Superellipse - Maple Help

Superellipse

Main Concept

A superellipse, also known as a Lamé curve, is a closed curve defined by the equation ${\left|\frac{x}{a}\right|}^{n}+{\left|\frac{y}{b}\right|}^{n}=1$, where n, a, and b are all positive numbers. The parameters a and b scale the figure along the axes and are called the semi-diameters of the curve.

If , the figure is called a hypoellipse.

If , the figure is called a hyperellipse.

If , the figure is an ordinary ellipse (or circle if ).

 Interesting Shapes When , the superellipse looks like a four-armed star with concave sides. In particular, when 2/3 and , the figure is called an "astroid" because it is a hypocycloid with four cusps. [To learn more about hypocycloids, see the "Epicycloid and Hypocycloid" Math App.] When , the superellipse is a diamond with corners  and $\left(0,±b\right)$. When , the superellipse looks like a rectangle with rounded corners. In particular, when  and , the figure is called a "squircle" because it has properties between those of a square and those of a circle.

The following graph shows a superellipse. Use the sliders to adjust the semi-diameters and exponents to see what shapes you can make.

 Exponent, n Semi-Diameter, a Semi-Diameter, b



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