Cyclocycloid
# Cyclocycloid

An 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 or negative depending on whether it is of an Epicycloid or Hypocycloid variety.

The classic Spirograph toy traces out these curves.

