Anisotropy analysis

The Minkowski tensors contain information about both the preferred direction and the amplitude of the anisotropy. The latter can be conveniently extracted by scalar anisotropy indices.

Here, we present some common indices in both 2D and 3D.

Ratio of Eigenvalues $\beta_\nu^{r,s}$

$\displaystyle \beta_\nu^{r,s} := \frac{|\mu_{\mathrm{min}}|}{|\mu_{\mathrm{max}}|} \in [0,1]$

$\beta_\nu^{r,s}$ equal to 0 indicates a “flat” body $K$ .

$\beta_\nu^{r,s}$ equal to 1 indicates an “isotropic” body $K$ , in the sense that it has a statistically identical mass distribution in any set of three orthogonal directions; this includes the sphere, but also regular polyhedra and the FCC, BCC and HCP Voronoi cells.