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

where is the body , translated by the vector .

## 2D

Using the position vector and the normal vector 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 .

### Minkowski Vectors

### Minkowski Tensors

with

= curvature

## 3D

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

### Minkowski Vectors

### Minkowski Tensors