View source for Superellipse

{{Use dmy dates|date=September 2015}}
[[File:Superellipse.svg|300px|thumb|Examples of superellipses for <math>a=1, \ b=0.75</math>]]
A '''superellipse''', also known as a '''Lamé curve''' after [[Gabriel Lamé]], is a closed curve resembling the [[ellipse]], retaining the geometric features of [[semi-major axis]] and [[semi-minor axis]], and symmetry about them, but a different overall shape.

In the [[Cartesian coordinate system]], the set of all points (''x'',&nbsp;''y'') on the curve satisfy the equation

:<math>\left|\frac{x}{a}\right|^n\! + \left|\frac{y}{b}\right|^n\! = 1,</math>

where ''n'', ''a'' and ''b'' are positive numbers, and the vertical bars {{abs|&nbsp;}} around a number indicate the [[absolute value]] of the number.

==Specific cases==

This formula defines a [[closed curve]] contained in the [[rectangle]] &minus;''a'' ≤ ''x''&nbsp;≤&nbsp;+''a'' and &minus;''b''&nbsp;≤&nbsp;''y''&nbsp;≤&nbsp;+''b''.  The parameters ''a'' and ''b'' are called the ''semi-diameters'' of the curve.

{|
|-
| <math> 0 < n < 1</math>
| The superellipse looks like a four-armed star with [[wikt:concave|concave]] (inwards-curved) sides.  
For ''n''&nbsp;=&nbsp;1/2, in particular, each of the four arcs is a segment of a [[parabola]].
|[[File:Superellipse star.svg|thumb|200px|right|The superellipse with ''n''&nbsp;=&nbsp;{{fraction|1|2}}, ''a''&nbsp;=&nbsp;''b''&nbsp;=&nbsp;1]]
|-
| <math> n = 1</math>
| The curve is a [[parallelogram]] with corners  (±''a'',&nbsp;0) and (0,&nbsp;±''b'').  
|
|-
| <math> 1< n < 2</math>
| The curve looks like a parallelogram with those same corners but with [[convex set|convex]] (outwards-curved) sides.  
The [[curvature]] increases without [[Limit of a sequence|limit]] as one approaches its extreme points.
|[[File:Superellipse rounded diamond.svg|thumb|200px|right|The superellipse with ''n''&nbsp;=&nbsp;{{fraction|3|2}}, ''a''&nbsp;=&nbsp;''b''&nbsp;=&nbsp;1]]
|-
| <math>n=2</math>
| The curve is an ordinary [[ellipse]] (in particular, a [[circle]] if ''a''&nbsp;=&nbsp;''b'').  
|
|-
| <math>n>2</math>
| The curve looks superficially like a [[rectangle]] with rounded corners.  
The curvature is zero at the points (±''a'',&nbsp;0) and (0,&nbsp;±''b'').
| [[File:Superellipse chamfered square.svg|thumb|200px|right|[[Squircle]], the superellipse with ''n''&nbsp;=&nbsp;4, ''a''&nbsp;=&nbsp;''b''&nbsp;=&nbsp;1]]
|}

If ''n''&nbsp;<&nbsp;2, the figure is also called a '''hypoellipse'''; if ''n''&nbsp;>&nbsp;2, a '''hyperellipse'''.

When ''n''&nbsp;≥&nbsp;1 and ''a''&nbsp;=&nbsp;''b'', the superellipse is the boundary of a [[ball (mathematics)|ball]] of '''R'''<sup>2</sup> in the [[norm (mathematics)#p-norm|''n''-norm]].

The extreme points of the superellipse are (±''a'',&nbsp;0) and (0,&nbsp;±''b''), and its four "corners" are (±''sa,&nbsp;±sb''), where <math>s=2^{-1/n}</math> (sometimes called the "superness"<ref>Donald Knuth: ''The METAFONTbook'', p. 126</ref>).

==Mathematical properties==
When ''n'' is a positive [[rational number]] ''p''/''q'' (in lowest terms), then each quadrant of the superellipse is a [[algebraic curve|plane algebraic curve]] of order ''pq''.<ref>For a derivation of the algebraic equation in the case where ''n'' = 2/3, see p. 3 of http://xahlee.info/SpecialPlaneCurves_dir/Astroid_dir/astroid.pdf.</ref> In particular, when ''a'' = ''b'' = 1 and ''n'' is an even integer, then it is a [[Fermat curve]] of degree ''n''. In that case it is non-singular, but in general it will be [[Singular point of an algebraic variety|singular]]. If the numerator is not even, then the curve is pieced together from portions of the same algebraic curve in different orientations.

