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* = 2*r* and *d* > *r* or *d* < *r* (*d* is not equal to *r*).^{[2]} (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]}

## See also

## References

**^**J. Dennis Lawrence (1972).*A catalog of special plane curves*. Dover Publications. pp. 165–168. ISBN 0-486-60288-5.**^**Gray, Alfred.*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. doi:10.1103/PhysRevE.100.010302.

## External links

- 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. - Hypotrochoid Web App Realtime Drawing App for Hypotrochoids & Epitrochoids