Parametric Equations


Brandon Samples


The parametric equation that I want to consider has the form

[cos(t), sin(nt)]

for different values of n. Let's begin by considering the following animation.

Okay, so what you are looking at is an animation of the above parametric equation when the t variable ranges between 0 and 2 pi and n ranges from -10 to 10 with 250 iterations. I wanted to start with the above parametric equation precisely because for certain values of n, we know exactly what the graph should look like. You'll notice that there are two instances for which the graph becomes a circle. Can you guess which ones they will be? Obviously, the case of n=1 should give you a circle, but which other one will it be?

Well, the trig functions cosine and sine are both periodic with period 2pi, so we should expect that the other circle occurs when n = -1.

Now, what's most interesting to me is to think about what happens as you step up n past n=1. Let's begin by looking at the case when n = 1.5.

Now, it appears that the curve moves from the point (1,0) and moves to the left ending at the point (-1,-1), but is this all that this curve does? Why do the above curves cover the interval twice but this most recent curve doesn't seem to? We might guess that the curve is double covered. Let's prove this.

The claim is that the point (cos(t),sin(3/2t)) = (cos(2pi-t),sin((3/2)(2pi-t)) = (cos(2pi-t),sin(3pi-3/2t)) = (cos(2pi-t), sin(pi-3/2t)) which is true on the interval from 0 to pi, so this is what we should expect.

Now, what about if we change n to 2, this doubles the periodicity of the sine term while leaving the cosine term alone, so we might expect some spiral curve which is connected. Is this what you expected?

Now, our last intersting observation is to think about when the curves will be connected? Should we expect this to happen only when the n term is an integer? Let's look at an example. this question is a bit more subtle.