Superellipse

A superellipse, also known as a Lamé curve after Gabriel Lamé, is a closed curve resembling the ellipse, retaining the geometric features of semi-major axis and semi-minor axis, and symmetry about them, but a different overall shape.

In the Cartesian coordinate system, the set of all points $(x,y)$ on the curve satisfy the equation

:$\left, \frac\^n\!\! + \left, \frac\^n\! = 1,$

where $n,a$ and $b$ are positive numbers, and the vertical bars around a number indicate the absolute value of the number.

Specific cases

This formula defines a closed curve contained in the rectangle −''a'' ≤ ''x'' ≤ +''a'' and −''b'' ≤ ''y'' ≤ +''b''. The parameters ''a'' and ''b'' are called the ''semi-diameters'' of the curve.
The overall shape of the curve is determined by the value of the exponent ''n'', as shown in the following table:

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 $s=2^$ (sometimes called the "superness").

Mathematical properties

When ''n'' is a positive rational number ''p''/''q'' (in lowest terms), then each quadrant of the superellipse is a plane algebraic curve of order ''pq''. In particular, when ''a'' = ''b'' = 1 and ''n'' is an even integer, then it is a Fermat curve of degree ''n''. In that case it is non-singular, but in general it will be singular. If the numerator is not even, then the curve is pieced together from portions of the same algebraic curve in different orientations.

The curve is given by the parametric equations (with parameter $t$ having no elementary geometric interpretation)

:$\left. \begin x\left(t\right) &= \plusmn a\cos^ t \\ y\left(t\right) &= \plusmn b\sin^ t \end \right\} \qquad 0 \le t \le \frac$

where each ± can be chosen separately so that each value of $t$ gives four points on the curve. Equivalently, letting $t$ range over $0\le t < 2\pi,$

:$\begin x\left(t\right) &= ^ \cdot a \sgn(\cos t) \\ y\left(t\right) &= ^ \cdot b \sgn(\sin t) \end$

where the sign function is
:$\sgn(w) = \begin -1, & w < 0 \\ 0, & w = 0 \\ +1, & w > 0 . \end$

Here $t$ is not the angle between the positive horizontal axis and the ray from the origin to the point, since the tangent of this angle equals ''y/x'' while in the parametric expressions $\frac = \frac (\tan t)^ \neq \tan t.$

The area inside the superellipse can be expressed in terms of the gamma function as

:$\mathrm = 4 a b \frac ,$

or in terms of the beta function as

:$\mathrm = \frac \Beta\!\left(\frac,\frac+1\right) .$

The pedal curve is relatively straightforward to compute. Specifically, the pedal of
: $\left, \frac\^n\! + \left, \frac\^n\! = 1,$
is given in polar coordinates by
: $(a \cos \theta)^+(b \sin \theta)^=r^.$

Generalizations

The superellipse is further generalized as:

:$\left, \frac\^m \!\!+ \left, \frac\^n\! = 1; \qquad m, n > 0.$

or

:$\begin x\left(t\right) &= ^ \cdot a \sgn(\cos t) \\ y\left(t\right) &= ^ \cdot b \sgn(\sin t). \end$

Note that $t$ is a parameter which is not linked to the physical angle through elementary functions.

History

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

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. 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.''The Superellipse''
in ''The Guide to Life, The Universe and Everything'' by BBC
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