REAL Services Analytical Almanac: epicycloid.html

	Application Notes:
	epicycloid is a plane curve produced by tracing the path of a chosen point of a circle — called epicycle — which rolls without slipping around a fixed 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: $x(\theta) = r (k+1) \left( \cos \theta - \frac{\cos((k+1)\theta)}{k+1} \right)$ $y(\theta) = r (k+1) \left( \sin \theta - \frac{\sin((k+1)\theta)}{k+1} \right)$ 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 Epicycloid Examples *REAL Services* 700 Portage Trail Cuyahoga Falls, OH 44221.3057 voice: 330.630.3700 fax: 330.630.3733 © 1995-2005 REAL Services_® U.S.A. - Analytical Almanac All Rights Reserved