Explorations of x=cos(at), y=sin(bt).

For various a and b, let's investigate x=cos(at), y=sin(bt) for 0 < t < 2 .

We can easily guess that if a and b are equal, a and b are related with the period of the circle with radius 1.

After fixing a value, let's change b value.

From the graphs, we can guess the role of b value.

After fixing b value, let's change a value.




Similarly, we can guess the role of a value from the graphs.

Exporations of x=(1-t^2)/(1+t^2), y= 2t/(1+t^2).

Let's explore the graphs of
   for    -< t < 
If we would change the domain of t,

                for   -3< t < 
                 for  -< t < 3
As the domain increase, the ends of the circle approaches point ( -1,0 ).