Editing Hypocycloid

{{Short description|Curve traced by a point on a circle rolling within another circle}}
{{More citations needed|date=November 2011}}

[[File:astroid2.gif|thumb|460px|The red path is a hypocycloid traced as the smaller black circle rolls around inside the larger black circle (parameters are R=4.0, r=1.0, and so k=4, giving an [[astroid]]).]]
In [[geometry]], a '''hypocycloid''' is a special [[plane curve]] generated by the trace of a fixed point on a small [[circle]] that rolls within a larger circle. As the radius of the larger circle is increased, the hypocycloid becomes more like the [[cycloid]] created by rolling a circle on a line.

==History==

The 2-cusped hypocycloid called [[Tusi couple]] was first described by the 13th-century [[Persian people|Persian]] [[Islamic astronomy|astronomer]] and [[Islamic mathematics|mathematician]] [[Nasir al-Din al-Tusi]] in ''Tahrir al-Majisti (Commentary on the Almagest)''.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Tusi Couple |url=https://mathworld.wolfram.com/ |access-date=2023-02-27 |website=mathworld.wolfram.com |language=en}}</ref><ref>{{Cite book |last=Blake |first=Stephen P. |url=https://books.google.com/books?id=3dJVDwAAQBAJ&dq=Tusi+couple+al+majisti&pg=PA67 |title=Astronomy and Astrology in the Islamic World |date=2016-04-08 |publisher=Edinburgh University Press |isbn=978-0-7486-4911-2 |language=en}}</ref> German painter and German Renaissance theorist [[Albrecht Dürer]] described [[epitrochoid]]s in 1525, and later Roemer and Bernoulli concentrated on some specific hypocycloids, like the astroid, in 1674 and 1691, respectively.<ref name=":1" />

==Properties==

If the rolling circle has radius {{mvar|r}}, and the fixed circle has radius {{math|1=''R'' = ''kr''}}, then the 
[[parametric equations]] for the curve can be given by either:
<math display="block">\begin{align}
& x (\theta) = (R - r) \cos \theta + r \cos \left(\frac{R-r}{r} \theta \right) \\
& y (\theta) = (R - r) \sin \theta - r \sin \left( \frac{R - r}{r} \theta \right)
\end{align}</math>
or:
<math display="block">\begin{align}
& x (\theta) = r (k - 1) \cos \theta + r \cos \left( (k - 1) \theta \right) \\
& y (\theta) = r (k - 1) \sin \theta - r \sin \left( (k - 1) \theta \right)
\end{align}</math>

If {{mvar|k}} is an integer, then the curve is closed, and has {{mvar|k}} [[Cusp (singularity)|cusps]] (i.e., sharp corners, where the curve is not 
[[Differentiable function|differentiable]]). Specially for {{math|1=''k'' = 2}} the curve is a straight line and the circles are called [[Tusi couple]]. Nasir al-Din al-Tusi was the first to describe these hypocycloids and their applications to high-speed [[printing press|printing]].<ref name=":0">{{citation|title=Epicyclic gears applied to early steam engines|journal=Mechanism and Machine Theory|volume=23|issue=1|year=1988|pages=25–37|first=G.|last=White|doi=10.1016/0094-114X(88)90006-7|quote=Early experience demonstrated that the hypocycloidal mechanism was structurally unsuited to transmitting the large forces developed by the piston of a steam engine. But the mechanism had shown its ability to convert linear motion to rotary motion and so found alternative low-load applications such as the drive for printing machines and sewing machines.}}</ref><ref>{{citation|title=Hermite interpolation by hypocycloids and epicycloids with rational offsets|first1=Zbyněk|last1=Šír|first2=Bohumír|last2=Bastl|first3=Miroslav|last3=Lávička|journal=Computer Aided Geometric Design|volume=27|issue=5|year=2010|pages=405–417|doi=10.1016/j.cagd.2010.02.001|quote=G. Cardano was the first to describe applications of hypocycloids in the technology of high-speed printing press (1570).}}</ref>

If <math>k</math> is a [[rational number]], say <math>k = p/q</math> expressed as [[irreducible fraction]], then the curve has <math>p</math> cusps. 

To close the curve and complete the 1st repeating pattern: 

* <math>\theta</math>=0 to q rotations
* <math>\alpha</math>=0 to p rotations
* total rotations of rolling circle=p-q rotations

If {{mvar|k}} is an [[irrational number]], then the curve never closes, and fills the space between the larger circle and a circle of radius {{math|''R'' − 2''r''}}.

