hypicycloid

is a special plane curve generated by the trace of a fixed point on a small circle that rolls within a larger circle. It is comparable to the cycloid but instead of the circle rolling along a line, it rolls within a circle.

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 within the larger circle except for a disk of radius R − r in the center of the larger circle

Hypicycloid Examples