Squircular Calculations

Abstract: The Fernandez-Guasti squircle is a plane algebraic curve that is an intermediate shape between the circle and the square. It has qualitative features that are similar to the more famous Lam\'e curve. However, unlike the Lam\'e curve which has unbounded exponents, the Fernandez-Guasti squircle is a low degree quartic curve. This makes it more amenable to algebraic manipulation and simplification. In this paper, we will analyze this squircle and derive formulas for its area, arc length, and polar form. We will also provide several parametric equations of this squircle. Finally, we extend the Fernandez-Guasti squircle to three dimensions by coming up with an analogous surface that is an intermediate shape between the sphere and the cube.