There are two incompatible definitions of the squircle.


The first defines the squircle as the quartic plane curve which is special case of the superellipse with a=b and r=4, namely


illustrated above. This curve as arc length

s=-(3^(1/4))/(16sqrt(2)pi^(7/2)Gamma(5/4))G_(5,5)^(5,5)(1|1/3,2/3,5/6,1,4/3; 1/(12),5/(12),7/(12),3/4,(13)/(12))a

(OEIS A186642), where G_(p,q)^(m,n)(x,...) is a Meijer G-function (M. Trott, pers. comm., Oct. 21, 2011), encloses area


and has area moment of inertia tensor

 I=a^4[pi/(2sqrt(2)) 0; 0 pi/(2sqrt(2))].

The second definition of the squircle was given by Fernandez Guasti (1992), but apparently not dubbed with the name "squircle" until later (Fernández Guasti et al. 2005). This curve has quartic Cartesian equation


with squareness parameter s, where s=0 corresponds to a circle with radius k and s=1 to a square of side length 2k. This curve is actually semialgebraic, as it must be restricted to |x|,|y|<=k to exclude other branches. This squircle encloses area


where E(x,k) is an elliptic integral of the second kind, which can be verified reduces to 4k^2 for s->1 and pik^2 for s->0.

Both versions somewhat resemble the shape of the region swept out by a Reuleaux triangle.

The generalization of the squircle to the case with unequal x- and y-dimensions might be dubbed the rectellipse.

See also

Circle, Ellipse, Rectellipse, Rounded Rectangle, Square, Superellipse

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Fernandez Guasti, M. "Analytic Geometry of Some Rectilinear Figures." Int. J. Educ. Sci. Technol. 23, 895-901, 1992.Fernández Guasti, M.; Meléndez Cobarrubias, A.; Renero Carrillo, F. J.; and Cornejo Rodríguez, A. "LCD Pixel Shape and Far-Field Diffraction Patterns." Optik 116, 265-269, 2005. Sloane, N. J. A. Sequence A186642 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

Weisstein, Eric W. "Squircle." From MathWorld--A Wolfram Web Resource.

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