Squircle

A squircle

A squircle is a mathematical shape with properties between those of a square and those of a circle. It is a special case of superellipse. The word "squircle" is a portmanteau of the words "square" and "circle".

Equation

In a Cartesian coordinate system, the squircle centered on the point (ab) with axes parallel to the coordinate axes is described by the equation:

$\left( x - a \right)^4 + \left( y - b \right)^4 = r^4$

where r is the minor radius of the squircle (cf. equation of a circle).

Generalisation

The squircle is a specific case (found by setting n = 4) of the class of shapes known as "supercircles", which have the equation

$\left| x - a \right|^n + \left| y - b \right|^n = |r|^n.\,$

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

$\left|\frac{x - a}{r_a}\right|^n\! + \left|\frac{y - b}{r_b}\right|^n\! = 1,\,$

where ra and rb are the semi-major and semi-minor axes. Superellipses were extensively studied and popularised by the Danish mathematician Piet Hein.

Similar shapes

A squircle (blue) compared with a rounded square (red). (Larger image)

A shape similar to a squircle, called a rounded square, may be generated by arranging four quarters of a circle and connecting their loose ends with straight lines. Such a shape is very similar but not identical to the squircle. Although constructing a rounded square may be conceptually and physically simpler, the squircle has the simpler equation and can be generalised much more easily. One consequence of this is that the squircle and other superellipses can be scaled up or down quite easily. This is useful where, for example, one wishes to create nested squircles.

Various forms of a truncated circle

Another similar shape is defined as the boolean intersection of a square and a concentric circle whose diameter is greater than the length of the side of the square, but less than the length of the diagonal of the square. Such truncated circle shapes lack the tangent continuity possessed by both superellipses and rounded squares.

Uses

Squircles are useful in optics. 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.[1]

Squircles have also been used to construct dinner plates. 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. This is even more true of a square plate, but there are various problems (such as fragility, and difficulty of wiping up sauce) associated with the corners of square plates.[2]

Sumvision manufactures an SD card-based MP3 player named Squircle for the budget market. It is not, however, a genuine squircle, although vaguely similar in shape.[3]

Nokia is closely associated with the squircle, having used it as a touchpad button in many phones.[4][5]