Squircle

Squircle is a portmanteau word used to refer to 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 centered on the point ("a", "b") with axes parallel to the coordinate axes is described by the equation:

: $left(\; x\; -\; a\; ight)^4\; +\; left(\; y\; -\; b\; ight)^4\; =\; r^4$

where "r" is the minor radius of the squircle (cf. equation of a circle). In other words, it is the boundary of the unit ball of
$L^4$.

Generalisation

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

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

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

:$left|frac\{left(\; x\; -\; a\; ight)\}\{r\_a\}\; ight|^n!\; +\; left|frac\{left(\; y\; -\; b\; ight)\}\{r\_b\}\; ight|^n!\; =\; 1,,$

where "r"_"a" and "r"_"b" are the semi-major and semi-minor axes. Superellipses were extensively studied and popularised by the Danish mathematician Piet Hein.

imilar shapes

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. 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.

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. [M. Fernández Guasti et al., "Optik" 116 p265–269, 2005. [http://investigacion.izt.uam.mx/mfg/pub/lcdpix_optik05.pdf Available online] , accessed 20 November 2006]

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. [Kitchen Contraptions, [http://www.kitchencontraptions.com/archives/006830.php Squircle Plate] , accessed 20 November 2006]

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. [Advanced MP3 Players, [http://www.advancedmp3players.co.uk/shop/product_info.php?products_id=1092 Squircle SD Card MP3 Player] , accessed 20 November 2006]

Microsoft's second iteration of the Zune portable media player sports a squircle as the button pad. [Zune Scene, [http://www.zunescene.com/zune-2-launch-date/ Zune 2 October Launch] , accessed 23 September 2007]

ee also

*Ellipse
*Astroid
*Ellipsoid
*Oval

References

External links

* [http://www.geocities.com/dougtclark/mySquircle.html Web based supercircle generator]