Squircle

File:Squircle.png

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.

Equation

In a Cartesian coordinate system, the squircle centred on the point $(a,b)$ 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 subgroup of the even more general superellipses, which have the equation

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

where $r_{a}$ and $r_{b}$ are the semimajor and semiminor axes. Superellipses were extensively studied and popularised by the Danish mathematician Piet Hein.

Similar shapes

File:Squircle rounded square.png

A squircle (black) compared with a rounded square (red) (see text)

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. As can be seen in the picture to the right, 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, making it far more interesting mathematically.

Uses

Squircles are primarily 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.

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. The same is true of a square plate, but there are various problems (such as wiping up sauce) associated with the corners of square plates.

Sum Vision manufacture an SD card based MP3 player named Squircle for the budget market. It is not, however, a genuine squircle, just similar in shape.

Other Uses

School children sometimes use the term Squircle to mean a 'squigly circle', such as if the teacher asked them to make a circle but it turned out to be a bit 'pear shaped' the students might call the shape a squircle.

See also

• Ellipse
• Astroid
• Ellipsoid
• Oval (geometry)

External links

• Web based supercircle generator