Superellipse

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 ${\displaystyle (x,y)}$ on the curve satisfy the equation

${\displaystyle \left|{\frac {x}{a}}\right|^{n}\!\!+\left|{\frac {y}{b}}\right|^{n}\!=1,}$

where ${\displaystyle n,a}$ and ${\displaystyle b}$ are positive numbers, and the vertical bars around a number indicate the absolute value of the number. The 3-dimensional generalization is called superellipsoid (some literatures also name it superquadrics).[1][2]

In the Polar coordinate system, the superellipse equation is (the set of all points ${\displaystyle (r,\theta )}$ on the curve satisfy the equation) :

${\displaystyle r=\left(\left|{\frac {cos(\theta )}{a}}\right|^{n}\!\!+\left|{\frac {sin(\theta )}{b}}\right|^{n}\!\right)^{-1/n}\!,}$