Author: Laurence D. Finston

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Last updated: May 28, 2022

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Superellipses |

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Superellipses are algebraic curves that mediate between ellipses and rectangles, according
to the values of certain parameters.

METAFONT's superellipses are described in **The METAFONTbook**, p. 126. (Knuth 1986).

In **Mathematical Carnival** (Martin Gardner 1989), Chapter 18 is titled
"Piet Hein's Superellipse" (pp. 240– 254).

Wikipedia article: Superellipse

Superellipses have both a Cartesian and a parametric equation, which may be found in the Wikipedia article cited above. "Ordinary" superellipses have only one value for β whereas a general version has one for the x-coordinate and one for the y-coordinate.

The *superellipse* macro in METAFONT's `plain` format does not, however, use
either of these equations. Instead, it creates a **path** using direction specifiers,
thereby taking advantage of properties of the superellipse regarding the tangents at
particular points on the curve.

The *superellipse* macro therefore only allows for
convex superellipses, whereas using appropriate parameter values results in
paths consisting of concave sections.

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