Hypotrochoid

{{short description|Curve traced by a point outside a circle rolling within another circle}}

[[Image:HypotrochoidOutThreeFifths.gif|thumb|400px|

The red curve is a hypotrochoid drawn as the smaller black circle rolls around inside the larger blue circle (parameters are {{math|1=R = 5, r = 3, d = 5}}).]]

In geometry, a hypotrochoid is a roulette traced by a point attached to a circle of radius {{mvar|r}} rolling around the inside of a fixed circle of radius {{mvar|R}}, where the point is a distance {{mvar|d}} from the center of the interior circle.

The parametric equations for a hypotrochoid are:{{cite book | author=J. Dennis Lawrence | title=A catalog of special plane curves | publisher=Dover Publications | year=1972 | isbn=0-486-60288-5 | pages=[https://archive.org/details/catalogofspecial00lawr/page/165 165–168] | url-access=registration | url=https://archive.org/details/catalogofspecial00lawr/page/165 }}

:$\backslash begin\{align\}$

& 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{align}

where {{mvar|θ}} is the angle formed by the horizontal and the center of the rolling circle (these are not polar equations because {{mvar|θ}} is not the polar angle). When measured in radian, {{mvar|θ}} takes values from 0 to $2\; \backslash pi\; \backslash times\; \backslash tfrac\{\backslash operatorname\{LCM\}(r,\; R)\}\{R\}$ (where {{math|LCM}} is least common multiple).

Special cases include the hypocycloid with {{math|1=d = r}} and the ellipse with {{math|1=R = 2r}} and {{math|d ≠ r}}.{{Cite book|url=https://books.google.com/books?id=-LRumtTimYgC&pg=PA906|title=Modern Differential Geometry of Curves and Surfaces with Mathematica|edition=Second|last=Gray|first=Alfred|date=29 December 1997 |authorlink=Alfred Gray (mathematician)|publisher=CRC Press|isbn=9780849371646|language=en|page=906}} The eccentricity of the ellipse is

:$e=\backslash frac\{2\backslash sqrt\{d/r\}\}\{1+(d/r)\}$

becoming 1 when $d=r$ (see Tusi couple).

Image:Ellipse as hypotrochoid.gif (drawn in red) may be expressed as a special case of the hypotrochoid, with {{math|1=R = 2r}} (Tusi couple); here {{math|1=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.{{Cite journal|last1=Aceituno|first1=Pau Vilimelis|last2=Rogers|first2=Tim|last3=Schomerus|first3=Henning|date=2019-07-16|title=Universal hypotrochoidic law for random matrices with cyclic correlations|url=https://link.aps.org/doi/10.1103/PhysRevE.100.010302|journal=Physical Review E|volume=100|issue=1|pages=010302|doi=10.1103/PhysRevE.100.010302|pmid=31499759 |arxiv=1812.07055 |bibcode=2019PhRvE.100a0302A |s2cid=119325369 }}