Editing Superellipse (section)

=== 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" />