Hypocycloid is defined by the following parametric equations:
$$x(\phi) = (a - b) \cos(\phi) + b \cos\big(\dfrac {a - b} b \phi\big)\\ y(\phi) = (a - b) \sin(\phi) - b \sin\big(\dfrac {a - b} b \phi\big)$$
where $a$ and $b$ are radii of the fixed and moving circles respectively and $\phi$ is an angle.
Hypocycloid is closed (has a period) if $\dfrac {a}{b}$ is rational. Am I correct that the period in this case equals
$$2\pi\big(\dfrac a {\gcd(a, b)}\big)$$
or, equivalently,
$$\phi \in \big[0;2\pi(\dfrac a {\gcd(a, b)})\big]$$
Thanks!