In geometry, an epicycloid is a plane curve produced by tracing the path of a chosen point of a circle — called an epicycle — which rolls without slipping around a fixed circle. It is a particular kind of roulette.
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 either:
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.