I am trying to find the parametric equations of a cycloidal curve, which, instead of using a circle, uses an ellipse to oscillate around a base circle. Below are equations of the standard, circular epicycloid and hypocycloid parametric equations generally used.
Does anybody have any ideas?
References:
Hypocycloid: http://mathworld.wolfram.com/Hypocycloid.html
$$\begin{align} x &= (a−b)\cos\phi + b \cos\left(\frac{a−b}{b}\;\phi\right) \\[6pt] y &= (a−b)\sin\phi − b \sin\left(\frac{a−b}{b}\;\phi\right) \end{align}$$
Epicycloid: http://mathworld.wolfram.com/Epicycloid.html
$$\begin{align} x &= (a+b) \cos\phi−b \cos\left(\frac{a+b}{b}\;\phi\right) \\[6pt] y &= (a+b) \sin\phi−b \sin\left(\frac{a+b}{b}\;\phi\right) \end{align}$$