Superellipse

A Super Ellipse, also Lame curve or Lamésches Oval, is a geometric figure (curve ) that a "middle thing" between ellipse and rectangle (or between circle and square → Super circle) represents. A superellipse may be described in a Cartesian coordinate system as a set of all points (x, y) for which:

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

The case n = 2 leads to a regular ellipse; larger n ( > 2 ) returns the actual Super Ellipse, which increasingly approaches a rectangle; n leads to Subellipsen that have corners in the x - and y- axes and converge as n approaches 0 the coordinate system below 2.

The term " super ellipse " 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.

Applications

The Danish scientist Piet Hein popularized the use of the super ellipse in architecture, urban planning and the (furniture ) design. He created in this context, the SUPER ELLIPSE ® brand ( n = 2.5 ).

In addition, Piet Hein designed the Super Egg (Super Egg), a three-dimensional SuperEllipsoid. There is a body of revolution based on a super- ellipse with n = 2.5:

Unlike a regular ellipsoid this SuperEllipsoid stands firmly upright on a flat surface ( wagging ).

The (closed ) inner capsule of surprise eggs shaped like - but a cylinder with a strong rounding of its edges (large radius of curvature) so that remains a flat floor space of about half the cylinder radius.

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

Thus the rectangles are possible, obtained with s = 1 ( n → ∞ ).

Special Super ellipses

If we choose n = 1, then a diamond or rhombus is formed ( for the special case a = b is a square) with the area of ​​a * b / 2 For n = 2/ 3 (and a = b ) there is an astroid. For a = b there is a super circle.

Related curves

• The super formula describes a bevy closed, rotating symmetrical curves.