Write- Up: Investigations with Parametric Equations

In this assignment I will work with equations of the form:

where a=b,a<b, a>b, and a varying n.

Let's begin with the simplest case n=1, then look at each subcase.


Now, let's look at the case of n=2:

Case n=3:

Case n=4 :

Upon further investigation I realized that there is not much of a difference beyond this until n is very large. Let' see an example of this.

Case n=1000:

After looking at several cases, we can see a clear pattern with our equation for differing values of n. The case a=b seems to always be circular in shape. The case a<b seems to always be that of a vertical ellipse. The case a>b seems to always be the case of a horizontal ellipse.