# 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{alignat}{2} \left | \frac{x}{R} \right | ^{2p} + \left | \frac{y}{R} \right | ^{2p} + \left | \frac{z}{R} \right | ^{2p} & \le 1 \\ | x | ^{2p} + | y | ^{2p} + | z | ^{2p} & \le |R|^{2p} \\ \end{alignat}

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.