Epitrochoid

The parametric equations for an epitrochoid are: The parameter θ is geometrically the polar angle of the center of the exterior circle.

Special cases include the limaçon with R = r and the epicycloid with d = r. 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

The orbit of the Moon, when centered around the Sun, approximates an epitrochoid.

The combustion chamber of the Wankel engine is an epitrochoid with R = 2, r = 1 and d = 1.