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:^{[citation needed]}

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

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

Where θ is the angle formed by the horizontal and the center of the rolling circle (note that these are not polar equations because θ 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

• Epitrochoid
• Cycloid
• Hypocycloid
• Epicycloid
• Spirograph

References

• J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 165–168. ISBN 0-486-60288-5. 

External links

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