# Superellipse

## Family of closed mathematical curves / From Wikipedia, the free encyclopedia

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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 dimentional Cartesian coordinate system, a superellipse is defined as the set of all points $(x,y)$ on the curve that satisfy the equation$\left|{\frac {x}{a}}\right|^{n}\!\!+\left|{\frac {y}{b}}\right|^{n}\!=1,$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.^{[1]} ^{[2]}^{[3]}

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