Hypotrochoid

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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.

Image:Hypotrochoid-1.gif

The red curve is a hypotrochoid traced as the smaller black circle rolls around inside the larger black circle (parameters are R = 1.0, r = 0.6, d = 1.2).

The parametric equations for a hypotrochoid are:

<math>x = (R - r)\cos\theta + d\cos\left({R - r \over r}\theta\right)</math>

<math>y = (R - r)\sin\theta - d\sin\left({R - r \over r}\theta\right)</math>

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

The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.