Parametric Plots: A Creative Outlet - Parameterizing Circles and Ellipses

Judy Holdener and Keith Howard
To start, consider the parameterization (x(t), y(t)) = (cos(t), sin(t)), with t varying from 0 to 2. At time t = 0, the particle is at the point (x(0),y(0)) = (cos(0), sin(0)) = (1, 0). As t increases from 0 to /2, cos(t) decreases in value from 1 to 0, and sin(t) increases from 0 to 1. Hence, for t /2, the particle moves from its starting point at (1, 0) to the point (0, 1). It is not hard to see that the result is a quarter circle of radius 1 traversed in the counter-clockwise direction.

 Parametric Plotter

Using the MAPLET: A MAPLET is a graphical interface that allows the user access to the power of MAPLE without interacting with the complex underlying code. The Parametric Plotter MAPLET, which can be opened by clicking the link at the right, is designed to plot parametric curves. Please be patient while the initial MAPLET loads -- you may not see any indication onscreen that anything is happening until the Java/MAPLET button appears on your tsk bar.

The MAPLET is composed of, from top to bottom, a plotter box, four text fields to enter the parametric equations and the parameter plotting range, and a sequence of buttons designed to plot a designated parametric curve, clear the last curve plotted, clear all plots, and exit the MAPLET. Upon opening the MAPLET, you will see that it has been initialized with the parameterization (x(t), y(t)) = (cos(t), sin(t)), varying t from 0 to 2. Notice that the terms in the text fields are written in MAPLE syntax. For example, 2 is entered as 2*Pi. The syntax required in all text fields must be consistent with the language of MAPLE. Thus, multiplication must be made explicit, and special functions, such as the sine function, must be written just as in the MAPLE program.

By clicking the "Plot" button, you can obtain a plot of the initialized parametric functions. To plot other parametric equations, simply make the necessary changes in the appropriate text fields. The plots of new parametric curves will overlay the plots of previous curves, unless you use the "Clear" button, which will clear all plots from the plotter, allowing you to start anew. Clicking the “Clear Last” button will erase just the most recent curve plotted. The plots produced by the MAPLET are not drawn on a one-to-one scale, thus special attention should be given to axes labeling, as images may appear slightly warped. When you are done using the Parametric Plotter MAPLET, you can close the MAPLET by clicking the "Exit" button.

NOTE: If the sizing of your MAPLET window is not optimal, and you are having difficulty seeing all of the buttons/features, then see the Notes to Instructor.

Exercise 2.1
  1. Using the Parametric Plotter MAPLET, plot the curve parameterized by
    (x(t), y(t)) = (cos(t), sin(t)) for 0t /2, then for 0 , and finally for 0t 3/2. What do you observe?

  2. Recall that the equation of the unit circle arises from the distance formula -- all points on the circle are located at a distance of exactly one unit from the origin. Use this equation to show that all points of the form (cos(t), sin(t)) do indeed fall on the unit circle.

Exercise 2.2
  1. Next use the MAPLET to plot the curve parameterized by (x(t), y(t)) = (cos(2t), sin(2t)) for 0t /2, then 0 , and finally 0t 3/2. What do you observe?

  2. Plot the curve parameterized by (x(t), y(t)) = (cos(t/2), sin(t/2)) for 0t /2, then 0 , and finally 0t 3/2. What do you observe?

  3. Plot the curve parameterized by (x(t), y(t)) = (cos(-t), sin(-t)) for 0t /2, then 0 , and finally 0t 3/2. What do you observe?

  4. Generalize your results from parts a) and b). How does the parameterization (cos(kt), sin(kt)) compare to (cos(t), sin(t))?

Exercise 2.3   

For each of a) through c), use the MAPLET to check your answer by producing the appropriate parametric plot.

  1. Find a parameterization of a circle centered at the origin having radius 3.

  2. Find a parameterization of a circle centered at (5, 3) having radius 3.

  3. Can you figure out how to plot ellipses? Produce the plot of an ellipse centered at the point (5, 3) having a major axis of length 10 and minor axis of length 6. The major (respectively, minor) axis is defined to be the line segment serving as the largest (respectively, smallest) possible diameter of the ellipse.