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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).

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

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[edit]


  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 links[edit]