Translational Covariant Minkowski Tensors

In addition to the T-invariant Minkowski tensors, T-covariant tensors
are required for a complete additive characterization. Alesker’s theorem
identifies the minimal set of tensors up to rank two. (TODO insert reference)

Translation-covariant, or T-covariant tensors for short, depend on the chosen origin and transform as follows when the object K is being translated:
TODO describe behavior under translation here
W_\nu^{r,s}(\hat T_x K) = ...
where \hat T_x K is the body K, translated by the vector \vec x.

2D

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.

Minkowski Vectors

W_0^{1,0} = \quad \int\limits_K \textbf r \, \mathrm d A\propto \Phi_2^{1,0}
W_1^{1,0} = \frac 12 \int\limits_{\partial K} \textbf r \, \mathrm d l\propto \Phi_1^{1,0}
W_2^{1,0} = \frac 12 \int\limits_{\partial K} \kappa \, \textbf r \, \mathrm d l\propto \Phi_0^{1,0}

Minkowski Tensors

W_0^{2,0} = \quad \int\limits_K \textbf r \otimes \textbf r \, \mathrm d A\propto \Phi_2^{2,0}
W_1^{2,0} = \frac 12 \int\limits_{\partial K} \textbf r \otimes \textbf r \, \mathrm d l\propto \Phi_1^{2,0}
W_2^{2,0} = \frac 12 \int\limits_{\partial K} \kappa \, \textbf r \otimes \textbf r \, \mathrm d l\propto \Phi_0^{2,0}
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}

with
\kappa = curvature

3D

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}