Cyclocycloid



A cyclocycloid is a roulette traced by a point attached to a circle of radius r rolling around, 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 a cyclocycloid are


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

where $$\theta$$ is a parameter (not the polar angle). And r can be positive (represented by a ball rolling outside of a circle) or negative (represented by a ball rolling inside of a circle) depending on whether it is of an epicycloid or hypocycloid variety.

The classic Spirograph toy traces out these curves.