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:
Special cases include the hypocycloid with d = r and the ellipse with R = 2r.
The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.