On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909293750517760 |
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| author | Edelsbrunner, Herbert Garber, Alexey Ghafari, Mohadese Heiss, Teresa Saghafian, Morteza |
| author_facet | Edelsbrunner, Herbert Garber, Alexey Ghafari, Mohadese Heiss, Teresa Saghafian, Morteza |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_01076 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane Edelsbrunner, Herbert Garber, Alexey Ghafari, Mohadese Heiss, Teresa Saghafian, Morteza Combinatorics Computational Geometry Metric Geometry Probability 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. |
| title | On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane |
| topic | Combinatorics Computational Geometry Metric Geometry Probability |
| url | https://arxiv.org/abs/2204.01076 |