Editing Superellipse (section)

==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}}