View source for Squircle

{{Short description|Shape between a square and a circle}}
{{Distinguish|squared circle (disambiguation){{!}}squared circle}}
{{Use British English|date=September 2013}}
[[File:Superellipse chamfered square.svg|thumb|200px|right|Squircle centred on the origin ({{math|''a'' {{=}} ''b'' {{=}} 0}}) with minor radius {{math|1=''r''&nbsp;=&thinsp;1}}: {{math|''x''<sup>4</sup> + ''y''<sup>4</sup> {{=}} 1}}]]

A '''squircle''' is a [[shape]] intermediate between a [[Square (geometry)|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]]. The word "squircle" is a [[portmanteau]] of the words "square" and "circle". Squircles have been applied in [[design]] and [[optics]].

==Superellipse-based squircle ==
In a [[Cartesian coordinate system]], the [[superellipse]] is defined by the equation
<math display="block">\left|\frac{x - a}{r_a}\right|^n + \left|\frac{y - b}{r_b}\right|^n = 1,</math>
where {{math|''r''<sub>''a''</sub>}} and {{math|''r''<sub>''b''</sub>}} are the [[Semi-major axis|semi-major]] and [[Semi-minor axis|semi-minor]] axes, {{mvar|a}} and {{mvar|b}} are the {{math|''x''}} and {{math|''y''}} coordinates of the centre of the ellipse, and {{mvar|n}} is a positive number. The squircle is then defined as the superellipse where {{math|1=''r''<sub>''a''</sub> = ''r''<sub>''b''</sub>}} and {{math|1=''n'' = 4}}. Its equation is:<ref name=Weisstein>{{MathWorld|Squircle}}</ref><math display="block">\left|x - a\right|^4 + \left|y - b\right|^4 = r^4</math>where {{math|''r''}} is the [[radius]] of the squircle. Compare this to the [[Circle#Equations|equation of a circle]]. When the squircle is centred at the origin, then {{math|1=''a'' = ''b'' = 0}}, and it is called [[Lamé's special quartic]].

The [[area]] inside the squircle can be expressed in terms of the [[beta function]] {{math|B}} or the [[gamma function]] {{math|Γ}} as<ref name=Weisstein/><math display="block"> \mathrm{Area} = 4\int_{0}^{r}\sqrt[4]{r^4-x^4}dx=4r\int_{0}^{r}\sqrt[4]{1-\frac{x^4}{r^4}}dx=4r^2\int_{0}^{1}\sqrt[4]{1-u^4}du=\sqrt[4]{4}r^2\int_{0}^{\frac{1}{4}}v^{-\frac{3}{4}}\sqrt[4]{1-4v}dv= </math><math display="block">
r^2\int_{0}^{1}(1-w)^{\frac{1}{4}}w^{-\frac{3}{4}}dw=r^2\int_{0}^{1}(1-w)^{\frac{5}{4}-1}w^{\frac{1}{4}-1}dw=r^2\cdot\text{B}\left(\frac{1}{4},\frac{5}{4}\right)=4 r^2 \frac{\left(\operatorname{\Gamma} \left(1+\frac14\right)\right)^2}{\operatorname{\Gamma} \left(1+\frac24\right)} = 
</math><math display="block">
\frac{8r^2 \left(\operatorname{\Gamma} \left(\frac54\right)\right)^2 }{ \sqrt{\pi} } = \varpi \sqrt{2}\, r^2 \approx 3.708149\, r^2,
</math>where {{mvar|r}} is the radius of the squircle, and <math> \varpi </math> is the [[lemniscate constant]].

===''p''-norm notation===
In terms of the [[Lp space#The p-norm in finite dimensions|{{math|''p''}}-norm]] {{math|‖ · ‖<sub>''p''</sub>}} on {{math|'''R'''<sup>2</sup>}}, the squircle can be expressed as:<math display="block"> \left\|\mathbf{x} - \mathbf{x}_c\right\|_p = r </math>where {{math|1=''p'' = 4}}, {{math|1='''x'''<sub>''c''</sub> = (''a'', ''b'')}} is the vector denoting the centre of the squircle, and {{math|1='''x''' = (''x'', ''y'')}}. Effectively, this is still a "circle" of points at a distance {{mvar|r}} from the centre, but distance is defined differently. For comparison, the usual circle is the case {{math|1=''p'' = 2}}, whereas the square is given by the {{math|''p'' → ∞}} case (the [[Uniform norm|supremum norm]]), and a rotated square is given by {{math|1=''p'' = 1}} (the [[taxicab norm]]). This allows a straightforward generalization to a '''spherical cube''', or '''[[#Sphube|sphube]]''', in {{math|'''R'''<sup>3</sup>}}, or '''hypersphube''' in higher dimensions.<ref name="fong">{{cite arXiv|title=Squircular Calculations|author=Chamberlain Fong|eprint=1604.02174|year=2016|class=math.GM }}</ref>

