then the curve has cusps.

Count the animation rotations to see and

If is an irrational number, then the curve never closes, and forms a dense subset of the space between the larger circle and a circle of radius .

The distance from the origin to the point on the small circle varies up and down as

:where
* = radius of large circle and
* = diameter of small circle .

The epicycloid is a special kind of epitrochoid.