A squircle is a shape intermediate between a 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.

In a Cartesian coordinate system, the superellipse is defined by the equation $${\displaystyle \left|{\frac {x-a}{r_{a}}}\right|^{n}+\left|{\frac {y-b}{r_{b}}}\right|^{n}=1,}$$ where r_a and r_b 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 prototypical squircle is then defined as the superellipse where r_a = r_b and n = 4. Its equation is:$${\displaystyle \left|x-a\right|^{4}+\left|y-b\right|^{4}=r^{4}}$$where r is the 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 Lamé's special quartic.

The area inside this squircle can be expressed in terms of the beta function B or the gamma function Γ as$${\displaystyle \mathrm {Area} =4\int _{0}^{r}{\sqrt[{4}]{r^{4}-x^{4}}}dx=4r\int _{0}^{r}{\sqrt[{4}]{1-{\frac {x^{4}}{r^{4}}}}}dx=4r^{2}\int _{0}^{1}{\sqrt[{4}]{1-u^{4}}}du={\sqrt[{4}]{4}}r^{2}\int _{0}^{\frac {1}{4}}v^{-{\frac {3}{4}}}{\sqrt[{4}]{1-4v}}dv=}$$$${\displaystyle r^{2}\int _{0}^{1}(1-w)^{\frac {1}{4}}w^{-{\frac {3}{4}}}dw=r^{2}\int _{0}^{1}(1-w)^{{\frac {5}{4}}-1}w^{{\frac {1}{4}}-1}dw=r^{2}\cdot {\text{B}}\left({\frac {1}{4}},{\frac {5}{4}}\right)=4r^{2}{\frac {\left(\operatorname {\Gamma } \left(1+{\frac {1}{4}}\right)\right)^{2}}{\operatorname {\Gamma } \left(1+{\frac {2}{4}}\right)}}=}$$$${\displaystyle {\frac {8r^{2}\left(\operatorname {\Gamma } \left({\frac {5}{4}}\right)\right)^{2}}{\sqrt {\pi }}}=\varpi {\sqrt {2}}\,r^{2}\approx 3.708149\,r^{2},}$$where r is the radius of the squircle, and ${\displaystyle \varpi }$ is the lemniscate constant.

In terms of the p-norm ‖ · ‖_p on R², the squircle can be expressed as:$${\displaystyle \left\|\mathbf {x} -\mathbf {x} _{c}\right\|_{p}=r}$$where p = 4, x_c = (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 hypersphube in higher dimensions. Different values of p 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. 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. This kind of squircle, centered at the origin, is defined by the equation:$${\displaystyle x^{2}+y^{2}-{\frac {s^{2}}{r^{2}}}x^{2}y^{2}=r^{2}}$$where r is the 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, it is a square. This equation allows a smooth parametrization of the transition to a square from a circle, without invoking infinity.

The FG squircle's radial distance ${\displaystyle \rho }$ from center to edge can be described parametrically in terms of the circle radius and rotation angle:

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

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