Squircle
A squircle is a shape intermediate between a square and a circle.
There are at least two definitions of "squircle" in use, one based on the superellipse, the other arising from work in optics.
In a Cartesian coordinate system, the superellipse is defined by the equation
where ra and rb are the semi-major and semi-minor axes, a and b are the x and y coordinates of the centre of the ellipse, and n is a positive number.
When the squircle is centred at the origin, then a = b = 0, and it is called Lamé's special quartic.
The area inside the squircle can be expressed in terms of the gamma function Γ as[1]
where p = 4, xc = (a, b) is the vector denoting the centre of the squircle, and x = (x, y).
For comparison, the usual circle is the case p = 2, whereas the square is given by the p → ∞ case (the supremum norm), and a rotated square is given by p = 1 (the taxicab norm).
This allows a straightforward generalization to a spherical cube, or sphube, in R3, or hypersphube in higher dimensions.
[2] This kind of squircle, centered at the origin, is defined by the equation:
where r is the minor radius of the squircle, s is the squareness parameter, and x and y are in the interval [−r, r].
This equation allows a smooth parametrization of the transition to a square from a circle, without involving infinity.
from center to edge can be described parametrically in terms of the circle radius and rotation angle:[5]
In practice, when plotting on a computer, a small value like 0.001 can be added to the angle argument
where r is the minor radius of the squircle, s is the squareness parameter, and x and y are in the interval [−r, r].
This shape can be visualized using online graphing calculators such as Desmos.
[7] A shape similar to a squircle, called a rounded square, may be generated by separating four quarters of a circle and connecting their loose ends with straight lines, or by separating the four sides of a square and connecting them with quarter-circles.
Another similar shape is a truncated circle, the boundary of the intersection of the regions enclosed by a square and by a concentric circle whose diameter is both greater than the length of the side of the square and less than the length of the diagonal of the square (so that each figure has interior points that are not in the interior of the other).
Such shapes lack the tangent continuity possessed by both superellipses and rounded squares.
A rounded cube can be defined in terms of superellipsoids.
Similar to the name squircle, a sphube is a portmanteau of sphere and cube.
In polar coordinates, the sphube is expressed parametrically as
in this case does not behave identically to its squircle counterpart, nevertheless the surface is a sphere when
If light is passed through a two-dimensional square aperture, the central spot in the diffraction pattern can be closely modelled by a squircle or supercircle.
If a rectangular aperture is used, the spot can be approximated by a superellipse.
A squircular plate has a larger area (and can thus hold more food) than a circular one with the same radius, but still occupies the same amount of space in a rectangular or square cupboard.
[8] Many Nokia phone models have been designed with a squircle-shaped touchpad button,[9][10] as was the second generation Microsoft Zune.
[11] Apple uses an approximation of a squircle (actually a quintic superellipse) for icons in iOS, iPadOS, macOS, and the home buttons of some Apple hardware.
[12] One of the shapes for adaptive icons introduced in the Android "Oreo" operating system is a squircle.
[14] Italian car manufacturer Fiat used numerous squircles in the interior and exterior design of the third generation Panda.
