Fernandez-Guasti's squircle

In 1992, Manuel Fernandez-Guasti introduced a smooth algebraic curve that lies between the circle and the square.

The curve has two parameters: radius r and squareness s

This is a quartic curve (i.e. degree 4) because of the x²y² term.

When s = 0, the FG-squircle is a circle with radius r.
When s = 1, the FG-squircle is a square with side length 2r.

In between, it is a shape that resembles both the circle and the square


Note: For this discussion, we are mainly interested with the curve located in the region
-r ≤ x ≤ r and -r ≤ y ≤ r

This quartic polynomial equation can have values for (x,y) outside this scope, but we can ignore those 4 extraneous regions.

See http://148.206.32.135/mfg/arti/90-94/agrec-ijmest92.pdf
for Fernandez-Guasti's original paper.

See also my paper https://arxiv.org/ftp/arxiv/papers/1604/1604.02174.pdf
for more details.