Epitrochoid
In geometry, an epitrochoid ( or ) 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: x ( θ ) = ( R + r ) cos θ − d cos ( R + r r θ ) y ( θ ) = ( R + r ) sin θ − d sin ( R + r r θ ) {\displaystyle {\begin{aligned}&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)\end{aligned}}} The parameter θ is geometrically the polar angle of the center of the exterior circle.