Editing Superellipse

{{short description|Family of closed mathematical curves}}
{{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 defined by an equation that allows for various shapes between a rectangle and an ellipse.

In two dimensional [[Cartesian coordinate system]], a superellipse is defined as the set of all points <math>(x,y)</math> on the curve that satisfy the equation<math display="block">\left|\frac{x}{a}\right|^n\!\! + \left|\frac{y}{b}\right|^n\! = 1,</math>where <math>a</math> and <math>b</math> are positive numbers referred to as [[Semidiameter|semi-diameters]] or [[Semi-major and semi-minor axes|semi-axes]] of the superellipse, and <math>n</math> is a positive parameter that defines the shape. When <math>n=2</math>, the superellipse is an ordinary ellipse. For <math>n>2</math>, the shape is more rectangular with rounded corners, and for <math>0<n<2</math>, it is more pointed.<ref>{{Cite journal |last1=Shi |first1=Pei-Jian |last2=Huang |first2=Jian-Guo |last3=Hui |first3=Cang |last4=Grissino-Mayer |first4=Henri D. |last5=Tardif |first5=Jacques C. |last6=Zhai |first6=Li-Hong |last7=Wang |first7=Fu-Sheng |last8=Li |first8=Bai-Lian |date=2015-10-15 |title=Capturing spiral radial growth of conifers using the superellipse to model tree-ring geometric shape |journal=Frontiers in Plant Science |language=English |volume=6 |page=856 |doi=10.3389/fpls.2015.00856 |doi-access=free |pmid=26528316 |pmc=4606055 |issn=1664-462X}}</ref><ref>{{Cite journal |last=Barr |date=1981 |title=Superquadrics and Angle-Preserving Transformations |url=https://ieeexplore.ieee.org/document/1673799 |journal=IEEE Computer Graphics and Applications |volume=1 |issue=1 |pages=11–23 |doi=10.1109/MCG.1981.1673799 |s2cid=9389947 |issn=1558-1756}}</ref><ref>{{Cite book |last1=Liu |first1=Weixiao |last2=Wu |first2=Yuwei |last3=Ruan |first3=Sipu |last4=Chirikjian |first4=Gregory S. |title=2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) |chapter=Robust and Accurate Superquadric Recovery: A Probabilistic Approach |date=2022 |chapter-url=https://ieeexplore.ieee.org/document/9878948 |pages=2666–2675 |doi=10.1109/CVPR52688.2022.00270|arxiv=2111.14517 |isbn=978-1-6654-6946-3 |s2cid=244715106 }}</ref>

In the [[polar coordinate system]], the superellipse equation is (the set of all points <math>(r,\theta)</math> on the curve satisfy the equation):<math display="block">r = \left(\left|\frac{\cos\theta}{a}\right|^n\!\! + \left|\frac{\sin\theta}{b}\right|^n\!\right)^{-1/n}\!.</math>

== Specific cases ==

This formula defines a [[closed curve]] contained in the [[rectangle]] {{math|&minus;''a'' ≤ ''x'' ≤ +''a''}} and {{math|&minus;''b'' ≤ ''y'' ≤ +''b''}}.  The parameters <math>a</math> and <math>b</math> are the semi-diameters or semi-axes of the curve. The overall shape of the curve is determined by the value of the exponent <math>n</math>, as shown in the following table:

{| class="wikitable"
|-
| <math> 0 < n < 1</math>
| width="390px" | The superellipse looks like a four-armed star with [[wikt:concave|concave]] (inwards-curved) sides.<br>For <math>n=1/2</math>, in particular, each of the four arcs is a segment of a [[parabola]].<br>An [[astroid]] is the special case <math>a=b</math> , <math>n=2/3</math>
|[[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 [[rhombus]] with corners  (<math>\pm a,0 </math>) and (<math>0,\pm b</math>).  
|
|-
| <math> 1< n < 2</math>
| The curve looks like a rhombus with the same corners but with [[convex set|convex]] (outwards-curved) sides.<br>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 <math>a=b</math>).  
|
|-
| <math>n>2</math>
| The curve looks superficially like a [[rectangle]] with rounded corners.<br>The curvature is zero at the points  (<math>\pm a,0 </math>) and (<math>0,\pm b</math>).
| [[File:Superellipse chamfered square.svg|thumb|200px|right|[[Squircle]], the superellipse with ''n''&nbsp;=&nbsp;4, ''a''&nbsp;=&nbsp;''b''&nbsp;=&nbsp;1]]
|}

