Squircle

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. In a Cartesian coordinate system, the superellipse is defined by the equation where ra and rb are the semi-major and semi-minor axes, a and b are the x and y coordinates of the center of the ellipse, and n is a positive number. The squircle can be defined as the superellipse with ra = rb and n = 4. Its equation is: where r is the minor radius of the squircle. Compare this to the equation of a circle. When the squircle is centered at the origin, then a = b = 0, and it is called Lamé's special quartic. In terms of the p-norm ‖ ⋅ ‖ p {displaystyle |cdot |_{p}} on R 2 {displaystyle mathbb {R} ^{2}} , the squircle can be expressed as: where p = 4, x c = ( a , b ) {displaystyle mathbf {x} _{c}=(a,b)} is the vector denoting the center of the squircle, and x = ( x , y ) {displaystyle mathbf {x} =(x,y)} . Effectively, this is still a 'circle' of points at a distance r from the center, but distance is defined differently. For comparison, the usual circle is the case p = 2, whereas the square is given by the p → ∞ {displaystyle p o infty } 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 R 3 {displaystyle mathbb {R} ^{3}} , or 'hypersphubes' in higher dimensions. The area inside the squircle can be expressed in terms of the gamma function Γ(x) as where r is the minor radius of the squircle, and S is the lemniscate constant.