super ellipse


last updated: 2005-10-21

It was Gabriel (1795-1870) who was the first to study this class of curves (in 1818), leading to the name of Lamé curve.
The curve can be seen as a generalisation of an ellipse, a super ellipse (or superellipse). Sometimes the name of super ellipse is only used when the curve is equal in both x- and y-directions. In that case the symmetrical case has the name of super circle (or supercircle).

For a < 2, the curve is also named a hypoellipse.
A circle goes via a square (a = 1) in the direction of a cross (a = 0).
A special case is the astroid, for a = 2/3. Some authors call the general case the astroid.

a = -2 gives the cross curve.









For a > 2, the curve is also named a hyperellipse.
A special case is Piet Hein's ellipse, for a = 5/2. This Danish poet and architect (1905-1996) used the curve for architecture objects as motorway bridges.
The Melior font, designed by Hermann Zapf (in 1952), was based on this curve.

Super ellipse have also been used for:

  • a roundabout in an old city square Sergels Torg in Stockholm, Sweden
  • a table for the Vietnam War negotiators in Paris, 1969. The story is that Piet Hein helped the parties agree on the shape for the negotiating table 
  • the Azteca Olympic Stadium in Mexico City, Mexico (1968)

The curve can be extended to the generalized super ellipse, removing the symmetry between x- and y-axis:

Some examples are the following:

This curve has been generalized to the Gielis curve.

The three-dimensional version of the hyper ellipse has been called the super egg, by Piet Hein.