A squircle is a shape intermediate between a square and a circle. There are at least two definitions of "squircle" in use, the most common of which is based on the superellipse. The word "squircle" is a portmanteau of the words "square" and "circle". Squircles have been applied in design and optics.
Squircle centred on the origin (a = b = 0) with minor radius r = 1: x4 + y4 = 1
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. The squircle is then defined as the superellipse with ra = rb and n = 4. Its equation is:[1]
where r is the minor radius of the squircle, and is the lemniscate constant.
p-norm notationedit
In terms of the p-norm‖ · ‖p on R2, the squircle can be expressed as:
where p = 4, xc = (a, b) is the vector denoting the centre of the squircle, and x = (x, y). Effectively, this is still a "circle" of points at a distance r from the centre, but distance is defined differently. 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 hyperspace in higher dimensions.[2]
Fernández-Guasti squircleedit
Another squircle comes from work in optics.[3][4] It may be called the Fernández-giovani squircle, after one of its authors, to distinguish it from the superellipse-related squircle above.[2] This kind of squircle, centred at the origin, can be 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]. If s = 0, the equation is a circle; if s = 1, this is a square. This equation allows a rounded parametrization of the transition to a square from a circle, without involving infinity.
Similar shapesedit
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 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. 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.
Various forms of a truncated circle
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.
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.[4]
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.[5]
Many Nokia phone models have been designed with a squircle-shaped touchpad button,[6][7] as was the second generation Microsoft Zune.[8]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.[9] One of the shapes for adaptive icons introduced in the Android "Oreo" operating system is a squircle.[10]Samsung uses squircle-shaped icons in their Android software overlay One UI, and in Samsung Experience and TouchWiz.[11]
Italian car manufacturer Fiat used numerous squircles in the interior and exterior design of the third generation Panda.[12]
^M. Fernández Guasti (1992). "Analytic Geometry of Some Rectilinear Figures". Int. J. Educ. Sci. Technol. 23: 895–901.
^ abM. Fernández Guasti; A. Meléndez Cobarrubias; F.J. Renero Carrillo; A. Cornejo Rodríguez (2005). "LCD pixel shape and far-field diffraction patterns" (PDF). Optik. 116 (6): 265–269. Bibcode:2005Optik.116..265F. doi:10.1016/j.ijleo.2005.01.018. Retrieved 20 November 2006.
^. Kitchen Contraptions. Archived from the original on 1 November 2006. Retrieved 20 November 2006.
^Nokia Designer Mark Delaney mentions the squircle in a video regarding classic Nokia phone designs: . Archived from the original on 6 January 2010. Retrieved 9 December 2009. See 3:13 in video
^"Clayton Miller evaluates shapes on mobile phone platforms". Retrieved 2 July 2011.
^Marsal, Katie. "Microsoft discontinues hard drives, "squircle" from Zune lineup". Apple Insider. Retrieved 25 August 2022.
^"The Hunt for the Squircle". Retrieved 23 May 2022.
