Assignment 10:  Explorations of Parametric Curves

by Kristina Dunbar, UGA

Parametric curves in the plane are a pair of functions of the form:

x = f(t)

y = g(t)

where the two continuous functions define ordered pairs (x,y).  These two equations are usually called the parametric equations of a curve.  The extent of the curve will depend on the range of t.

Let's graph:

x = cos (t)

y = sin (t)

Because we are dealing with cosine and sine functions, we know that the normal period is 2π, so let's let t range from 0 to 2π.

What if t didn't range from 0 to 2π?  What if it ranged from the following?

0 ≤ t ≤ 1                                0 ≤ t ≤ π                                π/2 ≤ t ≤ 3π/2

We see that t is the angle θ that the curve sweeps out, where θ = 0 has the coordinates (1,0), θ =  π/2 has the coordinates (0,1), and so on.

This should remind you of cut-outs of the unit circle.

Now let's investigate the equations

x = cos (at)

y = sin (bt)

where 0 ≤ t ≤ 2π.

====+=+=+=+=+=+=+=+=+=+====

1.  Let a = b, and a not 1.

It looks exactly the same as the earlier graph, where a = b = 1.

====+=+=+=+=+=+=+=+=+=+====

2.  Let a > b.

This looks like a parabola!

+++==========+++

Interesting, it looks like two boomerangs.

+++==========+++

Let's try it again:

+++==========+++

Once more:

====+=+=+=+=+=+=+=+=+=+====

3.  Let a < b

+++==========+++

+++==========+++

+++==========+++

These types of curves are actually called Lissajous curves, and there is a great website with an interactive lab you can explore here.

Now let's investigate the equations

x = a cos (t)

y = b sin (t)

where 0 ≤ t ≤ 2π.

====+=+=+=+=+=+=+=+=+=+====

1.  Let a = b, and a not 1.

It retains the same shape as the original equations, but the radius of the circle increases to the amount a.

+++==========+++

Let's put a few of these graphs together and see how it looks:

====+=+=+=+=+=+=+=+=+=+====

2.  Let a > b.

We get a straight line with a slope of .5, or b/a.

+++==========+++

The length of these straight lines is actually 2√(a2 + b2).

====+=+=+=+=+=+=+=+=+=+====

3.  Let a < b.

This looks like it will be the exact same thing as when a > b; the slope of the line is still b/a and the length of the line is still 2√(a2 + b2).

Some good links for Lissajous Curves:

Lissajous Curves - from Math World

B. Surendranath's Lissajous Page with Java Applet

Math.Com's Lissajous Lab