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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).

In geometry, 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 θ is the angle formed by the horizontal and the center of the rolling circle (these are not polar equations because θ is not the polar angle). When measured in radian, θ takes values from 0 to (where LCM is least common multiple).

Special cases include the hypocycloid with d = r and the ellipse with R = 2r and dr.[2] The eccentricity of the ellipse is

becoming 1 when (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.

Hypotrochoids describe the support of the eigenvalues of some random matrices with cyclic correlations[3]

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.
  3. ^ Aceituno, Pau Vilimelis; Rogers, Tim; Schomerus, Henning (2019-07-16). "Universal hypotrochoidic law for random matrices with cyclic correlations". Physical Review E. 100 (1): 010302. doi:10.1103/PhysRevE.100.010302.

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