Minkowski Tensors of Planar Anisotropic Voronoi Diagrams

Abstract

Anisotropic Voronoi diagrams, in which each cell is the region first reached by an ellipse growing from its site, generate non-convex cells whose integral geometry can be described using Minkowski tensors of arbitrary order. In this work, we derive semi-analytical formulas for Minkowski tensors of arbitrary order of the 0th, 1st and 2nd kind for planar anisotropic Voronoi cells that are star convex at their nucleation point. The resulting formulas involve integrals of contact functions, which describe the cell boundary through a diffeomorphic transformation. This transformation renders the geometry amenable to analytical treatment, enabling us to obtain expressions for the desired Minkowski tensors. Although most of the integrals lack closed-form solutions, they can be efficiently approximated numerically once the contact function and its derivatives are known along the boundary of the cell of interest.

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BibTeX RIS

BibTeX

@misc{venkovic2026minkowski,
  title = {{Minkowski Tensors of Planar Anisotropic Voronoi Diagrams}},
  author = {Nicolas Venkovic and Hartwig Anzt},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.08656},
  eprint = {2610.08656},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.08656},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Minkowski Tensors of Planar Anisotropic Voronoi Diagrams
AU  - Nicolas Venkovic
AU  - Hartwig Anzt
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.08656
UR  - https://arxiv.org/abs/2610.08656
AB  - Anisotropic Voronoi diagrams, in which each cell is the region first reached by an ellipse growing from its site, generate non-convex cells whose integral geometry can be described using Minkowski tensors of arbitrary order. In this work, we derive semi-analytical formulas for Minkowski tensors of arbitrary order of the 0th, 1st and 2nd kind for planar anisotropic Voronoi cells that are star convex at their nucleation point. The resulting formulas involve integrals of contact functions, which describe the cell boundary through a diffeomorphic transformation. This transformation renders the geometry amenable to analytical treatment, enabling us to obtain expressions for the desired Minkowski tensors. Although most of the integrals lack closed-form solutions, they can be efficiently approximated numerically once the contact function and its derivatives are known along the boundary of the cell of interest.
ER  -

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