== Fernández-Guasti squircle ==

Another squircle comes from work in optics.<ref>{{cite journal|journal=Int. J. Educ. Sci. Technol.| volume = 23|author = M. Fernández Guasti | title= Analytic Geometry of Some Rectilinear Figures| pages=895–901 |year=1992}}</ref><ref name="optik">{{cite journal |journal=[[Optik (journal)|Optik]] |volume=116 |pages=265–269 |year=2005 |url=http://investigacion.izt.uam.mx/mfg/pub/lcdpix_optik05.pdf |accessdate=20 November 2006 |doi=10.1016/j.ijleo.2005.01.018 |title=LCD pixel shape and far-field diffraction patterns |issue=6 |author1=M. Fernández Guasti |author2=A. Meléndez Cobarrubias |author3=F.J. Renero Carrillo |author4=A. Cornejo Rodríguez|bibcode=2005Optik.116..265F }}</ref> It may be called the Fernández-Guasti squircle or FG squircle, after one of its authors, to distinguish it from the superellipse-related squircle above.<ref name="fong" /> This kind of squircle, centered at the origin, is defined by the equation:<math display="block"> x^2 + y^2 - \frac{s^2}{r^2} x^2 y^2 = r^2 </math>where {{mvar|r}} is the radius of the squircle, {{mvar|s}} is the squareness parameter, and {{mvar|x}} and {{mvar|y}} are in the [[interval (mathematics)|interval]] {{closed-closed|−''r'', ''r''}}. If {{math|1=''s'' = 0}}, the equation is a circle; if {{math|1=''s'' = 1}}, it is a square. This equation allows a smooth [[Parametrization (geometry)|parametrization]] of the transition to a square from a circle, without invoking [[infinity]].

=== Polar form ===
The FG squircle's radial distance <math>\rho</math> from center to edge can be described parametrically in terms of the circle radius and rotation angle:<ref name="squircular">{{cite conference |title=Squircular Calculations |author=C. Fong |conference=Joint Mathematics Meeting 2018, SIGMAA-ARTS |year=2018 |eprint=1604.02174}}</ref>

<math display="block">\rho = \frac{r\sqrt{2}}{s|\sin{2\theta}|} \sqrt{1-\sqrt{1-s^2 \sin^2{2\theta}}}</math>

In practice, when plotting on a computer, a small value like 0.001 can be added to the angle argument <math>2\theta</math> to avoid the [[indeterminate form]] <math>\frac{0}{0}</math> when <math>\theta=\frac{n \pi}{2}</math> for any integer <math>n</math>, or one can set <math>\rho=r</math> for these cases.

=== Linearizing squareness ===
The squareness parameter <math>s</math> in the FG squircle, while bounded between 0 and 1, results in a nonlinear interpolation of the squircle "corner" between the inner circle and the square corner. If <math>s_L</math> is the intended liniearly-interpolated position of the corner, the following relationship converts <math>s_L</math> to <math>s</math> for use in the squircle formula to obtain correctly interpolated squircles:<ref name="squircular" />

<math display="block">s = 2 \frac{\sqrt{(3-2\sqrt{2})s_L^2-(2-\sqrt{2})s_L}}{(1-(1-\sqrt{2})s_L)^2}</math>

== Periodic squircle ==
Another type of squircle arises from [[trigonometry]].<ref name="fong2">{{cite arXiv|title=Visualizing Squircular Implicit Surfaces|author=C. Fong|eprint=2210.15232|year=2022|class=cs.GR }}</ref> This type of squircle is periodic in {{math|'''R'''<sup>2</sup>}} and has the equation

<math display="block">\cos\left(\frac{s \pi x}{2 r}\right) \cos\left(\frac{s \pi y}{2 r}\right) = \cos\left(\frac{s \pi }{2 }\right) </math>

where {{mvar|r}} is the minor radius of the squircle, {{mvar|s}} is the squareness parameter, and {{mvar|x}} and {{mvar|y}} are in the interval {{open-open|−r, r}}. As {{mvar|s}} approaches 0 in the [[limit (mathematics)|limit]], the equation becomes a circle. When {{math|1=''s'' = 1}}, the equation is a square.

== Similar shapes ==
[[File:squircle rounded square.svg|thumb|A squircle ({{color|blue|blue}}) compared with a rounded square ({{color|red|red}}).]]

===Rounded square===
A shape similar to a squircle, called a ''{{visible anchor|rounded square}}'', may be generated by separating four quarters of a circle and connecting their loose ends with straight [[line (geometry)|lines]], or by separating the four sides of a square and connecting them with quarter-circles. 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 a 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.

===Truncated circle===
[[File:Truncated circles.svg|thumb|Various forms of a truncated circle]]
Another similar shape is a ''[[Truncation (geometry)|truncated]] circle'', the boundary of the [[intersection (set theory)|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.

===Rounded cube===
A ''rounded cube'' can be defined in terms of [[superellipsoid]]s.

