Parametric Equations of Epicycloid
Parametric Equations
If the smaller circle has radius r, and the larger circle has radius R=kr,
x = (R+r)*cos(theta) - r*cos((R+r)/r*theta)
y = (R+r)*sin(theta) - r*sin((R+r)/r*theta)
If k is an integer, then the curve is closed, and has k cusps.
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 forms a dense subset of the space between the larger circle and a circle of radius R+2r.