Finding extreme values of a deltoid curve Take a look at hypocycloids (< If the larger circle has a radius of three times the radius of the smaller circle, a deltoid is created. Here, we consider a deltoid. **Calculating the parametric equations for $x(θ)$ and $y(θ)$ after simplification with $R = 3r$.** $$x(\theta)=2r\cos\theta+r\cos2\theta \quad\hbox{and}\quad y(\theta)=2r\sin\theta-r\sin2\theta.$$ How long does it take the small circle to walk along the complete larger circle?** Help is very much appreciated!

I'll answer only your 4th question, for the others see my comment above. If $\omega$ is the angular velocity of the small circle, then its center moves with linear velocity $v=\omega r$. On the other hand, the center goes once around a circle of radius $2r$, so the time it takes is $$ T={4\pi r\over v}={4\pi \over \omega }=2\tau, $$ where $\tau=2\pi/\omega$ is the time needed for the small circle to make a complete turn around itself.