Even and odd functions

2014-01-01T02:10:46Z

2001:470:0:A9:4DA0:291D:5BA1:5282: /* Basic properties */ one list item was missing a period

{{Refimprove|date=November 2011}}

[[File:Deltoid2.gif|thumb|460px|The red curve is a hypocycloid traced as the smaller black circle rolls around inside the larger blue circle (parameters are R=3.0, r=1.0, and so k=3, giving a [[deltoid curve|deltoid]]).]]
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. It is comparable to the [[cycloid]] but instead of the circle rolling along a line, it rolls within a circle.

==Properties==

If the smaller circle has radius ''r'', and the larger circle has radius ''R'' = ''kr'', then the
[[parametric equations]] for the curve can be given by either:
:<math>x (\theta) = (R - r) \cos \theta + r \cos \left( \frac{R - r}{r} \theta \right)</math>
:<math>y (\theta) = (R - r) \sin \theta - r \sin \left( \frac{R - r}{r} \theta \right),</math>
or:
:<math>x (\theta) = r (k - 1) \cos \theta + r \cos \left( (k - 1) \theta \right) \,</math>
:<math>y (\theta) = r (k - 1) \sin \theta - r \sin \left( (k - 1) \theta \right). \,</math>

If ''k'' is an integer, then the curve is closed, and has ''k'' [[Cusp (singularity)|cusps]] (i.e., sharp corners, where the curve is not
[[differentiable]]). Specially for k=2 the curve is a straight line and the circles are called Cardano circles. [[Girolamo Cardano]] was the first to describe these hypocycloids and their applications to high-speed [[printing press|printing]].<ref>{{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 ''k'' is a [[rational number]], say ''k'' = ''p''/''q'' expressed in simplest terms, then the curve has ''p'' cusps.

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

Each hypocycloid (for any value of ''r'') is a [[brachistochrone]] for the gravitational potential inside a homogeneous sphere of radius ''R''.<ref>{{Citation |last=Rana |first=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 |chapterurl=http://books.google.com/books?id=dptKVr-5LJAC&pg=PA230}}</ref>

==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
Image:Hypocycloid-6.svg| k=6
Image:Hypocycloid-2-1.svg| k=2.1
Image:Hypocycloid-3-8.svg| k=3.8
Image:Hypocycloid-5-5.svg| k=5.5
Image:Hypocycloid-7-2.svg| k=7.2
</gallery>

The hypocycloid is a special kind of [[hypotrochoid]], which are 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|Rolling Hypocycloids|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 give 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|accessdate=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|accessdate=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==

Curves similar to hypocyloids 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.

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>http://www.tvsquad.com/2007/08/21/a-glimpse-at-drew-careys-price-is-right/</ref>

==See also==
* Special cases: [[Astroid]], [[Deltoid curve|Deltoid]]
* [[List of periodic functions]]
* [[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>

==References==
{{Reflist}}

* {{cite book | author=J. Dennis Lawrence | title=A catalog of special plane curves | publisher=Dover Publications | year=1972 | isbn=0-486-60288-5 | pages=168, 171–173 }}

==External links==
* {{springer|title=Hypocycloid|id=p/h048530}}
* {{MacTutor | class=Curves | id=Hypocycloid | title=Hypocycloid}}
* [http://www.carloslabs.com/node/21 A free Javascript tool for generating Hypocyloid curves]
* [http://sourceforge.net/p/geofun/wiki/Home/ Plot Hypcycloid — 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

[[Category:Curves]]

[[de:Zykloide#Epi- und Hypozykloide]]
[[nl:Cycloïde#Hypocycloïde]]

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Even and odd functions