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 = 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 = 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.

We made a tool to draw a hypocycloid with a 3D printer. Please refer to the link below.