Hypotrochoid

File:HypotrochoidOutThreeFifths.gif
The red curve is a hypotrochoid drawn as the smaller black circle rolls around inside the larger blue circle (parameters are R = 5, r = 3, d = 5).

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.

The parametric equations for a hypotrochoid are:[1]

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

Where $\theta$ is the angle formed by the horizontal and the center of the rolling circle (note that these are not polar equations because $\theta$ is not the polar angle).

Special cases include the hypocycloid with d = r and the ellipse with R = 2r.

File:Ellipse as hypotrochoid.gif
The ellipse (drawn in red) may be expressed as a special case of the hypotrochoid, with R = 2r; here R = 10, r = 5, d = 1.

The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.