Superellipse - Maple Programming Help

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Superellipse

Main Concept

A superellipse, also known as a Lamé curve, is a closed curve defined by the equation xan+ybn=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 n < 2, the figure is called a hypoellipse.

If n &gt; 2, the figure is called a hyperellipse.

If n &equals; 2, the figure is an ordinary ellipse (or circle if a &equals; b).

Interesting Shapes

When 0  n  1, the superellipse looks like a four-armed star with concave sides. In particular, when n &equals; 2/3 and a &equals; b, 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 n &equals; 1, the superellipse is a diamond with corners ±a&comma; 0 and 0&comma;±b.

When n &gt; 2, the superellipse looks like a rectangle with rounded corners. In particular, when n &equals; 4 and a &equals;b, 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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