# Parametric Equation of Cycloidal Sine Curve

I am trying to find the parametric equation of a sine curve, which oscillates around a circle as it's $x$-axis. I have done preliminary approximations using Epicycloid parametric equations for the top portion of the wave and Hypocycloid parametric equations for the bottom portion while using the same exact same radii for the big and small circles for each case, respectively.

This actually worked pretty well, but I was wondering whether a sine curve, or something of the like, can be made in the same manner, since it is a little bit different. If you have any ideas or suggestions, it would be much appreciated.

References

Hypocycloid: $$x = (a-b)\cos \phi + b\cos \left( \frac{(a-b)}{b}\phi \right)$$ $$y = (a-b)\sin \phi - b\sin \left( \frac{(a-b)}{b}\phi \right)$$

Epicycloid: $$x = (a+b)\cos \phi - b\cos \left( \frac{(a+b)}{b}\phi \right)$$ $$y = (a+b)\sin \phi - b\sin \left( \frac{(a+b)}{b}\phi \right)$$

Replacing $\displaystyle \frac{b}{a}$ by $n$ yields any frequencies.