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 red curve is a cyclocycloid (in this case an hypocycloid) drawn as the smaller black circle rolls around inside the larger blue circle (parameters are R = 5, r = -3, d = 5).

The cyclocycloid (in this case an epicycloid) with R = 3, r = 1 and d = 1/2

The parametric equations for a cyclocycloid are

${\displaystyle x(\theta )=(R+r)\cos \theta -d\cos \left({R+r \over r}\theta \right),\,}$

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

where ${\displaystyle \theta }$ 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.