Special Plane Curves

Math 50C --- Multivariable Calculus
David Arnold

Introduction

One of the more important (and entertaining) topics in multivariable calculus is the study of parametric equations. If you are given a particular system of parametric equations, it is a simple task to draw the graph. For example, consider the following system of parametric equations.
x=3cos(5t)
y=5sin(7t)

The following Matlab code uses this set of parametric equations to produuce an image similar to that in Figure 1. 

t=linspace(0,2*pi,500);
x=3*cos(5*t);
y=5*sin(7*t);
plot(x,y)
Figure 1.

Finding a Parametrization for a Given Curve

Although it is quite easy to sketch the graph of a given set of parametric equations, it can be extremely difficult to engage in the opposite process: finding a parametrization for a given curve. For example, in class we defined the cycloid as the trace of a point on the rim of a wheel that rolls along a line without slippage (See Figure 2).
Figure 2.

As you saw in class, it was not an easy task to find a parametrization for the cycloid. If you recall, we found the following parametrizations in class. In each case, a measures the radius of the rolling wheel.
 

x = a(theta-sin(theta)) 

y = a(1-cos(theta))

 The parameter theta measures the angle of 

revolution from the vertical
x = s - a[sin(s/a)] 

y=a[1-cos(s/a)]

 The parameter s measures the arc length 

of the portion of curve already drawn.
x = a[wt - sin(wt)] 

y = a[1 - cos(wt)]

 The parameter t measures the time (s) and w is 

the angular velocity (rad/s) of the wheel.

Your Assignment

There is a huge collection of curves in the plane some of whose properties were known by the ancient Greeks mathematicians. I would like you to select one of these curves for extensive study and then write a report on your findings. Your report must be submitted in electronic form (so that we can publish it on our web site) and must contain the following components.
  1. A description or definition of the curve.
  2. You must develop a system of parametric equations describing the curve. No credit will be given for simply quoting a system of parametric equations that describe the curve. You must start from the definition and show all of the analysis and geometry that lead to the development of the parametric equations for your chosen curve. Think of this part of the assignment as follows: Suppose you are giving a lecture on the cycloid. You begin with a definition of the curve as the locus of a point on a wheel rolling along a line. You then proceed in your lecture to develop the parametric equations of the cycloid for your audience. Thus, anyone reading your paper should be able to follow your mathematical argument and fully understand the path leading to your parametric equations.
  3. Matlab-generated images of your chosen curve. You may also use other software to generate additional images, if you believe their inclusion will better illustrate your discussion of the curve. In particular, you might want to investigate use of  the Geometer's Sketchpad.
  4. Include some history of your curve in your report. What famous mathematicians worked with or are associated with your chosen curve? How has your curve been used and applied to other problems?
Use the Scientific Notebook to write your report or use the editor in Netscape to prepare your report in HTML. Note: There is a possiblity of creating Java applets using the Geometer's Sketchpad. See me for help. Save your file and submit your work to me on disk.

Grading Your Report

Your report will be graded and count the equivalent of one examination toward your final grade in the class. All reports are due no later than May 18, 2006.

How Can I Find Material for My Report?

There are a number of beautiful sources where you can find some fabulous possible plane curves for your report. First of all, I recommend the following textbooks. Each of these texts have excellent bibliographies with excellent additional resources. They are available in my office. The Humboldt State University is also an excellent source for books on plane curves.

There are also a number of beautiful sites on the internet that catalog a host of special plane curves. You definitely want to visit the following links.