Hypotrochoid

Curve traced by a point outside a circle rolling within another 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).

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]

x ( θ ) = ( R − r ) cos ⁡ θ + d cos ⁡ ( R − r r θ ) y ( θ ) = ( R − r ) sin ⁡ θ − d sin ⁡ ( R − r r θ ) {\displaystyle {\begin{aligned}&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)\end{aligned}}}

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 2 π × LCM ⁡ ( r , R ) R {\displaystyle 2\pi \times {\tfrac {\operatorname {LCM} (r,R)}{R}}} (where LCM is least common multiple).

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

e = 2 d / r 1 + ( d / r ) {\displaystyle e={\frac {2{\sqrt {d/r}}}{1+(d/r)}}}

becoming 1 when d = r {\displaystyle d=r} (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 (29 December 1997). 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. arXiv:1812.07055. Bibcode:2019PhRvE.100a0302A. doi:10.1103/PhysRevE.100.010302. PMID 31499759. S2CID 119325369.

External links