# 3D Functionals

For sufficiently smooth bodies $K$, the Minkowski Functionals can be intuitively defined via (weighted) integrals over the volume or boundary of the body $K$.

The scalar functionals can be interpreted as area, perimeter, or the Euler characteristic, which is a topological constant. The vectors are closely related to the centers of mass in either solid or hollow bodies. Accordingly, the second-rank tensors correspond the tensors of inertia, or they can be interpreted as the moment tensors of the distribution of the normals on the boundary.

### Minkowski Functionals

Volume
$W_0 = \quad \int\limits_K \mathrm d V \propto V_3$
Surface area
$W_1 = \frac 13 \int\limits_{\partial K} \mathrm d \mathcal O \propto V_2$
Integrated mean curvature
$W_2 = \frac 13 \int\limits_{\partial K} H \mathrm d \mathcal O \propto V_1$
Euler Characteristic
$W_3 = \frac 13 \int\limits_{\partial K} G \mathrm d \mathcal O\propto V_0$

with
$H= \frac 12 (\kappa_1 + \kappa_2)$
$G= \kappa_1 \kappa_2$
$\kappa_i$ = principal curvature

## Cartesian representation (Minkowski Tensors)

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 $(\textbf a^2)_{ij} = a_i a_j$.

### Minkowski Vectors

$W_0^{1,0} = \quad \int\limits_K \textbf r \, \mathrm d V\propto \Phi_3^{1,0}$
$W_1^{1,0} = \frac 13 \int\limits_{\partial K} \textbf r \, \mathrm d \mathcal O\propto \Phi_2^{1,0}$
$W_2^{1,0} = \frac 13 \int\limits_{\partial K} H \, \textbf r \, \mathrm d \mathcal O\propto \Phi_1^{1,0}$
$W_3^{1,0} = \frac 13 \int\limits_{\partial K} G \, \textbf r \, \mathrm d \mathcal O\propto \Phi_0^{1,0}$

### Minkowski Tensors

$W_0^{2,0} = \quad \int\limits_K \textbf r \otimes \textbf r \, \mathrm d V\propto \Phi_3^{2,0}$
$W_1^{2,0} = \frac 13 \int\limits_{\partial K} \textbf r \otimes \textbf r \, \mathrm d \mathcal O\propto \Phi_2^{2,0}$
$W_2^{2,0} = \frac 13 \int\limits_{\partial K} H \, \textbf r \otimes \textbf r \, \mathrm d \mathcal O\propto \Phi_1^{2,0}$
$W_3^{2,0} = \frac 13 \int\limits_{\partial K} G \, \textbf r \otimes \textbf r \, \mathrm d \mathcal O\propto \Phi_0^{2,0}$
$W_1^{0,2} = \frac 13 \int\limits_{\partial K} \textbf n \otimes \textbf n \, \mathrm d \mathcal O\propto \Phi_2^{0,2}$
$W_2^{0,2} = \frac 13 \int\limits_{\partial K} H \, \textbf n \otimes \textbf n \, \mathrm d \mathcal O\propto \Phi_1^{0,2}$

Figures are under construction

## Irreducible representation (Spherical Minkowski Tensors)

under construction