# 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 $p$ :

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

The general equation is parametrized by the size, $R$ , and the curvature $p$ :

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

## Volume

The normalized volume for a superball is:

${\frac {V_{\mathrm {sb} }}{R^{3}}}={\frac {2}{2p}}\mathrm {B} \left({\frac {1}{p}},{\frac {2p+1}{2p}}\right)\mathrm {B} \left({\frac {1}{2p}},{\frac {p+1}{p}}\right)$ Where $\mathrm {B} \left(x,y\right)=\Gamma (x)\Gamma (y)/\Gamma (x+y)$ and $\Gamma (x)$ is the usual Euler gamma function.

## Equations

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