Hypotrochoid
In geometry, a hypotrochoid is a roulette traced by a point attached to a circle of radius r rolling around the inside of a fixed circle of radius R, where the point is a distance d from the center of the interior circle. The parametric equations for a hypotrochoid 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}}} where θ is the angle formed by the horizontal and the center of the rolling circle (these are not polar equations because θ is not the polar angle).