If <math>n<2</math>, the figure is also called a '''hypoellipse'''; if <math>n>2</math>, a '''hyperellipse'''. When <math>n\geq1</math> and <math>a=b</math>, the superellipse is the boundary of a [[ball (mathematics)|ball]] of <math>\R^2</math> in the <math>n</math>[[norm (mathematics)#p-norm|-norm]]. The extreme points of the superellipse are (<math>\pm a,0 </math>) and (<math>0,\pm b</math>), and its four "corners" are (<math>\pm s_{a}</math>,<math>\pm s_{b}</math>), 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]] <math>p/q</math> (in lowest terms), then each quadrant of the superellipse is a [[algebraic curve|plane algebraic curve]] of order <math>p/q</math>.<ref>{{Cite web |title=Astroid |url=http://xahlee.info/SpecialPlaneCurves_dir/Astroid_dir/astroid.pdf |access-date=14 March 2023 |website=Xah Code}}</ref> In particular, when <math>a=b=1</math> 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 display="block">\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 <math>\pm</math> 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 display="block">
\begin{align}
 x\left(t\right) &= {\left|\cos t\right|}^{\frac{2}{n}} \cdot a \sgn(\cos t) \\
 y\left(t\right) &= {\left|\sin t\right|}^{\frac{2}{n}} \cdot b \sgn(\sin t)
\end{align}
</math>where the [[sign function]]  is<math display="block"> \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 <math>y/x</math> while in the parametric expressions <math display="inline">\frac{y}{x} = \frac{b}{a} (\tan t)^{2/n} \neq \tan t.</math>

===Area===

The [[area]] inside the superellipse can be expressed in terms of the [[gamma function]] as<math display="block"> \mathrm{Area} = 4 a b \frac{\left(\Gamma \left(1+\tfrac{1}{n}\right)\right)^2}{\Gamma \left(1+\tfrac{2}{n}\right)} , </math>or in terms of the [[beta function]] as
:<math> \mathrm{Area} = \frac{4 a b}{n} \Beta\!\left(\frac{1}{n},\frac{1}{n}+1\right) . </math><ref name=":1">{{Cite web |title=Ellipsoids in Higher Dimensions |url=https://analyticphysics.com/Higher%20Dimensions/Ellipsoids%20in%20Higher%20Dimensions.htm |access-date=2024-06-19 |website=analyticphysics.com}}</ref>

===Perimeter===

The [[perimeter]] of a superellipse, like that of an [[ellipse]], does not admit [[Closed-form expression|closed-form solution]] purely using [[elementary function]]s. Exact solutions for the perimeter of a superellipse exist using [[Series (mathematics)|infinite summations]];<ref>{{cite web |url=https://fractional-calculus.com/super_ellipse.pdf |title=Superellipse (Lame curve) |author=<!--Not stated--> |date= |website= |publisher= |archive-url=https://web.archive.org/web/20220331035622/https://fractional-calculus.com/super_ellipse.pdf |access-date=November 9, 2023 |archive-date=31 March 2022 |quote=}}</ref> these could be truncated to obtain approximate solutions. [[Numerical integration]] is another option to obtain perimeter estimates at arbitrary precision.

A closed-form approximation obtained via [[symbolic regression]] is also an option that balances parsimony and accuracy. Consider a superellipse centered on the origin of a 2D plane. Now, imagine that the superellipse (with shape parameter <math>n</math>) is stretched such that the first quadrant (e.g., <math>x>0</math>, <math>y>0</math>) is an arc from <math>(1, 0)</math> to <math>(0, h)</math>, with <math>h \geq 1</math>. Then, the arc length of the superellipse within that single quadrant is approximated as the following function of <math>h</math> and <math>n</math>:<ref>{{cite web |last1=Sharpe |first1=Peter |title=AeroSandbox |url=https://github.com/peterdsharpe/AeroSandbox/blob/5c3a8697fc377bc5d5c4881f08ee0a6f19ccd33c/aerosandbox/geometry/fuselage.py#L793 |publisher=GitHub |access-date=9 November 2023}}</ref>

<code>h + (((((n-0.88487077) * h + 0.2588574 / h) ^ exp(n / -0.90069205)) + h) + 0.09919785) ^ (-1.4812293 / n)</code>

This single-quadrant arc length approximation is accurate to within ±0.2% for across all values of <math>n</math>, and can be used to efficiently estimate the total perimeter of a superellipse.

