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

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