Class 6.2 (and 4.M, 6.M, and 7.M)

Parametric Equations and Tangent (Velocity) Vectors

Reading assignment for Wednesday, October 8

• Stewart 9.1 and 11.7
• Review the instruction sheet for Lab 4.M (Writing Letters Using Parametrized Curves)
• This Web page

Parametric representation of curves

1. Daredevil Bob raced his motorcycle through the desert faster and faster. Relative to his RV (and with distances measured in miles) his position as a function of time was given by

   x = 2 cos(1 + t²) (east-west),

   y = 2 sin(1 + t²) (north-south).

   He started at noon (t = 0). At t = sqrt(2*Pi - 1) he found himself back at his starting point and slammed on the brakes. We can see that Bob's path satisfies

   x² + y² = 4,

   and that therefore it is a circle with radius 2 miles (circumference 4*Pi miles).

2. To commemorate Bob's feat, the park service built a paved road along his route. When potholes need to be filled, the advance crew records their location by noting their distance along the road from the starting point. The point at distance s is located at

   = <2 cos(s/2), 2 sin(s/2)>.

   The domain of this vector-valued function is the interval of s between 0 and 4*Pi. (The same equations were used by the surveyors in building the road in the first place.)

3. Bob's sister, Sarah, stayed at home studying calculus. In one problem she encountered the circle with equation

   x² + y² = 4.

   She saw that it would be impossible to solve for y as a function of x, because in choosing the sign of the square root she would lose half of the circle. For the same reason, the circle can't be represented by x as a single function of y. Sarah realized that a good way to represent the circle is to treat x and y equally, writing each of them as a function of the angle, u, around the circle.

   x = 2 cos(u),     y = 2 sin(u).

There are several conceptual points to be noted here.

1. In the third scenario, we started with a curve (a geometrical object) and found a parametric representation of it. In the first story, we started with a parametrization, originating as the description of a motion, and derived the curve (or "orbit") from it. (The second scenario can be looked at either way, depending on whether you are building the road or repairing it.)
2. For a given curve, the parametric representation is not unique. We saw three different pairs of parametric equations that all describe the same circle. The second and third parametrizations seem simpler and more natural, but the first one was the one appropriate to a particular physical problem. (The distance parameter s is called arc length and is the geometrically natural way of parametrizing an arbitrary curve for which "angle" has no meaning.)
3. In the first scenario the parameter had the physical significance of time, a variable as "real" as the coordinates x and y themselves. In the other cases the parameter was introduced more arbitrarily, mainly as a tool for describing the curve. Our intuition for motion is so strong that one often uses the language of a point "moving" along the curve even in more abstract situations where the curve does not have a mechanical interpretation (just as we talk of a scalar- valued function "changing" even when the independent variable is not time).
4. In the second story we wrote the parametrization as a vector-valued function of the parameter, while in the other two cases we wrote two separate scalar equations. The freedom to choose either of these points of view is valuable, as we'll see in the discussion of velocity below.

It is easy to write down simple parametric equations, such as

x = t²,     y = t⁵,

which describe curves that are not terribly familiar. Conversely, there are familiar curves, such as hyperbolas, for which a parametrization may not be immediately obvious. You will see a variety of examples of parametrized curves in the homework and in Lab 6.M (The Flight of a Baseball). Our primary interest, however, is in (a) parametrized curves in the abstract, as generic two-dimensional motions in physics; and (b) the very simplest cases, straight lines and circles. We have probably said enough about circles already.

A line has the parametric equations

x = x₀ + v_x t,     y = y₀ + v_y t,

where x₀, y₀, v_x, and v_y are constants. If t varies through all real numbers, the image curve is the entire line; smaller intervals for t yield line segments. A nice feature of the parametric representation is that vertical lines do not need to be treated as a separate case: they are just those for which v_x =0. The two parametric equations can be combined into the vectorial parametric equation

r = r₀ + vt.

With this choice of notation it should be clear that this formula describes the motion of a free particle with initial position r₀ = < x₀, y₀> and velocity v = < v_x, v_y>.

Tangent vectors; vectorial velocity and acceleration

In the foregoing example the velocity vector, v, points along the line. (In the lab handout it was constructed by subtracting the coordinates of one point on the line from those of another.) We should also note that its components can be obtained by differentiating the parametric equations:

v_x = dx/dt,     v_y = dy/dt.

These derivatives happen to be constants (the same for all t) in this special case.

The derivatives of a more general set of parametric equations have a similar significance. For the example

x = t²,     y = t⁵,

we find

dx/dt = 2t,     dy/dt = 5t⁴.

We can put these together into the vector

v(t) = dr/dt = <2t, 5t⁴>.

Let's plot the curve and attach the vector v(0.8) at the point r(0.8).

