The TNB Frame

This section helps reinforce why having a unit speed parameterization, as was found in the previous section, is useful. The binormal vector is treated minimally, and instructors who wish to omit it can do so. We still recommend doing , three parts of which can be answered without consideration for \(\vB\).

This section can likely be done during a short class period and may be combined with an introduction to curvature. is a good place to see how the direct calculation of \(\vN\) is usually quite difficult and is a short conceptual activity that is focused on when the unit normal and unit tangent vectors do not exist. is a good activity to show why it is not typical to use the definition to calculate the unit normal. This motivates our later work on efficient calculation of \(/vN\) using the vector tools and splitting acceleration in . For this reason, we do not have many exercises that involve \(\vN\), but instead look at using calculus tools to find \(\vT\) and other vector calculus quantities.

In , we saw how a vector-valued function of one variable will graphically correspond to a curve in space. While the application of derivatives, integrals, and limits apply to these functions componentwise, we will need to apply our calculus and vector tools together to measure important properties of these curves. Conceptually, you should think about the parameterization of the curve in space as a description on how to travel through those points in space. Remember that parameterizations are not unique: there is more than one way to walk the same path!

For example, consider the curve in space that we are exploring to be a race track as shown in . Not every driver will travel along this racetrack in the same way, but every driver has to go through these same points in space, since we assume that everyone will stay on the track and not take shortcuts. In the vocabulary of this chapter, any parameterization of this curve has to contain the same points, but can possibly go through these \((x,y,z)\)-coordinates at different parameter values. More generally, some properties we measure will depend on the particular way that a driver goes along the course and some properties will be the same for all drivers when measured at the same location on the track.

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

Based on our work in , if \(\vr(t)\) is a parameterization of a curve in space, then the velocity vector, \(\vv(t)=\vr\, '(t)\), is in the direction of travel. Since the speed, \(\vecmag{\vv(t)}\), is not always 1 (the velocity is not always a unit vector), we define the direction of travel using a unit vector as follows.

By definition, the unit tangent vector exists as long as the velocity exists and is not the zero vector. If the velocity vector is \(\vec{0}\), then the object is not moving, thus the direction of travel would not make sense at that instant. If there is a jump or discontinuity in the derivative of \(\vr\) so that \(\vv\) does not exist, the direction of travel at that instant does not make sense because there is not a consistent way to define how the object is moving.

The definition of \(\vT\) allows us to separate the velocity of a moving object into its magnitude (speed) and direction (unit tangent). Using a unit vector allows us to express the direction of travel as a vector that is separate from the length of any related measurements. Remember that \(\vT\) changes along the curve and is calculated as a function of the parameter.

The next activity gives you an opportunity to practice with the velocity and unit tangent vectors and speed.

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

Recalling our analogy in which we think of a parameterization as describing how someone drives along a racetrack, we say that \(\vT\) measures the direction that the car is heading at any given instant. If we use the curve plotted in as an example of the racetrack, then \(\vT(2)\approx \langle -0.8813, -0.4691, 0.0568 \rangle\). The plot in shows how the direction of travel corresponds to the curve at the point \(\vr(2)\). What direction is the car turning at \(t=2\)? You can see by the plot that the car will be be turning left, but that is a direction relative to the car. We wish to measure the direction that the car is turning at this instant in terms of the locations given by the parameterization in an \(xyz\)-coordinate system?

We first consider \(\vT\, '=\frac{d\vT}{dt}\), which measures how quickly \(\vT\) is changing in terms of the parameter. The vector \(\vT\, '=\frac{d\vT}{dt}\) will measure the rate of change of the direction of travel. Since vectors have magnitude and direction, the derivative of a vector-valued function measures both the change in the magnitude of \(\vT\) and the change in the direction of \(\vT\). Since \(\vT\) will always have length 1 (when \(\vT\) exists), all of the change in \(\vT\) corresponds to a change in direction. Thus, the direction of \(\vT'=\frac{d\vT}{dt}\) will be exactly what we are looking for: \(\vT\, '\) will point in the direction of turning. For now, we don't care about how fast the car is turning; we want to find the direction of the turning. This leads us to define the unit normal vector, denoted \(\vN\), as the unit vector in the direction of \(\vT\, '\).

We are able to define the unit normal without the classic calculus approach because our vector tools allowed us to make sense of this measurement rather than requiring a new tool.

Based on the definition of the unit normal vector, there are a few different characteristics a curve can have that would cause \(\vN\) to not exist. The next activity leads you to consider three different cases:

1. \(\vN\)\(\vT\)
2. \(\vN\)\(\vT\, '\)
3. \(\vN\)\(\vT\, '=\vec{0}\)

is the exceptional case where \(\vN\) is easily calculated directly from the definition and the calculations are fairly clean. In the next activity, we will look at a curve with a simple polynomial parameterization that will demonstrate how easily the calculation of \(\vN\) can go off the rails.

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

Since the title of this section is and we have defined \(\vT\) and \(\vN\), you likely are expecting the definition of a vector called \(\vB\), which we do in this subsection. Since we saw in that the unit tangent and unit normal vectors are orthogonal, this final vector will be defined in a way that completes a right-handed coordinate system with \(\vT\) and \(\vN\).

This definition of \(\vB\) creates a three-dimensional right-handed coordinate system that is relative to the motion along the curve. Notice that while we had to divide by magnitude to ensure that \(\vT\) and \(\vN\) were unit vectors, the binormal vector is a unit vector because \(\vecmag{\vT \times \vN } = \vecmag{\vT} \vecmag{\vN} \sin(\theta)\) where \(\theta\) is the angle between \(\vT\) and \(\vN\). Since \(\vT\) and \(\vN\) are unit vectors and orthogonal to each other, we see that \(\vecmag{\vT}\vecmag{\vN} \sin(\theta)= 1\). The binormal vector \(\vB\) gives the axis of rotation for the motion along the curve. This measure of rotation follows our right-handed measure of rotation. If you put the fingers of your right hand in the direction of travel (\(\vT\)) and curl your fingers in the direction of turning (\(\vN\)), then your thumb will correspond to the axis of rotation based on how you are moving along the curve.

The TNB frame is also called the Frenet frame and is frequently used in fields such as aviation and space travel because much of the important flight information is expressed in terms of relative measurements to the craft and its motion. The plane determined by the vectors \(\vN(t)\) and \(\vB(t)\) is called the normal plane to the curve at \(t\) and consists of all lines that are orthogonal to the tangent line to the curve at \(t\). The plane determined by \(\vT(t)\) and \(\vN(t)\) is called the osculating plane and is the plane the comes closest to containing the curve for inputs near \(t\). In the remaining sections of this chapter, we will return to these ideas of what is happening on the osculating plane.

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

To conclude this section, we consider whether the measurements of \(\vT\), \(\vN\), and \(\vB\) are properties of the driver or the road. A parameterized curve has a stated orientation (direction of travel). Remember that a measurement involving a parameterized curve is a property of the driver if two different parameterizations can have different measurements at the same location on the curve. It is important to remember that this does not mean that you look at the same parameter value on the curve, but rather the same location on the curve. A measurement on a parameterized curve is a property of the road if every parameterization of this curve must have the same measurement for a fixed location on the curve. These are the same ideas that were first discussed in . In the analogy of a parameterization being a description for how a particular race car travels around a track as a function of time, a measurement is a property of the driver if that measurement can have different values for different drivers when measured at the same point on the racetrack. A measurement is a property of the road if that measurement must have the same value for all drivers when measured at the same point on the racetrack.

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