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.
where is the angle formed by the horizontal and the center of the rolling circle (these are not polar equations because is not the polar angle). When measured in radian, takes values from to where LCM is least common multiple.
Hypotrochoids describe the support of the eigenvalues of some random matrices with cyclic correlations 
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- Flash Animation of Hypocycloid
- Hypotrochoid from Visual Dictionary of Special Plane Curves, Xah Lee
- Interactive hypotrochoide animation
- O'Connor, John J.; Robertson, Edmund F., "Hypotrochoid", MacTutor History of Mathematics archive, University of St Andrews