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.

The red curve is a hypotrochoid drawn as the smaller black circle rolls around inside the larger blue circle (parameters are R = 5.0, r = 3, d = 5).
The red curve is a hypotrochoid drawn as the smaller black circle rolls around inside the larger blue circle (parameters are R = 5.0, r = 3, d = 5).

The parametric equations for a hypotrochoid are:

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

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

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

The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.

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