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 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 ${\displaystyle (x,y)}$ on the curve that satisfy the equation$${\displaystyle \left|{\frac {x}{a}}\right|^{n}\!\!+\left|{\frac {y}{b}}\right|^{n}\!=1,}$$where ${\displaystyle a}$ and ${\displaystyle b}$ are positive numbers referred to as semi-diameters or semi-axes of the superellipse, and ${\displaystyle n}$ is a positive parameter that defines the shape. When ${\displaystyle n=2}$, the superellipse is an ordinary ellipse. For ${\displaystyle n>2}$, the shape is more rectangular with rounded corners, and for ${\displaystyle 0<n<2}$, it is more pointed.^{[1] [2][3]}

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