Author: Laurence D. Finston
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Last updated: May 28, 2022
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.