[Insert drawing.] (temporary screen with both drawings)

We see that the vector is tangent to the curve (and points in the direction in which the parameter increases). This is true in general, because it can be shown (see p. 554 (Sec. 9.2) of Stewart) that the slope of the vector v(t) is the same as the slope of the curve at the point r(t):

(dy/dt)/(dx/dt) = dy/dx.

(This fact will be easier to understand after we have studied the "chain rule" next week.)

Side remark: If dx/dt = 0 and dy/dt is not zero, then the tangent line and tangent vector are vertical and the slope of the curve is undefined (infinite) at that point. If both derivatives are zero (as at t = 0 on our example curve), the tangent vector does not have a well-defined direction; in physical terms this indicates that the moving point has "slowed down" to a halt at the time t. A better parametrization should then be chosen (see below) to obtain a more useful tangent vector.

The tangent vector v(t) = dr/dt is also called the derivative of the vector-valued function r(t). Instead of assembling it from the derivatives of the scalar-valued coordinate functions x(t) and y(t), one can define it directly as the limit of a difference quotient (see p. 727 of Stewart).

The tangent vector to a curve at a point is not unique; it depends upon the parametrization. Sarah's tangent vector to the circle at the point at angle u, calculated from her parametrization

r = <2 cos(u), 2 sin(u)>,

is

v = <-2 sin(u), 2 cos(u)>.

Her brother's parametrization is

r = <2 cos(1 + t²), 2 sin(1 + t²)>.

Using Maple, or peeking ahead at next week's reading to learn the chain rule, we see that his tangent vector is

v = <-4t sin(1 + t²), 4t cos(1 + t²))>.

Even after Sarah substituted u = 1 + t² into her formula (so that she and Bob would be labeling the points in the same way), their formulas are different. The two tangent vectors point in the same direction, but they have different lengths. The road crew's arc-length parametrization yields the unit tangent vector, one with length 1; this choice is preferred for some purposes.

[Insert drawing] (temporary screen with both drawings)

On the other hand, Bob's tangent vector is the velocity of his physical motion. (More generally, whenever the parametrization of a curve is not arbitrary but has some significance of its own, then the same is true of the corresponding tangent vector.) Since the velocity vector is itself a function of t, one can differentiate it again to get the acceleration vector, whose components are the (scalar) accelerations in the x and y directions.

In fact, from Daredevil Bob's point of view, the acceleration vector is the most fundamental, since that is what he controls with his accelerator pedal (and handlebars). In mechanics we usually are given the acceleration and need to find the velocity and position (Newton's second law). In a vectorial situation, this means that we start from a vector-valued function a(t) and end up with a parametric curve describing the motion. [example]

More about parametric equations

Parametric equations provide a useful way of looking at two graphs simultaneously. Suppose one has two functions, f(t) and g(t). Then the usual way of graphing them would be to use two graphs, f(t) vs. t and g(t) vs. t. But parametric plotting allows the suppression of t to show the quantities represented by f and g plotted on one coordinate system.

For example, the population F of foxes in an area might vary with year approximately like F = 50 + 20 sin(t), and the population H of hares in the area might vary like H = 500 + 300 cos(t). Then, plotting these functions as one parametric curve in the F-H plane (incidentally, it's an ellipse centered at (F,H) = (50,500)) gives an additional way of visualizing the interaction of foxes and hares. It might suggest a causal relationship between the two, such as that more foxes cause a decline in the hare population, while fewer hares causes a decline in the fox population.

Example 4 in the lab handout constructed a rectangle as four parametrized line segments, each with the parameter domain $0 < t < 1$. Translated from Maple into standard mathematical notation, those parametric equations are

[insert equations] (temporary screen with both sets of equations)

This way of proceeding is quite adequate and standard for computer graphics applications. However, sometimes it might be important to consider the entire rectangle as a single parametrized curve. This situation will arise automatically if the rectangle is the path of a moving body. (It starts at the corner with coordinates (1,1), moves along the first segment to (3/2,1), then along the fourth segment to (3/2,3/2), then along the third segment in the reverse direction to (1,1/2), then along the second segment in the reverse direction to (1,1).) In that case the parametric equations involve two piecewise-defined functions instead of eight simple functions. We can easily construct the new parametric equations from the old ones by remembering "shifting and rescaling" from the first two weeks of this course. First note that to move in the reverse direction along a segment parametrized by 0 < t < 1, we merely need to replace t by 1 - t in the formulas. Shifting each interval of length 1/4 to make it start at t = 0 and then scaling by 4 to make it of length 1, we then get these formulas:

[insert equations] (temporary screen with both sets of equations)

Of course, these expressions can be simplified, but we have left them in the form that makes clearer how they arose from the original parametrizations of the four line segments.