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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* = 2*r* and *d* ≠ *r*.^{[2]} The eccentricity of the ellipse is

becoming 1 when (see Tusi couple).

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]}

**^**J. Dennis Lawrence (1972).*A catalog of special plane curves*. Dover Publications. pp. 165–168. ISBN 0-486-60288-5.**^**Gray, Alfred (29 December 1997).*Modern Differential Geometry of Curves and Surfaces with Mathematica*(Second ed.). CRC Press. p. 906. ISBN 9780849371646.**^**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. arXiv:1812.07055. Bibcode:2019PhRvE.100a0302A. doi:10.1103/PhysRevE.100.010302. PMID 31499759. S2CID 119325369.

- Weisstein, Eric W. "Hypotrochoid".
*MathWorld*. - Flash Animation of Hypocycloid
- Hypotrochoid from Visual Dictionary of Special Plane Curves, Xah Lee
- Interactive hypotrochoide animation
- O'Connor, John J.; Robertson, Edmund F., "Hypotrochoid",
*MacTutor History of Mathematics Archive*, University of St Andrews