Editing Superellipse (section)

===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.