Epicycloid refers to the trajectory drawn by a vertex on a small circle when a small circle is rolled outside a large circle.

The radius of a large circle is denoted by R, the radius of a small circle is denoted by r, and the following properties are given by k = R / r.

- If k is an integer number, the curve becomes a closed curve and has k points.
- If k is a rational number, it can have p points if it can be simplified to k = p / q.
- If k is a irrational number, the curve is not closed, filling all the space between a large circle and a circle with radius R-2r.