Superellipse

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A superellipse (or Lamé curve) is a geometric figure defined in the Cartesian coordinate system as the set of all points (xy) with

where n, a and b are positive numbers.

This formula defines a closed curve contained in the rectangleax ≤ +a and −b ≤ y ≤ +b. The parameters a and b are called the semi-diameters of the curve.

When n is between 0 and 1, the superellipse looks like a four-armed star with concave (inwards-curved) sides. For n = 1/2, in particular, the sides are arcs of parabolas.

When n is 1 the curve is a diamond with corners (±a, 0) and (0, ±b). When n is between 1 and 2, it looks like a diamond with those same corners but with convex (outwards-curved) sides. The curvature increases without limit as one approaches the corners.

When n is 2, the curve is an ordinary ellipse (in particular, a circle if a = b). When n is greater than 2, it looks superficially like a rectangle with chamfered (rounded) corners. The curvature is zero at the points (±a, 0) and (0, ±b).

If n < 2 the figure is also called an hypoellipse; if n > 2, a hyperellipse.

When n ≥ 1 and a = b, the superellipse is the boundary of a ball of R2 in the n-norm.

The extreme points of the superellipse are (±a,0) and (0,±b), and its four "corners" are (±sa,±sb), where s = 2 − 1 / n (sometimes called the "superness" [1]).

Contents

Algebraic properties

When n is a nonzero rational number p / q (in lowest terms), then the superellipse is a plane algebraic curve. For positive n the order is pq; for negative n the order is 2pq. In particular, when a and b are both one and n is an even integer, then it is a Fermat curve of degree n. In that case it is nonsingular, but in general it will be singular. If the numerator is not even, then the curve is pasted together from portions of the same algebraic curve in different orientations.

For example, if x4/3 + y4/3 = 1, then the curve is an algebraic curve of degree twelve and genus three, given by the implicit equation

or by the parametric equations

or

The area inside the superellipse can be expressed in terms of the gamma function, Γ(x), as

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