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 a rational number, say k=p/q expressed in simplest terms, then the curve has p cusps.
The epicycloid is a special kind of epitrochoid.
An epicycle with one cusp is a cardioid.
We assume that the position of is what we want to solve, is the radian from the tangential point to the moving point , and is the radian from the starting point to the tangential point.
Since there is no sliding between the two cycles, then we have that
By the definition of radian (which is the rate arc over radius), then we have that
From these two conditions, we get the identity
By calculating, we get the relation between and , which is
From the figure, we see the position of the point clearly.
See also 
- Special cases: Cardioid, Nephroid
- List of periodic functions
- Deferent and epicycle
- Epicyclic gearing
- J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 161,168–170,175. ISBN 0-486-60288-5.