Hypotrochoid

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:

[math]\displaystyle{ x (\theta) = (R - r)\cos\theta + d\cos\left({R - r \over r}\theta\right) }[/math]

[math]\displaystyle{ y (\theta) = (R - r)\sin\theta - d\sin\left({R - r \over r}\theta\right). }[/math]

Where [math]\displaystyle{ \theta }[/math] is the angle formed by the horizontal and the center of the rolling circle (these are not polar equations because [math]\displaystyle{ \theta }[/math] is not the polar angle). When measured in radian, [math]\displaystyle{ \theta }[/math] takes value from [math]\displaystyle{ 0 }[/math] to [math]\displaystyle{ 2 * \pi * \frac{LCM(r, R)}{r} }[/math]where LCM is least common multiple.

Special cases include the hypocycloid with d = r and the ellipse with R = 2r. (see Tusi couple)

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

The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.

See also

• Cycloid
• Cyclogon
• Epicycloid
• Rosetta (orbit)
• Apsidal precession

External links

• Weisstein, Eric W.. "Hypotrochoid". http://mathworld.wolfram.com/Hypotrochoid.html. 
• Flash Animation of Hypocycloid
• Hypotrochoid from Visual Dictionary of Special Plane Curves, Xah Lee
• Interactive hypotrochoide animation
• O'Connor, John J.; Robertson, Edmund F., "Hypotrochoid", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Curves/Hypotrochoid.html .

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