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.

Isoperimetric ratio Q

The isoperimetric ratio Q is defined as the quotient of the area and the circumference of an object. It is normalized so that Q = 1 for a circle.

In 2D: Q = 4 \pi \frac{W_0}{(2\,W_1)^2}

In 3D: Q = 36 \pi \frac{W_0^2}{(3\,W_1)^3}

Q-and-beta

Circular Minkowski indices q_l

In 2D, q_l are the normalized Fourier coefficients of the density of the edge normals; for details see Circular Minkowski Tensors.

Spherical Minkowski indices

In 3D, …

under construction