement is a property of the driver, that measurement can take different values for different parameterizations. If a measurement is a property of the road, then every parameterization must have the same measured value when considered at the same location on the curve. In short, parameterizations are drivers and the curve in space is the racetrack.