# Form Factor:Superball

A superball is a general mathematical shape that can be used to describe rounded cubes. In fact, it is a general parametrization that can describe, via a parameter ${\displaystyle p}$:

• Empty space (${\displaystyle p=0.0}$)
• Concave octahedra (${\displaystyle p<0.5}$)
• Octahedra (${\displaystyle p=0.5}$)
• Convex octahedra (${\displaystyle 0.5)
• Spheres (${\displaystyle p=1}$)
• Rounded cubes (${\displaystyle p>1}$)
• Cubes (${\displaystyle p\to \infty }$)

The general equation is parametrized by the size, ${\displaystyle R}$, and the curvature ${\displaystyle p}$:

{\displaystyle {\begin{alignedat}{2}\left|{\frac {x}{R}}\right|^{2p}+\left|{\frac {y}{R}}\right|^{2p}+\left|{\frac {z}{R}}\right|^{2p}&\leq 1\\|x|^{2p}+|y|^{2p}+|z|^{2p}&\leq |R|^{2p}\\\end{alignedat}}}

Obviously for ${\displaystyle p=1}$, we recover the equation for a sphere. In the limit of large ${\displaystyle p}$, we obtain a cube.

## Volume

The normalized volume for a superball is:

${\displaystyle {\frac {V_{\mathrm {sb} }}{R^{3}}}={\frac {2}{2p}}\mathrm {B} \left({\frac {1}{p}},{\frac {2p+1}{2p}}\right)\mathrm {B} \left({\frac {1}{2p}},{\frac {p+1}{p}}\right)}$

Where ${\displaystyle \mathrm {B} \left(x,y\right)=\Gamma (x)\Gamma (y)/\Gamma (x+y)}$ and ${\displaystyle \Gamma (x)}$ is the usual Euler gamma function.

## Equations

The form factor for a superball is likely not analytic. However, it can be computed numerically.