**Write-Up #10 **

**Parametric Equations**

By Jaepil Han

2. For various

anda, investigateb

(1) Varying

from 1 to 5 for integer valuesa

If

goes large, then the curve of the parametric equation has more frequency than before. However, the curves located only between y=-1 and y=1. As we may know, when the parameterais 1, xa^{2}+y^{2}=1 because (cost)^{2}+(sint)^{2}=1. Also, When the parameteris 2, x=1-2ya^{2}. And so on. Interstingly, when the parameteris even the graph has a continuous loop. When the parameterais odd the graph hasn't a continuous loop.a

Here's the graph of the parametric equation when

.a=50

(2) Varying

afrom 1 to 5 for integer values

Similarly, the value of

bgetting greater, the curve of the parametric equation has more frequency.

Here's the graph of the parametric equation when

b=50.

Even we change the values of a and b, it only changes the frequency of the curves. All the curves generated by the values of a and b make a sort of square.

Here's the graph.

Here's useful links releated to Lissajous-curves.

Lissajous-curves by WolframMathworld