**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√(a^{2} + b^{2}).

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

**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√(a^{2} + b^{2}).

Return to my Home Page.

*Some good links for Lissajous Curves:*

Lissajous Curves - from Math World