Each hypocycloid (for any value of {{mvar|r}}) is a [[brachistochrone]] for the gravitational potential inside a homogeneous sphere of radius {{mvar|R}}.<ref>{{Citation |last1=Rana |first1=Narayan Chandra |last2=Joag |first2=Pramod Sharadchandra |year=2001 |title=Classical Mechanics |publisher=Tata McGraw-Hill |isbn=0-07-460315-9 |pages=230–2 |chapter=7.5 Barchistochrones and tautochrones inside a gravitating homogeneous sphere |chapter-url=https://books.google.com/books?id=dptKVr-5LJAC&pg=PA230}}</ref>

The area enclosed by a hypocycloid is given by:
<ref name=":1">{{Cite web |url=https://geometryexpressions.com/downloads/lessons/student%20projects/Area%20Enclosed%20by%20a%20General%20Hypocycloid.pdf
|title=Area Enclosed by a General Hypocycloid |website=Geometry Expressions |access-date=Jan 12, 2019}}</ref>
<ref name=Wolfram>{{Cite web |url=http://mathworld.wolfram.com/Hypocycloid.html
|title=Hypocycloid |website=Wolfram Mathworld |access-date=Jan 16, 2019}}</ref>

<math display="block">A = \frac {(k - 1)(k - 2)} {k^2} \pi R^2 = (k - 1)(k - 2) \pi r^2 </math>

The [[arc length]] of a hypocycloid is given by:<ref name=Wolfram />

<math display="block">s = \frac {8(k - 1)} {k} R = 8(k - 1) r </math>

==Examples==

<gallery caption="Hypocycloid Examples">
Image:Hypocycloid-3.svg| k=3 → a [[deltoid curve|deltoid]]
Image:Hypocycloid-4.svg| k=4 → an [[astroid]]
Image:Hypocycloid-5.svg| k=5 → a pentoid
Image:Hypocycloid-6.svg| k=6 → an exoid
Image:Hypocycloid-2-1.svg| k=2.1 = 21/10
Image:Hypocycloid-3-8.svg| k=3.8 = 19/5
Image:Hypocycloid-5-5.svg| k=5.5 = 11/2
Image:Hypocycloid-7-2.svg| k=7.2 = 36/5
</gallery>

The hypocycloid is a special kind of [[hypotrochoid]], which is a particular kind of [[Roulette (curve)|roulette]].

A hypocycloid with three cusps is known as a [[deltoid curve|deltoid]].

A hypocycloid curve with four cusps is known as an [[astroid]].

The hypocycloid with two "cusps" is a degenerate but still very interesting case, known as the [[Tusi couple]].

==Relationship to group theory==

[[File:Rolling Hypocycloids.gif|thumb|Hypocycloids "rolling" inside one another.  The cusps of each of the smaller curves maintain continuous contact with the next-larger hypocycloid.]]

Any hypocycloid with an integral value of ''k'', and thus ''k'' cusps, can move snugly inside another hypocycloid with ''k''+1 cusps, such that the points of the smaller hypocycloid will always be in contact with the larger.  This motion looks like 'rolling', though it is not technically rolling in the sense of classical mechanics, since it involves slipping.

Hypocycloid shapes can be related to [[special unitary group]]s, denoted SU(''k''), which consist of ''k'' × ''k'' unitary matrices with determinant 1.  For example, the allowed values of the sum of diagonal entries for a matrix in SU(3), are precisely the points in the complex plane lying inside a hypocycloid of three cusps (a deltoid).  Likewise, summing the diagonal entries of SU(4) matrices gives points inside an astroid, and so on.

Thanks to this result, one can use the fact that SU(''k'') fits inside SU(''k+1'') as a [[subgroup]] to prove that an [[epicycloid]] with ''k'' cusps moves snugly inside one with ''k''+1 cusps.<ref>{{cite web|last=Baez|first=John|title=Deltoid Rolling Inside Astroid|url=http://blogs.ams.org/visualinsight/2013/12/01/deltoid-rolling-inside-astroid/|work=AMS Blogs|publisher=American Mathematical Society|access-date=22 December 2013}}</ref><ref>{{cite web|last=Baez|first=John|title=Rolling hypocycloids|url=http://johncarlosbaez.wordpress.com/2013/12/03/rolling-hypocycloids/|work=Azimuth blog|date=3 December 2013|access-date=22 December 2013}}</ref>

==Derived curves==

The [[evolute]] of a hypocycloid is an enlarged version of the hypocycloid itself, while
the [[involute]] of a hypocycloid is a reduced copy of itself.<ref>{{cite web |author=Weisstein, Eric W. |title=Hypocycloid Evolute |work=MathWorld |publisher=Wolfram Research |url=http://mathworld.wolfram.com/HypocycloidEvolute.html}}</ref>

The [[pedal curve|pedal]] of a hypocycloid with pole at the center of the hypocycloid is a [[rose curve]].

