Super ellipse

Examples of super ellipses for ${\displaystyle a=1,\ b=0.75}$

A superellipse , also known as Lamé's curve or Lamé's oval , is a geometric figure (curve) that represents a "middle thing" between an ellipse and a rectangle (or between a circle and a square → super circle ). A superellipse can be described in a Cartesian coordinate system as a set of all points ( x , y ) for which the following applies:

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

with the real values n ≥ 0 and a , b : semi-axes.

The case n = 2 leads to a normal ellipse; larger n (> 2) supplies the actual superellipse, which increasingly approximates a rectangle; n below 2 leads to sub-ellipses that have corners in the direction of the x and y axes and approach the axis cross for n towards 0.

The term “superellipse” goes back to the Danish scientist, inventor and writer Piet Hein (1905–1996). The general Cartesian description comes from the French physicist and mathematician Gabriel Lamé (1795-1870), who generalized the equation of the ellipse in this way.

Parametric representation

The following parameter representation results from the property of the sine and cosine functions (analogous to an ellipse): ${\displaystyle \cos ^{2}t+\sin ^{2}t=1}$

${\displaystyle \left.{\begin{aligned}x\left(t\right)&=\pm a\cos ^{\frac {2}{n}}t\\y\left(t\right)&=\pm b\sin ^{\frac {2}{n}}t\end{aligned}}\right\}\qquad 0\leq t\leq \pi /2\ .}$

Applications

Architecture & design

Piet Hein's super egg

Bruno Mathssons and Piet Hein's "Superellips" table

The Danish scientist Piet Hein popularized the use of the superellipse in architecture , urban planning and (furniture) design. In this context he registered the brand Superellipse ( n = 2.5).

Piet Hein also designed the Super- Egg, a three-dimensional superellipsoid. There is a rotation body formed on a superellipse with n based = 2.5:

${\displaystyle \left|\,{\frac {\sqrt {x^{2}+y^{2}}}{3}}\,\right|^{2{,}5}+\;\left|\,{\frac {z}{4}}\,\right|^{2{,}5}=1}$

Unlike a regular ellipsoid , this superellipsoid stands upright on a flat surface (wobbling).

The (closed) inner capsule of surprise eggs is shaped in a similar way - but a cylinder with strongly rounded edges (large radius of curvature), so that a flat base of about half the cylinder radius remains.

Fonts

Donald Knuth uses super ellipses in the Computer Modern fonts and the Metafont and Metapost programs with which these fonts were created. The difference between the letter O and the number 0 (zero) in Computer Modern Typewriter is mainly due to the different superness . This parameter superness (short s ) has the following relationship with the parameter n mentioned above :

${\displaystyle s^{n}={\frac {1}{2}}}$

With this, rectangles are also possible that are obtained with s = 1 ( n → ∞ ).

Aircraft construction

In the construction of wings for gliders , super-elliptical floor plans are used in some models (see Schempp-Hirth Quintus ).

Special super ellipses

If you choose n = 1, a rhombus or rhombus is created (for the special case a = b a square) with the area a * b / 2. If n = 2/3 (and a = b ) there is an astroid . For a = b there is a supercircle.

Generalizations

Variation of a superellipse with different exponents

Superellipsoids in three dimensions

Curves with different exponents

If different exponents are allowed for x and y coordinates, curves with equations are obtained

${\displaystyle \left|{\frac {x}{a}}\right|^{m}\!+\left|{\frac {y}{b}}\right|^{n}\!=1\ ,\quad m,n>0.}$

Substantially new curves result when one exponent is> 1 and the other is <1 (see figure).

Superellipsoids (surfaces)

A generalization of the superellipse to space provides the superellipsoids with the equations:

${\displaystyle \left|{\frac {x}{a}}\right|^{n}\!+\left|{\frac {y}{b}}\right|^{n}\!+\left|{\frac {z}{c}}\right|^{n}\!=1.}$

Here, too, you can increase the variety by choosing a different exponent for each coordinate.

Related curves

• The super formula describes a family of closed, rotationally symmetrical curves.

Web links

Commons : Superellipse  - collection of images, videos and audio files

• John J. O'Connor, Edmund F. Robertson :  Lame Curves. In: MacTutor History of Mathematics archive .
• Eric W. Weisstein : Superellipse . In: MathWorld (English).