Consider the parametric equations

These class of parametric curves are called the Lissajous curves.

First, I will investigate graphes for

Let's compare with ,

I found that a difference between Lissajous curves and second graphes is the size. Second graphes are smaller than the Lissajous curves.

The Lissajous curves when 0 < t < 1000

, when 0 < t < 1000

We can know

if the range of t is changed; that is, t is getting larger, the graphes of Lissajous curves and second graphes will be deferent.

Second, I will investigate Lissajous curves and graphes for

When 0 < t < 1000

We can get the same result as first investigation. When , I found that a difference between Lissajous curves and second graphes is the size. Second graphes are smaller than the Lissajous curves.

In addition, if the range of t is changed; that is, t is getting larger, the graphes of Lissajous curves and second graphes will be deferent.

I think that graphes look like a plane when 0< t< 1000.