Translational Invariant Minkowski Tensors

Introduction: TODO

Irreducible Minkowski Tensors

2D – Circular Minkowski Tensors

Due to the different rotational symmetries of, for example, a rectangle and a triangle, their anisotropy is encoded in Minkowski tensors of different rank, for example, in W_1^{0,2} for the rectangle and in W_1^{0,3} for the triangle. To define rotational invariants that quantify the degree of anisotropy of different ranks (different rotational symmetries), we need the irreducible representation. Loosely speaking, it is a decomposition into tensors with an s-fold rotational symmetries.

Consider a polygon K with edge lengths L_k:
\vec L_k = L_k \cdot \vec n_k.

Distribution of normal directions:
\rho(\varphi) = \sum\limits_k L_k \, \delta(\varphi-\varphi_k)

Fourier analysis:
\psi_s = \int\limits_{0}^{2\pi} \textnormal d \varphi \,\exp(\I s \varphi) \, \rho(\varphi) = \sum\limits_k L_k \, \exp(\I s \varphi_k)

The zeroth Fourier component is the circumference of the polygon: \psi_0 = L = 2\,W_1
The first Fourier component vanishes for all closed polygons: \psi_1 = 0

As \normaldensity is a real function, we have \psi_{-s} = \psi^*_s.

Only for the circle vanish all irreducible tensors \psi_s with s\geq1 (left). If there is only a single Fourier component \psi_s non-zero, the choice of this rank s defines a convex shapes with an s-fold rotational symmetry.

Explain Phase <-> Rotation

Counterclockwise rotation by \theta: \psi_s(\hat R(\theta) K) = \psi_s(K) \exp(\I s\theta)

3D – Spherical Minkowski Tensors

under construction: explain 3D version here!

Spherical Harmonics

Classical Minkowski Tensors (Cartesian representation)
Using the position vector \textbf r and the normal vector \textbf n on the boundary, the Minkowski Vectors can be defined in the Cartesian representation.

The second-rank Minkowski tensors are defined using the symmetric tensor product (\textbf a\otimes \textbf a)_{ij} = a_i a_j.


W_1^{0,2} = \frac 12 \int\limits_{\partial K} \textbf n \otimes \textbf n \, \mathrm d l\propto \Phi_1^{0,2}
W_2^{0,2} = \frac 12 \int\limits_{\partial K} \kappa \, \textbf n \otimes \textbf n \, \mathrm d l\propto \Phi_0^{0,2}


W_1^{0,2} = \frac 13 \int\limits_{\partial K} \textbf n \otimes \textbf n \, \mathrm d \mathcal O\propto \Phi_2^{0,2}
W_2^{0,2} = \frac 13 \int\limits_{\partial K} H \, \textbf n \otimes \textbf n \, \mathrm d \mathcal O\propto \Phi_1^{0,2}