Superellipse

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Squircle, the superellipse for n = 4, a = b = 1, approximates a chamfered square.
Squircle, the superellipse for n = 4, a = b = 1, approximates a chamfered square.

The superellipse (or Lamé curve) is the geometric figure defined in the Cartesian coordinate system as the set of all points (x, y) with

\left|\frac{x}{a}\right|^n\! + \left|\frac{y}{b}\right|^n\! = 1

where n > 0 and a and b are the semi-major and semi-minor axes of the oval shape. The case n = 2 yields an ordinary ellipse; increasing n beyond 2 yields the hyperellipses, which increasingly resemble rectangles; decreasing n below 2 yields hypoellipses which develop pointy corners in the x and y directions and increasingly resemble crosses. The case n = 1 yields a line with x-intercept of a and y-intercept of b. The case n = 1/2 is a star each of whose sides is an arc of a parabola. A superellipse with a = b is also the unit circle in R2 when distance is defined by the n-norm.

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[edit] Effects of n

n = 3⁄2, a = b = 1 produces a rounder shape that somewhat resembles a chamfered square.
n = 3⁄2, a = b = 1 produces a rounder shape that somewhat resembles a chamfered square.
n = 1⁄2, a = b = 1 produces a four-pointed star. Each segment is part of a parabola.
n = 1⁄2, a = b = 1 produces a four-pointed star. Each segment is part of a parabola.

When n is a nonzero rational number \frac{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

(x^4+y^4)^3-3(x^4-3x^2y^2+y^4)(x^4+3x^2y^2+y^4)+3(x^4+y^4)-1=0 , \,\!

or by the parametric equations

\left. \begin{align} x\left(\theta\right) &= \plusmn a\cos^{\frac{2}{n}} \theta \\ y\left(\theta\right) &= \plusmn b\sin^{\frac{2}{n}} \theta \end{align} \right\} \qquad 0 \le \theta < \frac{\pi}{2}

or

\begin{align} x\left(\theta\right) &= {|\cos \theta|}^{\frac{2}{n}} \cdot a \sgn(\cos \theta) \\ y\left(\theta\right) &= {|\sin \theta|}^{\frac{2}{n}} \cdot b \sgn(\sin \theta) \end{align}

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

\mathrm{Area} = 4 a b \frac{\left(\Gamma \left(1+\tfrac{1}{n}\right)\right)^2}{\Gamma \left(1+\tfrac{2}{n}\right)} .

[edit] Generalizations

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Example of the generalized superellipse with m ≠ n.
Example of the generalized superellipse with m ≠ n.

The superellipse is further generalized as:

\left|\frac{x}{a}\right|^m + \left|\frac{y}{b}\right|^n = 1; \qquad m, n > 0.

or

\begin{align} x\left(\theta\right) &= {|\cos \theta|}^{\frac{2}{n}} \cdot a \sgn(\cos \theta) \\ y\left(\theta\right) &= {|\sin \theta|}^{\frac{2}{m}} \cdot b \sgn(\sin \theta) \end{align}

[edit] Superellipsoid

Brass superegg by Piet Hein.
Brass superegg by Piet Hein.

In three-dimensions, a superellipsoid or superegg can be made by revolving a superellipse into a surface of revolution and scaling. Following Barr (1992), it is convenient to distinguish a north-south parameter n, an east-west parameter e, and length, width, and depth parameters ax, ay, az. Then an implicit equation for the surface is

\left( |x/a_x|^{2/e} + |y/a_y|^{2/e} \right)^{e/n} + |z/a_z|^{2/n} = 1 . \,\!

Parametric equations in terms of surface parameters u and v (longitude and latitude) are

\begin{align} x(u,v) &{}= a_x c(v,n) c(u,e) \\ y(u,v) &{}= a_y c(v,n) s(u,e) \\ z(u,v) &{}= a_z s(v,n) \\ & -\pi/2 \le v \le \pi/2, \quad -\pi \le u < \pi , \end{align}

where the auxiliary functions are

\begin{align} c(\omega,m) &{}= \sgn(\cos \omega) |\cos \omega|^m \\ s(\omega,m) &{}= \sgn(\sin \omega) |\sin \omega|^m \end{align}

and the signum function sgn(x) is

\sgn(x) = \begin{cases} -1, & x < 0 \\ 0, & x = 0 \\ +1, & x > 0 . \end{cases}

The volume inside this surface can be expressed in terms of beta functions, β(m,n) = Γ(m)Γ(n)/Γ(m+n), as

V = \frac23 a_x a_y a_z e n \beta \left( \frac{e}2,\frac{e}2 \right) \beta \left(n,\frac{n}2 \right) .

[edit] History

Though he is often credited with its invention, the Danish poet and scientist Piet Hein (1905–1996) did not discover the super-ellipse. The general Cartesian notation of the form comes from the French mathematician Gabriel Lamé (1795–1870) who generalized the equation for the ellipse. However, Piet Hein did popularize the use of the superellipse in architecture, urban planning, and furniture making, and he did invent the super-egg or superellipsoid by starting with the superellipse

\left|\frac{x}{4}\right|^{2.5} + \left|\frac{y}{3}\right|^{2.5} = 1

and revolving it about the x-axis. Unlike a regular ellipsoid, the super-ellipsoid can stand upright on a flat surface.

City planners in Stockholm, Sweden needed a solution for a roundabout in their city square Sergels Torg. Piet Hein's superellipse provided the needed aesthetic and practical solution. In 1968, when negotiators in Paris for the Vietnam War could not agree on the shape of the negotiating table, Balinski and Holt suggested a superelliptical table in a letter to the New York Times (Gardner 1977:251). The superellipse was used for the shape of the 1968 Azteca Olympic Stadium, in Mexico City.

Hermann Zapf's typeface Melior, published in 1952, uses superellipses for letters such as o. Many web sites say Zapf actually drew the shapes of Melior by hand without knowing the mathematical concept of the superellipse, and only later did Piet Hein point out to Zapf that his curves were extremely similar to the mathematical construct, but these web sites do not cite any primary source of this account. Thirty years later Donald Knuth built into his Computer Modern type family the ability to choose between true ellipses and superellipses (both approximated by cubic splines).

Man is the animal that draws lines which he himself then stumbles over. In the whole pattern of civilization there have been two tendencies, one toward straight lines and rectangular patterns and one toward circular lines. There are reasons, mechanical and psychological, for both tendencies. Things made with straight lines fit well together and save space. And we can move easily — physically or mentally — around things made with round lines. But we are in a straitjacket, having to accept one or the other, when often some intermediate form would be better. To draw something freehand — such as the patchwork traffic circle they tried in Stockholm — will not do. It isn't fixed, isn't definite like a circle or square. You don't know what it is. It isn't esthetically satisfying. The super-ellipse solved the problem. It is neither round nor rectangular, but in between. Yet it is fixed, it is definite — it has a unity. —Piet Hein

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