Vienna Standard Mean Ocean Water

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:Template:Fact

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

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

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.

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.

See also

• List of periodic functions
• Epitrochoid
• Cycloid
• Hypocycloid
• Epicycloid
• Spirograph

References

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External links

• Flash Animation of Hypocycloid
• Hypotrochoid from Visual Dictionary of Special Plane Curves, Xah Lee
• Interactive hypotrochoide animation
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