Mathematics 36, section 5 – AP Calculus Lab 7 – Parametric Curves November 30, 1999

Goals

Today, we want to use Maple to plot parametric curves and get more familiar with the concept by seeing a collection of interesting examples.

Parametric Curve Plotting in Maple

The plot command that we have used to draw graphs y = f(x) in the plane can also be used for parametric curve plotting. The format is somewhat different however:

                     plot([x-coord,y-coord,t-range]);

where x-coord and y-coord are the expressions for the x and y coordinates on the curve as functions of the parameter t, and t-range is the range of t values that you want to see plotted. For example, try the command:

                  plot([3*cos(t)+1,3*sin(t),t=0..2*Pi]);

Maple’s default in plotting is to try to “fill up” the screen as much as possible. This means that different scales are often used on the two coordinate axes. Look carefully at your plot and notice that this has happened. To prevent this from happening and get a true 1:1 ratio between the x-axis and y-axis scales, you can add an option to the plot command scaling=CONSTRAINED. Try this and reexecute your plot command to see the effect.

To plot more than one parametric curve together on the same set of axes, you can put the specifications for the different curves into a set of curly brackets {, }, and the options outside. For instance, try

 plot({[cos(t),sin(t),t=0..Pi],[2-3*t,2*t,t=0..1]}, scaling=CONSTRAINED);

Which curve is which?

Lab Questions

A) How would you use parametric curve plotting to plot a circle of any given radius r with center at any given point (h, k)? Illustrate your method by showing the circles: center (0, 0), radius r = 1, center (0, 0), radius r = 3, center (−1, 2), radius r = 4 together in the same plot.

B) An ellipse \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 can be thought of as a circle of radius 1 “stretched or shrunk” by a factor of a in the x-direction, and by a factor of b in the y-direction. Explain how this leads to parametric equations for ellipses. Also explain how to shift the whole ellipse so that its center is at a general given point (h, k) rather than at (0, 0). Illustrate by showing ellipses: center (0, 0), a = 1, b = 2 center (1, 0), a = 3, b = 2 center (2, −1), a = 4, b = 1 together in the same plot.

C) Plot the parametric curve x = t^{2} − 1, y = t^{3} − t, for t ∈ [−2, 2]. Why does the curve cross over itself at the origin (how many different t-values give (x, y) = (0, 0))?

D) Lissajous figures. The Lissajous figures are the parametric curves given by

x = cos(at) y = sin(bt)

where the constants a, b are any positive numbers.

1) If a, b are integers, then the Lissajous figure forms a nice closed curve (it returns to its starting point after some interval of t values). Plot the Lissajous figures with

\{ \begin{matrix}a = 2, b = 5 \\ a = 2, b = 4 \\ a = 3, b = 7 \end{matrix}

for t ∈ [0, 2π] (but not on the same axes – that would be a real mess!) What happens in the first one? Why the two “horns”? (It may help to plot x versus t and y versus t separately to see what is going on here.) Experiment with some different a, b, including the case a = 1, b = 2. Is that one the same as one you saw before? What is the general pattern of the shape of the Lissajous figure with integer a, b? Does it depend on whether a, b are odd or even? Explain.

2) Now try the case a = 2, b = \sqrt{3}. Take larger intervals [0, L] for L > 2π. What about a = π, b = e? How are these cases different?

Assignment

Individual lab reports, due Monday, December 6.