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 (sometimes called the "superness").
When n is a positive rational numberp/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 having no elementary geometric interpretation)
where each ± can be chosen separately so that each value of gives four points on the curve. Equivalently, letting range over
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. 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.
^For a derivation of the algebraic equation in the case where n = 2/3, see p. 3 of http://xahlee.info/SpecialPlaneCurves_dir/Astroid_dir/astroid.pdf.
^J. Edwards (1892). Differential Calculus. London: MacMillan and Co. pp. 164.
^ abGardner, Martin (1977), "Piet Hein's Superellipse", Mathematical Carnival. A New Round-Up of Tantalizers and Puzzles from Scientific American, New York: Vintage Press, pp. 240–254, ISBN 978-0-394-72349-5
^The Superellipse, in The Guide to Life, The Universe and Everything by BBC (27 June 2003)
^Tobler, Waldo (1973), "The hyperelliptical and other new pseudocylindrical equal area map projections", Journal of Geophysical Research, 78 (11): 1753–1759, Bibcode:1973JGR....78.1753T, CiteSeerX10.1.1.495.6424, doi:10.1029/JB078i011p01753.