===Pedal curve===

The [[pedal curve]] is relatively straightforward to compute. Specifically, the pedal of<math display="block">\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=[https://archive.org/details/in.ernet.dli.2015.109607/page/n182 164]| year=1892 | url=https://archive.org/details/in.ernet.dli.2015.109607}}</ref><math display="block">(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}}

The generalization of these shapes can involve several approaches.The generalizations of the superellipse in higher dimensions retain the fundamental mathematical structure of the superellipse while adapting it to different contexts and applications.

=== Higher dimensions ===
The generalizations of the superellipse in higher dimensions retain the fundamental mathematical structure of the superellipse while adapting it to different contexts and applications.<ref>{{Cite book |last1=Boult |first1=Terrance E. |last2=Gross |first2=Ari D. |chapter=Recovery of Superquadrics from 3-D Information |editor-first1=David P. |editor-first2=Ernest L. |editor-last1=Casasent |editor-last2=Hall |date=1988-02-19 |title=Intelligent Robots and Computer Vision VI |chapter-url=http://dx.doi.org/10.1117/12.942759 |series=SPIE Proceedings |volume=0848 |page=358 |publisher=SPIE |doi=10.1117/12.942759|bibcode=1988SPIE..848..358B }}</ref>

* A [[superellipsoid]] extends the superellipse into three dimensions, creating shapes that vary between ellipsoids and rectangular solids with rounded edges. The superellipsoid is defined as the set of all points <math>(x,y,z)</math> that satisfy the equation:<math display="block">\left|\frac{x}{a}\right|^n\!\! + \left|\frac{y}{b}\right|^n\! + \left|\frac{z}{c}\right|^n\! = 1,</math>where <math>a,b</math> and <math>c</math> are positive numbers referred to as the semi-axes of the superellipsoid, and <math>n</math> is a positive parameter that defines the shape.<ref name=":1" />
* A [[hyperellipsoid]] is the <math>d</math>-dimensional analogue of an [[ellipsoid]] (and by extension, a superellipsoid). It is defined as the set of all points <math>(x_1,x_2,\ldots, x_d)</math> that satisfy the equation:<math display="block">\left|\frac{x_1}{a_1}\right|^n\!\! + \left|\frac{x_2}{a_2}\right|^n\! +\ldots+ \left|\frac{x_d}{a_d}\right|^n\! = 1,</math>where <math>a_1,a_2,\ldots, a_d</math> are positive numbers referred to as the semi-axes of the hyperellipsoid, and <math>n</math> is a positive parameter that defines the shape.<ref name=":0">{{Cite journal |last1=Ni |first1=B. Y. |last2=Elishakoff |first2=I. |last3=Jiang |first3=C. |last4=Fu |first4=C. M. |last5=Han |first5=X. |date=2016-11-01 |title=Generalization of the super ellipsoid concept and its application in mechanics |url=https://www.sciencedirect.com/science/article/pii/S0307904X16303225 |journal=Applied Mathematical Modelling |volume=40 |issue=21 |pages=9427–9444 |doi=10.1016/j.apm.2016.06.011 |issn=0307-904X}}</ref>

=== Different exponents ===
[[File:Superellipse-m-n.svg|thumb|Variations of a superellipse with different exponents]]
Using different exponents for each term in the equation, allowing more flexibility in shape formation.<ref>{{Cite journal |last1=Cheng |first1=Xinyu |last2=Li |first2=Chengbo |last3=Peng |first3=Yixue |last4=Zhao |first4=Chuang |date=2021-04-17 |title=Discrete element simulation of super-ellipse systems |url=https://doi.org/10.1007/s10035-021-01107-4 |journal=Granular Matter |language=en |volume=23 |issue=2 |pages=50 |doi=10.1007/s10035-021-01107-4 |issn=1434-7636}}</ref>

