# .

An epitrochoid (/ɛpɨˈtrɒkɔɪd/ or /ɛpɨˈtroʊkɔɪd/) 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 a distance d from the center of the exterior circle.

The parametric equations for an epitrochoid are

$$x (\theta) = (R + r)\cos\theta - d\cos\left({R + r \over r}\theta\right),\,$$
$$y (\theta) = (R + r)\sin\theta - d\sin\left({R + r \over r}\theta\right).\,$$

where \theta is a parameter (not the polar angle).

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 combustion chamber of the Wankel engine is an epitrochoid.

Cycloid
Epicycloid
Hypocycloid
Hypotrochoid
Spirograph
List of periodic functions

References

J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 160–164. ISBN 0-486-60288-5.

Epitrochoid generator
Epitrochoid at 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