Hypocycloid

A hypocycloid is a special plane curve generated by the trace of a fixed point on a small circle that rolls within a larger circle.
If the radius of a large circle is $R$, and the radius of a small circle is $r$, the following properties are shown depending on the value of $k = \frac{R}{r}$.

• If $k$ is an integer number, the curve becomes a closed curve and has $k$ curves.
• If $k$ is a rational number, it has $p$ curves if it can be simplified to $k = \frac{p}{q}$.
• If $k$ is an irrational number, the curve is not closed, filling all the space between the large circle and the circle with radius $R-2r$.