While experimenting with program synthesis, I found an interesting mathematical curiosity, squircles. Squircle is a curve that interpolates between a square and a circle.
The most obvious way to get a squircle is just to linearly interpolate between a square and a circle in polar coordinates. However, the resulting intermediate curves are not smooth!
The most famous smooth squircle is probably the Lamé curve, also known as superellipse. This squircle is actually an exact circle, but with respect to the non-Euclidean p-norm (it's Euclidean for p=2)!
In a 1992 paper, Fernández-Guasti describes an algebraic squircle. Algebraic means that it's described by an equation consisting only of variables, numbers, addition and multiplication.
Full solutions of the Fernández-Guasti's equation actually contain 4 unbounded arms in addition to the squircular figure in the middle:
You could have discovered the Fernández-Guasti squircles by looking at the function F(x,y)=1-x^2-y^2+x^2y^2. Its level sets are expanding squircles: the unit square is the 0-set, and near the center x^2y^2 is too small, so the level sets tend to be circles in the center.
Additional material:
- Original paper by Fernández-Guasti. Paywalled, accessible via Sci-Hub.
- In Squircular Calculations Chamberlain Fong generalizes Fernández-Guasti squircle to 3d.