Curvature

This section helps reinforce why having a unit speed parameterization, as was found in , is useful. The preview activity sets up the measurement of curvature as a measure of how quickly the path turns (regardless of parameterization). We also bring your attention to the classic calculus approach that is used to define curvature, but a separate theorem describes the efficient way to calculate curvature.

The second half of this section also includes a brief description of the osculating circle and how that relates to the radius of curvature. This section concludes with our follow-up activity on the driver versus the road analogy.

In , we defined \(\vT\) and \(\vN\) to measure the direction of travel and the direction of turning for an oriented curve in space. Both \(\vT\) and \(\vN\) are unit vectors that are used to capture direction aspects of motion. In the next couple of sections, we will look at a few different ways to measure the magnitude of various aspects of our motion and turning on a curve in space.

Given a parameterization of a curve in space, the speed will give us an idea about how fast an object moves along the curve at any given parameter value. We learned in the previous section that \(\vN\) tells us the direction of turning, but we have not yet developed a way to measure how much turning is happening at a given location. For example, consider the three curves shown in . We would like to be able to measure how much turning is happening at the indicated point on each curve. Right now, you probably intuitively would say there is no turning in the first curve, a small amount in the second, and considerably more in the third. However, how can we assign a number to measure this amount of turning?

In this section, we focus on measuring how quickly a curve is turning. Keeping our running example in mind, we can think of this as measuring a property of the road. How quickly a curve is turning is a different measurement from how fast an object is turning as it moves along a path. That is a property of the driver to which we will return in .

The Preview Activity showed how looking at the change in \(\vT\), the direction of travel, for small steps in terms of arc length along our curve will allow us to measure the rate at which a curve is turning. This is different than measuring how quickly an object traveling along the path needs to turn in order to stay on the path. The arguments in the Preview Activity depend only on intrinsic geometric properties of the path (road) rather than how an object (car) moves along that path.

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

Hopefully it is not surprising that the unit speed parameterization of a curve \(\vr_1(s)\) is unique. One way to think about this is that if two drivers follow the same path, each at a speed of exactly 1 meter per second, then for any point on the path, the two drivers both reach that point in the same amount of time. For instance, it will take both drivers exactly 15 seconds to reach the point that is 15 meters down the path. (Remember the most useful property of a unit speed parameterization is that the parameter value for \(\vr_1\) is equal to the numeric value of the distance traveled along the curve, albeit with different units.) Conceptually, you can see how using the unit speed parameterization allows us to measure geometric properties of the curve independent of how an object moves along the path.

We have already explored how \(\vN\) points in the direction of turning. Now, however, we are looking for a scalar quantity that measures how quickly a curve turns. To measure how quickly a path turns, we examine how quickly the direction of travel changes in terms of the unit speed parameterization. We apply the classic calculus approach to this measurement in order to obtain the exact value of this measurement.

For step 1 of the CCA, we want approximate how quickly the curve turns by looking at the difference quotient for two nearby direction of travel vectors. Suppose \(\vr_1(s)\) is the unit speed parameterization of a curve \(C\) and consider a section of \(C\) from \(\vr_1(s_0)\) to \(\vr_1(s_0+\Delta s)\). We first consider how different the direction of travel is over this curve of length \(\Delta s\). (Remember, the change in parameter value here corresponds to the arc length along the curve because \(\vr_1\) is a unit speed parameterization.) shows a plot of a sample curve and two points separated by an arc length of \(\Delta s\).

We want to know how fast the direction of travel is changing at the point \(\vr_1(s_0)\). To do this, we approximate the change in the direction of travel on an interval around this point and take the limit as the size of the interval goes to zero. We compute the difference between the directions of travel at the beginning and end of our interval by putting both \(\vT(s_0+\Delta s)\) and \(\vT(s_0)\) starting at the same location. This allows us to use vector subtraction to find \(\vT(s_0+\Delta s)-\vT(s_0)\), which is the change in the direction of travel over the given interval. This vector subtraction is shown using the triangle of vectors in .

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