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 defined by an equation that allows for various shapes between a rectangle and an ellipse.

In two dimensional Cartesian coordinate system, a superellipse is defined as the set of all points

(in lowest terms), then each quadrant of the superellipse is a plane algebraic curve of order

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.

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

Numerical integration is another option to obtain perimeter estimates at arbitrary precision.

A closed-form approximation obtained via symbolic regression is also an option that balances parsimony and accuracy.

Then, the arc length of the superellipse within that single quadrant is approximated as the following function of

:[8] h + (((((n-0.88487077) * h + 0.2588574 / h) ^ exp(n / -0.90069205)) + h) + 0.09919785) ^ (-1.4812293 / n) This single-quadrant arc length approximation is accurate to within ±0.2% for across all values of

[10] Using different exponents for each term in the equation, allowing more flexibility in shape formation.

, the curve possesses more flexibility of behavior, and is better possible fit to describe some experimental information.

If any two or all three powers differ from each other, a solid is obtained that may possess more flexibility in representing real structural data than the super ellipsoid.

represents the structure of the National Centre for the Performing Arts in China.

[11] Superquadrics are a family of shapes that include superellipsoids as a special case.

They are used in computer graphics and geometric modeling to create complex, smooth shapes with easily adjustable parameters.

[13] While not a direct generalization of superellipses, hyperspheres also share the concept of extending geometric shapes into higher dimensions.

These related shapes demonstrate the versatility and broad applicability of the fundamental principles underlying superellipses.

This approach can be applied to superellipses, superellipsoids, and their higher-dimensional analogues to produce a wider variety of forms and better fit specific requirements in applications such as computer graphics, structural design, and data visualization.

For instance, anisotropic scaling allows the creation of shapes that can model real-world objects more accurately by adjusting the proportions along each axis independently.

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.

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.

Meanwhile, Piet Hein went on to use the superellipse in other artifacts, such as beds, dishes, tables, etc.

[16] 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.

[15] The superellipse was used for the shape of the 1968 Azteca Olympic Stadium, in Mexico City.

The second floor of the original World Trade Center in New York City consisted of a large, superellipse-shaped overhanging balcony.

The logo for news company The Local consists of a tilted superellipse matching the proportions of Sergels Torg.

In computing, mobile operating system iOS uses a superellipse curve for app icons, replacing the rounded corners style used up to version 6.