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arXiv · 2204.01076

On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane

Herbert Edelsbrunner·Alexey Garber·Mohadese Ghafari·Teresa Heiss·Morteza Saghafian
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Abstract

For a locally finite set in $\mathbb{R}^2$, the order-$k$ Brillouin tessellations form an infinite sequence of convex face-to-face tilings of the plane. If the set is coarsely dense and generic, then the corresponding infinite sequences of minimum and maximum angles are both monotonic in $k$. As an example, a stationary Poisson point process in $\mathbb{R}^2$ is locally finite, coarsely dense, and generic with probability one. For such a set, the distribution of angles in the Voronoi tessellations, Delaunay mosaics, and Brillouin tessellations are independent of the order and can be derived from the formula for angles in order-$1$ Delaunay mosaics given by Miles in 1970.

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Herbert Edelsbrunner, Alexey Garber, Mohadese Ghafari, Teresa Heiss, Morteza Saghafian. 2022-04-03. On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane. https://doi.org/10.1007/s00454-023-00566-1

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