Hypotrochoid

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 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).

The parametric equations for a hypotrochoid are:[1]

${\displaystyle x(\theta )=(R-r)\cos \theta +d\cos \left({R-r \over r}\theta \right)}$

${\displaystyle y(\theta )=(R-r)\sin \theta -d\sin \left({R-r \over r}\theta \right)}$

where ${\displaystyle \theta }$ is the angle formed by the horizontal and the center of the rolling circle (these are not polar equations because ${\displaystyle \theta }$ is not the polar angle). When measured in radian, ${\displaystyle \theta }$ takes values from ${\displaystyle 0}$ to ${\displaystyle 2\pi \times {\frac {\operatorname {LCM} (r,R)}{R}}}$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.

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

See also

• Cycloid
• Cyclogon
• Epicycloid
• Rosetta (orbit)
• Apsidal precession
• Spirograph

References

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.

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