Superellipse

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The superellipse with n = 12, a = b = 1
The superellipse with n = 32, a = b = 1
Squircle, the superellipse with n = 4, a = b = 1

A superellipse, also known as a Lamé curve after Gabriel Lamé is a geometric figure defined in the Cartesian coordinate system as the set of all points (xy) with

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

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, each of the four arcs is a Quadratic Bézier curve defined by the two axes; as a result, each arc is a segment of a parabola.

When n is 1 the curve is a rhombus with corners (±a, 0) and (0, ±b). When n is between 1 and 2, it looks like a rhombus 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 a 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]).

Mathematical properties[edit]

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.

Animation

The curve is given 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 superellipse 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)} .

The pedal curve is relatively straightforward to compute. Specifically, the pedal of

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

is given in polar coordinates by[2]

(a \cos \theta)^{\tfrac{n}{n-1}}+(b \sin \theta)^{\tfrac{n}{n-1}}=r^{\tfrac{n}{n-1}}.

Generalizations[edit]

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

(Note that \theta is not a physical angle of the figure, but just a parameter.)

History[edit]

The general Cartesian notation of the form comes from the French mathematician Gabriel Lamé (1795–1870) who generalized the equation for the ellipse.

The outer outlines of the letters 'o' and 'O' in Zapf's Melior typeface are described by superellipses with n = log(1/2) / log (7/9) ≈ 2.758

Hermann Zapf's typeface Melior, published in 1952, uses superellipses for letters such as o. Thirty years later Donald Knuth would build the ability to choose between true ellipses and superellipses (both approximated by cubic splines) into his Computer Modern type family.

The superellipse was named by the Danish poet and scientist Piet Hein (1905–1996) though he did not discover it as it is sometimes claimed. In 1959, city planners in Stockholm, Sweden announced a design challenge for a roundabout in their city square Sergels Torg. Piet Hein's winning proposal was based on a superellipse with n = 2.5 and a/b = 6/5.[3] As he explained it:

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.

Sergels Torg was completed in 1967. Meanwhile Piet Hein went on to use the superellipse in other artifacts, such as beds, dishes, tables, etc.[4] By rotating a superellipse around the longest axis, he created the superegg, a solid egg-like shape that could stand upright on a flat surface, and was marketed as a novelty toy.

In 1968, when negotiators in Paris for the Vietnam War could not agree on the shape of the negotiating table, Balinski, Kieron Underwood and Holt suggested a superelliptical table in a letter to the New York Times.[3] The superellipse was used for the shape of the 1968 Azteca Olympic Stadium, in Mexico City.

Waldo R. Tobler developed a map projection, the Tobler hyperelliptical projection, published in 1973,[5] in which the meridians are arcs of superellipses.

Three connected superellipses are used in the logo of the Pittsburgh Steelers.

See also[edit]

References[edit]

  1. ^ Donald Knuth: The METAFONTbook, p. 126
  2. ^ J. Edwards (1892). Differential Calculus. London: MacMillan and Co. p. 164. 
  3. ^ a b Gardner, Martin (1977), "Piet Hein’s Superellipse", Mathematical Carnival. A New Round-Up of Tantalizers and Puzzles from Scientific American, New York: Vintage Press, pp. 240–254, ISBN 978-0-394-72349-5 
  4. ^ The Superellipse, in The Guide to Life, The Universe and Everything by BBC (27th June 2003)
  5. ^ Tobler, Waldo (1973), "The hyperelliptical and other new pseudocylindrical equal area map projections", Journal of Geophysical Research 78 (11): 1753–1759, Bibcode:1973JGR....78.1753T, doi:10.1029/JB078i011p01753. 

External links[edit]