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Hypotrochoid

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:

$x = \left(R - r\right)costheta + dcosleft\left(\left\{R - r over r\right\}thetaright\right)$
$y = \left(R - r\right)sintheta - dsinleft\left(\left\{R - r over r\right\}thetaright\right)$

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

The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.