The curve is given by the [[parametric equation]]s (with parameter <math>t</math> having no elementary geometric interpretation)

:<math>\left.
\begin{align}
 x\left(t\right) &= \plusmn a\cos^{\frac{2}{n}} t \\
 y\left(t\right) &= \plusmn b\sin^{\frac{2}{n}} t
\end{align} \right\} \qquad 0 \le t \le \frac{\pi}{2} </math>

where each ± can be chosen separately so that each value of <math>t</math> gives four points on the curve. Equivalently, letting <math>t</math> range over <math>0\le t < 2\pi,</math>

:<math>
\begin{align}
 x\left(t\right) &= {|\cos t|}^{\frac{2}{n}} \cdot a \sgn(\cos t) \\
 y\left(t\right) &= {|\sin t|}^{\frac{2}{n}} \cdot b \sgn(\sin t)
\end{align}
</math>

where the [[sign function]]  is 
:<math> \sgn(w) = \begin{cases}
 -1, & w < 0 \\
  0, & w = 0 \\
 +1, & w > 0 .
\end{cases}</math>

Here <math>t</math> is not the angle between the positive horizontal axis and the ray from the origin to the point, since the tangent of this angle equals ''y/x'' while in the parametric expressions ''y/x'' = (''b/a'')(tan <math>t)</math><sup>2/''n''</sup> ≠ tan <math>t.</math>

The area inside the superellipse can be expressed in terms of the [[gamma function]], Γ(''x''), as

:<math> \mathrm{Area} = 4 a b \frac{\left(\Gamma \left(1+\tfrac{1}{n}\right)\right)^2}{\Gamma \left(1+\tfrac{2}{n}\right)} . </math>

The [[pedal curve]] is relatively straightforward to compute. Specifically, the pedal of 
: <math>\left|\frac{x}{a}\right|^n\! + \left|\frac{y}{b}\right|^n\! = 1,</math>
is given in [[polar coordinate]]s by<ref>{{cite book | author=J. Edwards | title=Differential Calculus
| publisher= MacMillan and Co.| location=London | pages=164| year=1892
|url=https://books.google.com/books?id=unltAAAAMAAJ&pg=PA164#v=onepage&q&f=false}}</ref>
: <math>(a \cos \theta)^{\tfrac{n}{n-1}}+(b \sin \theta)^{\tfrac{n}{n-1}}=r^{\tfrac{n}{n-1}}.</math>{{clear}}

==Generalizations==
{{Expand section|date=June 2008}}
[[File:Superellipse-m-n.svg|thumb|Variations of a superellipse with different exponents]]
The superellipse is further generalized as:

:<math>\left|\frac{x}{a}\right|^m + \left|\frac{y}{b}\right|^n = 1; \qquad m, n > 0.</math>

or

:<math>
\begin{align}
 x\left(t\right) &= {|\cos t|}^{\frac{2}{m}} \cdot a \sgn(\cos t) \\
 y\left(t\right) &= {|\sin t|}^{\frac{2}{n}} \cdot b \sgn(\sin t).
\end{align}
</math>

Note that <math>t</math> is a parameter which is not linked to the physical angle through elementary functions.

== History ==
The general Cartesian notation of the form comes from the French mathematician [[Gabriel Lamé]] (1795&ndash;1870), who generalized the equation for the ellipse.

[[File:MeliorSuperEllipse.svg|thumb|176px|The outer outlines of the letters 'o' and 'O' in Zapf's Melior typeface are described by superellipses with ''n'' = {{nowrap|log(1/2) / log (7/9)}} ≈ 2.758]]
[[Hermann Zapf]]'s [[typeface]] [[Melior (typeface)|Melior]], published in 1952, uses superellipses for letters such as ''o''. <!--Many web sites say Zapf actually drew the shapes of Melior by hand without knowing the mathematical concept of the superellipse, and only later did Piet Hein point out to Zapf that his curves were extremely similar to the mathematical construct, but these web sites do not cite any primary source of this account.-->  Thirty years later [[Donald Knuth]] would build the ability to choose between true ellipses and superellipses (both approximated by [[cubic spline]]s) into his [[Computer Modern]] type family.

The superellipse was named by the [[Denmark|Danish]] poet and scientist [[Piet Hein (Denmark)|Piet Hein]] (1905–1996) though he did not discover it as it is sometimes claimed. In 1959, city planners in [[Stockholm]], [[Sweden]] announced a design challenge for a [[roundabout]] in their city square [[Sergels Torg]]. Piet Hein's winning proposal was based on a superellipse with ''n'' = 2.5 and ''a''/''b'' = 6/5.<ref name=gardner>{{Citation | last=Gardner | first=Martin | author-link=Martin Gardner | chapter=Piet Hein’s Superellipse | year=1977 | title=Mathematical Carnival. A New Round-Up of Tantalizers and Puzzles from Scientific American | location=New York | publisher=[[Vintage Press]] | pages=240–254 | isbn=978-0-394-72349-5 | url-access=registration | url=https://archive.org/details/mathematicalcarn00gard }}</ref> As he explained it:

:''Man is the animal that draws lines which he himself then stumbles over. In the whole pattern of civilization there have been two tendencies, one toward straight lines and rectangular patterns and one toward circular lines. There are reasons, mechanical and psychological, for both tendencies. Things made with straight lines fit well together and save space. And we can move easily &mdash; physically or mentally &mdash; around things made with round lines. But we are in a straitjacket, having to accept one or the other, when often some intermediate form would be better. To draw something freehand &mdash; such as the patchwork traffic circle they tried in Stockholm &mdash; will not do. It isn't fixed, isn't definite like a circle or square. You don't know what it is. It isn't esthetically satisfying. The super-ellipse solved the problem. It is neither round nor rectangular, but in between. Yet it is fixed, it is definite &mdash; it has a unity.''

