Squircle
A squircle is a mathematical shape intermediate between a square and a circle. It is a special case of a superellipse. The word "squircle" is a portmanteau of the words "square" and "circle".
In a Cartesian coordinate system, the squircle centred on the point (a, b) with axes parallel to the coordinate axes is described by the equation:
where r is the minor radius of the squircle (cf. equation of a circle).
The case that is centred on the origin (that is, with a = b = 0) is called Lamé's special quartic.
The same equation expressed in polar coordinates, using the sign function , is:
The squircle is a specific case (found by setting n = 4) of the class of shapes known as "supercircles", which have the equation
Unfortunately, the taxonomy is not consistent - some authors refer to the class as "supercircles" and the specific case as a squircle, while others adopt the opposite naming convention. Supercircles in turn are a subset of the even more general "superellipses", which have the equation
where ra and rb are the semi-major and semi-minor axes. Superellipses were extensively studied and popularized by the Danish mathematician Piet Hein.
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