How to convert this Parametric Equation into AE expressions?

[hexart]

[hexart]

And, how to control the curvature or swings?

[wyz]

I don't get exactly what you are trying to do but to control the opacity with a Sine wave (or Cosine) I would do this:

p=Math.sin(time)

and then, since opacity is expressed as a percentile you just add this:

p*100

If you want to control the amplitude and frequency of the sine wave you can try this:

freq=2;
amplitude=10;
p=amplitude*Math.sin(freq*time);
p*100

[hexart]

Thank you, wyz;
I solved my problem with your help.

Here is mine:

freq=0.05;
amplitude=1;
p=amplitude*Math.cos(freq*YRotation+6.3);
p*100