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: \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).
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