Introduction: TODO

## 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 for the rectangle and in 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 -fold rotational symmetries.

Consider a polygon with edge lengths :

.

Distribution of normal directions:

Fourier analysis:

The zeroth Fourier component is the circumference of the polygon:

The first Fourier component vanishes for all closed polygons:

As is a real function, we have .

Explain Phase <-> Rotation

Counterclockwise rotation by :

## 3D – Spherical Minkowski Tensors

under construction: explain 3D version here!

Spherical Harmonics

The second-rank Minkowski tensors are defined using the symmetric tensor product .

### 2D

### 3D