Sergels Torg was completed in 1967.  Meanwhile, Piet Hein went on to use the superellipse in other artifacts, such as beds, dishes, tables, etc.<ref name=bbc>[http://www.bbc.co.uk/dna/h2g2/A1053884 ''The Superellipse''], in ''The Guide to Life, The Universe and Everything'' by [[British Broadcasting Corporation|BBC]] (27 June 2003)</ref>  By rotating a superellipse around the longest axis, he created the [[superegg]], a solid egg-like shape that could stand upright on a flat surface, and was marketed as a [[novelty item|novelty toy]].

In 1968, when negotiators in [[Paris]] for the [[Vietnam War]] could not agree on the shape of the negotiating table, Balinski, [[Kieron Underwood]] and Holt suggested a superelliptical table in a letter to the [[New York Times]].<ref name=gardner/>  The superellipse was used for the shape of the 1968 [[Estadio Azteca|Azteca Olympic Stadium]], in [[Mexico City]].

[[Waldo R. Tobler]] developed a [[map projection]], the [[Tobler hyperelliptical projection]], published in 1973,<ref>{{Citation|
last=Tobler|
first=Waldo|
title=The hyperelliptical and other new pseudocylindrical equal area map projections|
journal=Journal of Geophysical Research|
volume=78|
issue=11|
pages=1753–1759|
year=1973|
doi=10.1029/JB078i011p01753|
postscript=.
|bibcode=1973JGR....78.1753T
|citeseerx=10.1.1.495.6424
}}
</ref> in which the [[Meridian (geography)|meridians]] are arcs of superellipses.

The logo for news company [[The Local]] consists of a tilted superellipse matching the proportions of Sergels Torg. Three connected superellipses are used in the logo of the [[Pittsburgh Steelers]].

In computing, mobile operating system [[iOS]] uses a superellipse curve for app icons, replacing the [[rounded corner]]s style used up to version 6.<ref>http://iosdesign.ivomynttinen.com/</ref>

== See also ==
* [[Astroid]], the superellipse with ''n''&nbsp;=&nbsp;{{fraction|2|3}} and ''a''&nbsp;=&nbsp;''b'', is a hypocycloid with four cusps.
** [[Deltoid curve]], the hypocycloid of ''three'' cusps.
* [[Squircle]], the superellipse with ''n''&nbsp;=&nbsp;4 and ''a''&nbsp;=&nbsp;''b'', looks like "The Four-Cornered Wheel."
** [[Reuleaux triangle]], "The Three-Cornered Wheel."
* [[Superformula]], a generalization of the superellipse.
* [[Superquadrics]] and [[superellipsoid]]s, the three-dimensional "relatives" of superellipses.
* [[Superelliptic curve]], equation of the form ''Y''<sup>''n''</sup> = ''f''(''X'').
* [[Lp space|L<sup>p</sup> spaces]]

== References ==
{{Reflist}}
* {{citation | last=Barr | first=Alan H. | title=Geometric Modeling and Fluid Dynamic Analysis of Swimming Spermatozoa | year=1983 | publisher=[[Rensselaer Polytechnic Institute]] }} (Ph.D. dissertation using superellipsoids)
* {{citation | last=Barr | first=Alan H. | contribution=Rigid Physically Based Superquadrics | title=[[Graphics Gems]] III | year=1992 | editor-last=Kirk | editor-first=David | publisher=[[Academic Press]] | isbn=978-0-12-409672-1 | pages=137–159 ([http://www.graphicsgems.org/gemsiii/sqfinal.c code]: 472–477)}}
* {{citation | last=Gielis | first=Johan | title=Inventing the Circle: The Geometry of Nature | year=2003 | location=Antwerp | publisher=Geniaal Press | isbn=978-90-807756-1-9}}

== External links ==
{{commons category}}
* {{SpringerEOM|title=Lamé curve|id=Lam%C3%A9_curve&oldid=11566|last=Sokolov|first=D.D.}}
* [http://www.mathcurve.com/courbes2d.gb/lame/lame.shtml "Lamé Curve"] at MathCurve.
* {{MathWorld|title=Superellipse|urlname=Superellipse}}
* {{MacTutor|class=history/Curves|id=Lame|title=Lame Curves}}
* [http://www.2dcurves.com/power/powerse.html "Super Ellipse"] on 2dcurves.com
* [http://www.procato.com/superellipse/ Superellipse Calculator & Template Generator]
* [http://users.cs.cf.ac.uk/Paul.Rosin/fitting.html C code for fitting superellipses]

[[Category:Curves]]