Asquircle is ashape intermediate between asquare and acircle. There are at least two definitions of "squircle" in use, one based on thesuperellipse, the other arising from work inoptics. The word "squircle" is aportmanteau of the words "square" and "circle". Squircles have been applied indesign andoptics.

In aCartesian coordinate system, thesuperellipse is defined by the equation$${\displaystyle {\left|\frac{x-a}{r_a}\right|}^n+{\left|\frac{y-b}{r_b}\right|}^n=1,}$$wherer_a andr_b are thesemi-major andsemi-minor axes,a andb are thex andy coordinates of the centre of the ellipse, andn is a positive number. The prototypical squircle is then defined as the superellipse wherer_a =r_b andn = 4. Its equation is:^[1]$${\displaystyle {\left|x-a\right|}^4+{\left|y-b\right|}^4=r^4}$$wherer is theradius of the squircle. Compare this to theequation of a circle. When the squircle is centred at the origin, thena =b = 0, and it is calledLamé's special quartic.

Thearea inside this squircle can be expressed in terms of thebeta functionB or thegamma functionΓ as^[1]wherer is the radius of the squircle, andϖ is thelemniscate constant.

In terms of thep-norm‖ · ‖_p onℝ², the squircle can be expressed as:$${\displaystyle {\Vert \mathbf{x}-{\mathbf{x}}_c\Vert }_p=r}$$wherep = 4,x_c = (a,b) is the vector denoting the centre of the squircle, andx = (x,y). Effectively, this is still a "circle" of points at a distancer from the centre, but distance is defined differently. For comparison, the usual circle is the casep = 2, whereas the square is given by thep → ∞ case (thesupremum norm), and a rotated square is given byp = 1 (thetaxicab norm). This allows a straightforward generalization to aspherical cube (sphube), inℝ³, orhypersphube in higher dimensions.^[2] Different values ofp may be used for a more general squircle, from which an analog to trigonometry ("squigonometry") has been developed.

Another squircle comes from work in optics.^[3][4] 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.^[2] This kind of squircle, centered at the origin, is defined by the equation$${\displaystyle x^2+y^2-\frac{s^2}{r^2}{x}^2{y}^2=r^2}$$wherer is the radius of the squircle,s is the squareness parameter, andx andy are in theinterval[−r,r]. Ifs = 0, the equation is a circle; ifs = 1, it is a square. This equation allows a smoothparametrization of the transition to a square from a circle, without invokinginfinity.

The FG squircle's radial distanceρ from center to edge can be described parametrically in terms of the circle radius and rotation angle:^[5]

$${\displaystyle \rho =\frac{r\sqrt{2}}{s|\sin 2\theta |}\sqrt{1-\sqrt{1-s^2{\sin }^{2}2\theta }}}$$

In practice, when plotting on a computer, a small value like 0.001 can be added to the angle argument2θ to avoid theindeterminate form⁠0/0⁠ whenθ =⁠nπ/2⁠ for any integern, or one can setρ =r for these cases.

The squareness parameters 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. Ifs_L is the intended linearly-interpolated position of the corner, the following relationship convertss_L tos for use in the squircle formula to obtain correctly interpolated squircles:^[5]

$${\displaystyle s=2\frac{\sqrt{\left(3-2\sqrt{2}\right)s_{\mathrm{L}}^2-2\left(1-\sqrt{2}\right)s_{\mathrm{L}}}}{(1-\left(1-\sqrt{2}\right)s_{\mathrm{L}}{)}^2}}$$

Another type of squircle arises fromtrigonometry.^[6] This type of squircle is periodic inℝ² and has the equation

$${\displaystyle \cos \left(\frac{s\pi x}{2r}\right)\cos \left(\frac{s\pi y}{2r}\right)=\cos \left(\frac{s\pi }{2}\right)}$$

wherer is the minor radius of the squircle,s is the squareness parameter, andx andy are in the interval(−r,r). Ass approaches 0 in thelimit, the equation becomes a circle. Whens = 1, the equation is a square.

A shape similar to a squircle, called arounded square, may be generated by separating four quarters of a circle and connecting their loose ends with straightlines, 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.

Another similar shape is atruncated circle, the boundary of theintersection of the regions enclosed by a square and by a concentric circle whosediameter 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.

Arounded cube can be defined in terms ofsuperellipsoids.

Similar to the namesquircle, asphube 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:^[5]

$${\displaystyle x^2+y^2+z^2-\frac{s^2}{r^2}\left(x^2{y}^2+y^2{z}^2+x^2{z}^2-\frac{s^2}{r^2}{x}^2{y}^2{z}^2\right)=r^2}$$

In polar coordinates, the sphube is expressed parametrically as

While the squareness parameters in this case does not behave identically to its squircle counterpart, nevertheless the surface is a sphere whens equals 0, and approaches a cube with sharp corners ass approaches 1.^[5]

Squircles are useful inoptics. If light is passed through a two-dimensional square aperture, the central spot in thediffraction pattern can be closely modelled by a squircle or supercircle. If a rectangular aperture is used, the spot can be approximated by asuperellipse.^[4]

Squircles have also been used to constructdinner 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.^[7]

ManyNokia phone models have been designed with a squircle-shaped touchpad button,^[8][9] as was the second generationMicrosoft Zune.^[10]Apple uses an approximation of a squircle (actually a quintic superellipse) for icons iniOS,iPadOS,macOS, and the home buttons of some Apple hardware.^[11] One of the shapes for adaptive icons introduced in theAndroid "Oreo" operating system is a squircle.^[12]Samsung uses squircle-shaped icons in their Android software overlayOne UI, and inSamsung Experience andTouchWiz.^[13]

Italian car manufacturerFiat used numerous squircles in the interior and exterior design of the third generationPanda.^[14]

• Squigonometry
• Astroid
• Ellipse
• Ellipsoid
• L^p spaces
• Oval
• Squround
• Superegg

Wikimedia Commons has media related toSquircle.

• What is the area of a Squircle? onYouTube byMatt Parker
• Online Calculator for supercircle and super-ellipse
• Web based supercircle generator

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• Quartic curves

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