# Epitrochoid

The epitrochoid with R = 3, r = 1 and d = 1/2

An epitrochoid (/ɛpˈtrɒkɔɪd/ or /ɛpˈtrkɔɪd/) 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

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

where ${\displaystyle \theta }$ 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.