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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:[1]

where is the angle formed by the horizontal and the center of the rolling circle (these are not polar equations because is not the polar angle). When measured in radian, takes values from to where LCM is least common multiple.

Special cases include the hypocycloid with d = r is a line or flat ellipse and the ellipse with R = 2r and d > r or d < r (d is not equal to r).[2] (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 alsoEdit


  1. ^ J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 165–168. ISBN 0-486-60288-5.
  2. ^ Gray, Alfred. Modern Differential Geometry of Curves and Surfaces with Mathematica (Second ed.). CRC Press. p. 906. ISBN 9780849371646.

External linksEdit