Hypotrochoid

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]

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

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

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

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

[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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Hypotrochoid

See also

References

External links