The [[isoptic]] of a hypocycloid is a hypocycloid.

==Hypocycloids in popular culture==
[[File:Steelmark logo.svg|alt=A circle with three hypocycloids inside|thumb|right|The Steelmark logo, featuring three hypocycloids]]
Curves similar to hypocycloids can be drawn with the [[Spirograph]] toy.  Specifically, the Spirograph can draw [[hypotrochoid]]s and [[epitrochoid]]s.

The [[Pittsburgh Steelers]]' logo, which is based on the [[Steelmark]], includes three [[astroid]]s (hypocycloids of four [[cusp (singularity)|cusp]]s). In his weekly NFL.com column "Tuesday Morning Quarterback," [[Gregg Easterbrook]] often refers to the Steelers as the Hypocycloids. Chilean soccer team [[CD Huachipato]] based their crest on the Steelers' logo, and as such features hypocycloids.

The first Drew Carey season of ''[[The Price is Right (U.S. game show)|The Price Is Right]]'''s set features astroids on the three main doors, giant price tag, and the turntable area. The astroids on the doors and turntable were removed when the show switched to [[high-definition television|high definition]] broadcasts starting in 2008, and only the giant price tag prop still features them today.<ref>{{Cite web |last=Keller |first=Joel |date=21 August 2007 |title=A glimpse at Drew Carey's Price is Right |url=http://www.tvsquad.com/2007/08/21/a-glimpse-at-drew-careys-price-is-right/ |archive-url=https://web.archive.org/web/20100527074915/http://www.tvsquad.com/2007/08/21/a-glimpse-at-drew-careys-price-is-right |archive-date=27 May 2010 |website=TV Squad}}</ref>

==See also==
* [[Roulette (curve)]]
* Special cases: [[Tusi couple]], [[Astroid]], [[Deltoid curve|Deltoid]]
* [[List of periodic functions]]
* [[Cyclogon]]
* [[Epicycloid]]
* [[Hypotrochoid]]
* [[Epitrochoid]]
* [[Spirograph]]
* [[Flag of Portland, Oregon]], featuring a hypocycloid<ref>{{citation|title=Reading Portland: The City in Prose|editor1-first=John|editor1-last=Trombold|editor2-first=Peter|editor2-last=Donahue|publisher=Oregon Historical Society Press|year=2006|isbn=9780295986777|page=xvi|quote=At the center of the flag lies a star — technically, a hypocycloid — which represents the city at the confluence of the two rivers.}}</ref>
* [[Murray's Hypocycloidal Engine]], utilising a [[tusi couple]] as a substitute for a [[crank (mechanism)|crank]]

{{-}}
==References==
{{Reflist}}

==Further reading==
* {{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/168 168, 171–173] | url-access=registration | url=https://archive.org/details/catalogofspecial00lawr/page/168 }}

==External links==
{{commons category|Hypocycloid}}
* {{MathWorld|Hypocycloid|Hypocycloid}}
* {{springer|title=Hypocycloid|id=p/h048530}}
* {{MacTutor | class=Curves | id=Hypocycloid | title=Hypocycloid}}
* [https://web.archive.org/web/20081211012241/http://www.carloslabs.com/node/21 A free Javascript tool for generating Hypocyloid curves]  
*[http://www.v-jaekel.de/animate-trochoid-en.html Animation of Epicycloids, Pericycloids and Hypocycloids]
* [http://sourceforge.net/p/geofun/wiki/Home/ Plot Hypcycloid &mdash; GeoFun] 
* {{cite web |first=John |last=Snyder |title=Sphere with Tunnel Brachistochrone  |work=Wolfram Demonstrations Project |url=http://demonstrations.wolfram.com/SphereWithTunnelBrachistochrone/}} Iterative demonstration showing the brachistochrone property of Hypocycloid
{{Authority control}}
[[Category:Roulettes (curve)]]