Superellipse¶

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A superellipse is a planar, closed curve in the rectangle $x \in [-R_x,R_x] , y \in [-R_y,R_y]$ . It can be viewed as a generalization of the ellipse as it retains the symmetry properties about the semi-major and semi-minor axes. All points on the curve satisfy the following equation

$\begin{eqnarray*} \Big\vert \frac{x}{R_x} \Big\vert^n + \Big\vert \frac{y}{R_y} \Big\vert^n = 1 \end{eqnarray*}$

The ellipse is a special case of this equation with $n=2$ . For $n>=1$ convex curves are obtained. Examples are the rhombus ( $n=1$ ) or a curve resembling a rounded rectangle ( $n=4$ ) shown below. Concave curves, like the four armed star shown below, are a result of exponents $n<1$ .