I have tried to change the traditional hypocycloid a bit. What I've basically done is that a circle now rolls inside an ellipse. I am trying to find the equation for the same. I am mostly done, except the last part. Where I've to find $\phi$ in terms of $\theta$. Could anyone please help me regarding this?
Also, I would appreciate if somebody could post the whole answer, so that I can double check my steps.
Thanks a lot for your help :)
P.S.: Can we do this without elliptical integrals? Because that's too complicated for my level.