===Sphube===
Similar to the name ''squircle'', a ''sphube'' is a portmanteau of 'sphere' and 'cube'. It is the three-dimensional counterpart to the squircle. The equation for the FG-squircle in three dimensions is:<ref name="squircular" />

<math display="block">x^2 + y^2 + z^2 - \frac{s^2}{r^2}(x^2 y^2 + y^2 z^2 + x^2 z^2 - \frac{s^2}{r^2}x^2 y^2 z^2) = r^2</math>

In polar coordinates, the sphube is expressed parametrically as

<math display="block">x = \frac{r \cos\theta\ \cos\phi}{\sqrt{1-s\cos^2\theta\sin^2\phi - s\sin^2\theta}}</math><math display="block">y = \frac{r \cos\theta\ \sin\phi}{\sqrt{1-s\cos^2\theta\cos^2\phi - s\sin^2\theta}}</math><math display="block">z = \frac{r \sin\theta}{\sqrt{1 - s\cos^2\theta}}</math>

While the squareness parameter {{mvar|s}} in this case does not behave identically to its squircle counterpart, nevertheless the surface is a sphere when {{mvar|s}} equals 0, and approaches a cube with sharp corners as {{mvar|s}} approaches 1.<ref name="squircular" />

==Uses==
[[File:Squircle dishes.jpg|thumb|Squircle-shaped porcelain dishes]]
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]].<ref name="optik"/>

Squircles have also been used to construct [[Plate (dishware)|dinner plate]]s. 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.<ref>{{cite web |publisher=Kitchen Contraptions |url=http://www.kitchencontraptions.com/archives/006830.php |title=Squircle Plate |access-date=20 November 2006 |url-status=dead |archive-url=https://web.archive.org/web/20061101032925/http://www.kitchencontraptions.com/archives/006830.php |archive-date=1 November 2006 }}</ref>

Many [[Nokia]] phone models have been designed with a squircle-shaped touchpad button,<ref>Nokia Designer Mark Delaney mentions the squircle in a video regarding classic Nokia phone designs:<br/>{{cite video |url=http://conversations.nokia.com/2009/06/17/nokia-6700-the-little-black-dress-of-phones/ |title=Nokia 6700 – The little black dress of phones |quote=See 3:13 in video |access-date=9 December 2009 |url-status=dead |archive-url=https://web.archive.org/web/20100106035910/http://conversations.nokia.com/2009/06/17/nokia-6700-the-little-black-dress-of-phones/ |archive-date=6 January 2010 |df= }}</ref><ref>{{cite web |title=Clayton Miller evaluates shapes on mobile phone platforms |url=http://interuserface.net/2011/06/own-a-shape/ |accessdate=2 July 2011}}</ref> as was the second generation [[Zune Pad|Microsoft Zune]].<ref>{{cite web |last1=Marsal |first1=Katie |title=Microsoft discontinues hard drives, "squircle" from Zune lineup |url=https://appleinsider.com/articles/09/09/02/microsoft_discontinues_hard_drives_squircle_from_zune_lineup |website=Apple Insider |date=2 September 2009 |access-date=25 August 2022}}</ref> [[Apple Inc|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.<ref>{{cite web |title=The Hunt for the Squircle |url=https://applypixels.com/blog/the-hunt-for-the-squircle |accessdate=23 May 2022}}</ref> One of the shapes for adaptive icons introduced in the [[Android Oreo|Android "Oreo"]] operating system is a squircle.<ref>{{cite web |url=https://developer.android.com/guide/practices/ui_guidelines/icon_design_adaptive.html | title=Adaptive Icons | accessdate=15 January 2018}}</ref> [[Samsung]] uses squircle-shaped icons in their Android software overlay [[One UI]], and in [[Samsung Experience]] and [[TouchWiz]].<ref>{{Cite web |title=OneUI |url=https://developer.samsung.com/OneUI/iconography/background.html |access-date=2022-04-14 |website=Samsung Developers |language=en}}</ref>

Italian car manufacturer [[Fiat]] used numerous squircles in the interior and exterior design of the third generation [[Fiat Panda|Panda]].<ref>{{cite web |title=PANDA DESIGN STORY |url=http://www.fiatpress.com/download/2011/FIAT/FILES/111011_F_panda_design_story_en.pdf |accessdate=30 December 2018}}</ref>

==See also==
* [[Astroid]]
* [[Ellipse]]
* [[Ellipsoid]]
* [[Lp space|{{mvar|L<sup>p</sup>}} spaces]]
* [[Oval]]
* [[Squround]]
* [[Superegg]]

==References==
{{Reflist|30em}}

==External links==
{{Commons category}}
* {{YouTube|gjtTcyWL0NA|What is the area of a Squircle?}} by [[Matt Parker]]
* [http://www.procato.com/superellipse/ Online Calculator for supercircle and super-ellipse]
* [https://web.archive.org/web/20071203135224/http://www.geocities.com/dougtclark/mySquircle.html Web based supercircle generator]

[[Category:Geometric shapes]]
[[Category:Plane curves]]
[[Category:Quartic curves]]