An epitrochoid (/ɛpɪˈtrɒkɔɪd/ or /ɛpɪˈtroʊkɔɪd/) is a roulette traced by a point attached to a circle of radius r rolling around the outside of a fixed circle of radius R, where the point is at a distance d from the center of the exterior circle.
The parametric equations for an epitrochoid are
The parameter is geometrically the polar angle of the center of the exterior circle. (However, is not the polar angle of the point on the epitrochoid.)
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 orbits of planets in the once popular geocentric Ptolemaic system are epitrochoids.
The orbit of the moon, when centered around the sun, approximates an epitrochoid.
The combustion chamber of the Wankel engine is an epitrochoid.
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Epitrochoid in Animation Nodes I Blender 2.8 Tutorial

Deriving the Equations of an Epicycloid

Blender Animation Nodes Tutorial: Epitrochoid in 3D
Transcription
See also
 Cycloid
 Cyclogon
 Epicycloid
 Hypocycloid
 Hypotrochoid
 Spirograph
 List of periodic functions
 Rosetta (orbit)
 Apsidal precession
References
 J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 160–164. ISBN 0486602885.
External links
 Epitrochoid generator
 Weisstein, Eric W. "Epitrochoid". MathWorld.
 Visual Dictionary of Special Plane Curves on Xah Lee 李杀网
 Interactive simulation of the geocentric graphical representation of planet paths
 O'Connor, John J.; Robertson, Edmund F., "Epitrochoid", MacTutor History of Mathematics archive, University of St Andrews
 Plot Epitrochoid  GeoFun