Editing Hypotrochoid (section)

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

[[File:HypotrochoidOutThreeFifths.gif|thumb|upright=1|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 (curve)|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 equation]]s for a hypotrochoid are:<ref>{{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 }}</ref>

:<math>\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}</math>

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 <math>2 \pi \times \tfrac{\operatorname{LCM}(r, R)}{R}</math> (where {{math|LCM}} is [[least common multiple]]).

Special cases include the [[hypocycloid]] with {{math|1=''d'' = ''r''}} and the [[ellipse]] with {{math|1=''R'' = 2''r''}} and {{math|''d'' ≠ ''r''}}.<ref>{{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}}</ref>  The eccentricity of the ellipse is

:<math>e=\frac{2\sqrt{d/r}}{1+(d/r)}</math>

becoming 1 when <math>d=r</math> (see [[Tusi couple]]).
[[Image:Ellipse as hypotrochoid.gif|right|400px|thumb|The [[ellipse]] (drawn in red) may be expressed as a special case of the hypotrochoid, with {{math|1=''R'' = 2''r''}} ([[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.<ref>{{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 }}</ref>