If the smaller circle has radius r, and the larger circle has radius R = kr, then the parametric equations for the curve can be given by:
If k is an integer, then the curve is closed, and has k cusps (i.e., sharp corners, where the curve is not differentiable).
If k is a rational number, say k=p/q expressed in simplest terms, then the curve has p cusps.
If k is an irrational number, then the curve never closes, and fills the space between the larger circle and a circle of radius R+2r.
The epicycloid is a special kind of epitrochoid.
An epicycle with one cusp is a cardioid.