Online Calculator for supercircle and super-ellipse
October 27, 2023
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Not to be confused with squared circle A squircle is a shape intermediate between a square and a circle There are at least two definitions of squircle in use the most common of which is based on the superellipse The word squircle is a portmanteau of the words square and circle Squircles have been applied in design and optics Squircle centred on the origin a b 0 with minor radius r 1 x4 y4 1 Contents 1 Superellipse based squircle 1 1 p norm notation 2 Fernandez Guasti squircle 3 Similar shapes 4 Uses 5 See also 6 References 7 External linksSuperellipse based squircle editIn a Cartesian coordinate system the superellipse is defined by the equation x a r a n y b r b n 1 displaystyle left frac x a r a right n left frac y b r b right n 1 nbsp 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 The squircle is then defined as the superellipse with ra rb and n 4 Its equation is 1 x a 4 y b 4 r 4 displaystyle left x a right 4 left y b right 4 r 4 nbsp where r is the minor radius of the squircle Compare this to the equation of a circle When the squircle is centred at the origin then a b 0 and it is called Lame s special quartic The area inside the squircle can be expressed in terms of the gamma function G as 1 A r e a 4 r 2 G 1 1 4 2 G 1 2 4 8 r 2 G 5 4 2 p ϖ 2 r 2 3 708149 r 2 displaystyle mathrm Area 4r 2 frac left operatorname Gamma left 1 frac 1 4 right right 2 operatorname Gamma left 1 frac 2 4 right frac 8r 2 left operatorname Gamma left frac 5 4 right right 2 sqrt pi varpi sqrt 2 r 2 approx 3 708149 r 2 nbsp where r is the minor radius of the squircle and ϖ displaystyle varpi nbsp is the lemniscate constant p norm notation edit In terms of the p norm p on R2 the squircle can be expressed as x x c p r displaystyle left mathbf x mathbf x c right p r nbsp where p 4 xc a b is the vector denoting the centre of the squircle and x x y Effectively this is still a circle of points at a distance r from the centre but distance is defined differently 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 hyperspace in higher dimensions 2 Fernandez Guasti squircle editAnother squircle comes from work in optics 3 4 It may be called the Fernandez giovani squircle after one of its authors to distinguish it from the superellipse related squircle above 2 This kind of squircle centred at the origin can be defined by the equation x 2 y 2 s 2 r 2 x 2 y 2 r 2 displaystyle x 2 y 2 frac s 2 r 2 x 2 y 2 r 2 nbsp where r is the minor radius of the squircle s is the squareness parameter and x and y are in the interval r r If s 0 the equation is a circle if s 1 this is a square This equation allows a rounded parametrization of the transition to a square from a circle without involving infinity Similar shapes edit nbsp 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 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 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 nbsp Various forms of a truncated circleAnother 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 Uses editSquircles 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 4 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 5 Many Nokia phone models have been designed with a squircle shaped touchpad button 6 7 as was the second generation Microsoft Zune 8 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 9 One of the shapes for adaptive icons introduced in the Android Oreo operating system is a squircle 10 Samsung uses squircle shaped icons in their Android software overlay One UI and in Samsung Experience and TouchWiz 11 Italian car manufacturer Fiat used numerous squircles in the interior and exterior design of the third generation Panda 12 See also editAstroid Ellipse Ellipsoid Lp spaces Oval Squround SupereggReferences edit a b Weisstein Eric W Squircle MathWorld a b Chamberlain Fong 2016 Squircular Calculations arXiv 1604 02174 Bibcode 2016arXiv160402174F a href Template Cite journal html title Template Cite journal cite journal a Cite journal requires journal help M Fernandez Guasti 1992 Analytic Geometry of Some Rectilinear Figures Int J Educ Sci Technol 23 895 901 a b M Fernandez Guasti A Melendez Cobarrubias F J Renero Carrillo A Cornejo Rodriguez 2005 LCD pixel shape and far field diffraction patterns PDF Optik 116 6 265 269 Bibcode 2005Optik 116 265F doi 10 1016 j ijleo 2005 01 018 Retrieved 20 November 2006 Squircle Plate Kitchen Contraptions Archived from the original on 1 November 2006 Retrieved 20 November 2006 Nokia Designer Mark Delaney mentions the squircle in a video regarding classic Nokia phone designs Nokia 6700 The little black dress of phones Archived from the original on 6 January 2010 Retrieved 9 December 2009 See 3 13 in video Clayton Miller evaluates shapes on mobile phone platforms Retrieved 2 July 2011 Marsal Katie Microsoft discontinues hard drives squircle from Zune lineup Apple Insider Retrieved 25 August 2022 The Hunt for the Squircle Retrieved 23 May 2022 Adaptive Icons Retrieved 15 January 2018 OneUI Samsung Developers Retrieved 2022 04 14 PANDA DESIGN STORY PDF Retrieved 30 December 2018 External links edit nbsp Wikimedia Commons has media related to Squircle What is the area of a Squircle on YouTube by Matt Parker Online Calculator for supercircle and super ellipse Web based supercircle generator Retrieved from https en wikipedia org w index php title Squircle amp oldid 1181394077 Similar shapes, wikipedia, wiki, book, books, library,