For two-dimensional case the equation is <math>\left|\frac{x}{a}\right|^m\!\! + \left|\frac{y}{b}\right|^n\! = 1;m,n>0,</math> where <math>m</math> either equals to or differs from ''<math>n</math>''. If <math>m=n</math>, it is the Lamé's superellipses. If <math>m\neq n</math>, the curve possesses more flexibility of behavior, and is better possible fit to describe some experimental information.<ref name=":0" />
[[File:National Centre for the Performing Arts (China) 8.jpg|left|thumb|National Centre for the Performing Arts]]
For the three-dimensional case, three different positive powers <math>m</math>'', <math>n</math>'' and <math>p</math> can be used in the equation <math>\left|\frac{x}{a}\right|^m\!\! + \left|\frac{y}{b}\right|^n\! + \left|\frac{z}{c}\right|^p\! = 1</math>. If <math>m=n=p</math>, a super-ellipsoid is obtained. If any two or all three powers differ from each other, a solid is obtained that may possess more flexibility in representing real structural data than the super ellipsoid. A three-dimensional super-ellipsoid with <math>m=n=2.2</math>,  <math>p=2.4</math> and the semi-diameters <math>a=b=1</math>,  <math>c=0.5</math> represents the structure of the National Centre for the Performing Arts in China.<ref name=":0" />

In the general <math>N</math>–dimensional case, the equation is <math>\left|\frac{x_1}{a_1}\right|^{N_1}\!\! + \left|\frac{x_2}{a_2}\right|^{N_2}\! +\ldots+ \left|\frac{x_N}{a_N}\right|^{N_N}\! = 1</math>, where In general, <math>n_1,n_2,\ldots, n_N</math> may differ from each other. It is the superellipsoid only if <math>n_1=n_2=\ldots= n_N=n</math>.<ref name=":0" />

=== Related shapes ===
[[Superquadrics]] are a family of shapes that include superellipsoids as a special case. They are used in computer graphics and geometric modeling to create complex, smooth shapes with easily adjustable parameters.<ref>{{Cite web |title=SuperQuadrics - Applications |url=https://www.cs.mcgill.ca/~dudek/SQ/research.html |access-date=2024-06-18 |website=www.cs.mcgill.ca}}</ref> While not a direct generalization of superellipses, [[hyperspheres]] also share the concept of extending geometric shapes into higher dimensions. These related shapes demonstrate the versatility and broad applicability of the fundamental principles underlying superellipses.

=== Anisotropic scaling ===
[[Anisotropy|Anisotropic]] scaling involves scaling the shape differently along different axes, providing additional control over the geometry. This approach can be applied to superellipses, superellipsoids, and their higher-dimensional analogues to produce a wider variety of forms and better fit specific requirements in applications such as computer graphics, structural design, and data visualization. For instance, anisotropic scaling allows the creation of shapes that can model real-world objects more accurately by adjusting the proportions along each axis independently.<ref>{{Cite journal |last1=Land |first1=Richard |last2=Foley |first2=James D. |last3=Dam |first3=Andries Van |date=1984 |title=Fundamentals of Interactive Computer Graphics |url=http://dx.doi.org/10.2307/1574879 |journal=Leonardo |volume=17 |issue=1 |pages=59 |doi=10.2307/1574879 |jstor=1574879 |issn=0024-094X}}</ref>

== 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.

[[File:Sergels torg.jpg|thumb|The central fountain of Sergels Torg is outlined by a superellipse with ''n'' = 2.5 and ''a''/''b'' = 6/5.]]

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=[https://archive.org/details/mathematicalcarn00gard/page/240 240–254] | isbn=978-0-394-72349-5 | chapter-url-access=registration | chapter-url=https://archive.org/details/mathematicalcarn00gard/page/240 }}</ref> As he explained it:
{{blockquote|''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.''}}

[[File:The_Local_logo.png|thumb|The Local's logo, based on Stockholm's Sergels Torg, with the L representing the glass obelisk]]

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>[https://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]].

The second floor of the original [[World Trade Center (1973–2001)|World Trade Center]] in New York City consisted of a large, superellipse-shaped overhanging balcony.

[[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>{{cite web |last1=Mynttinen |first1=Ivo |title=The iOS Design Guidelines |url=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]]: [[superellipsoid]]s and [[supertoroid]]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}}

== 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]
* [https://github.com/bmlklwx/Robust-superellipse-fitting Superellipse fitting toolbox in MATLAB]
* [http://users.cs.cf.ac.uk/Paul.Rosin/fitting.html C code for fitting superellipses]

[[Category:Plane curves]]