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.
The shape indices are defined as
for closed polygons.
is the quadrupole component of the normal density
can detect anisotropy in a three-fold symmetric system (e.g. suitable for detecting equilateral triangles)
can detect anisotropy in a four-fold symmetric system (e.g. suitable for detecting rectangles)
A morphometric distance of a polygon to a reference structure can be quantified by considering the pseudo distance function
equal to 0 indicates a „flat“ body .
equal to 1 indicates an „isotropic“ body , 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.