In geometry, an epitrochoid (or) is a roulette traced by a point attached to a circle of radius rolling around the outside of a fixed circle of radius, where the point is at a distance from the center of the exterior circle.
The parametric equations for an epitrochoid are:
\begin{align} &x(\theta)=(R+r)\cos\theta-d\cos\left({R+r\overr}\theta\right)\\ &y(\theta)=(R+r)\sin\theta-d\sin\left({R+r\overr}\theta\right) \end{align}
(x(\theta),y(\theta))
Special cases include the limaçon with and the epicycloid with .
The classic Spirograph toy traces out epitrochoid and hypotrochoid curves.
The paths of planets in the once popular geocentric system of deferents and epicycles are epitrochoids with
d>r,
The orbit of the Moon, when centered around the Sun, approximates an epitrochoid.
The combustion chamber of the Wankel engine is an epitrochoid.