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 > 2, the figure is called a hyperellipse.
If n = 2, the figure is an ordinary ellipse (or circle if a = b).
When 0 ≤ n ≤ 1, the superellipse looks like a four-armed star with concave sides. In particular, when n = 2/3 and a = 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 = 1, the superellipse is a diamond with corners ±a, 0 and 0,±b.
When n > 2, the superellipse looks like a rectangle with rounded corners. In particular, when n = 4 and a =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.
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