Voronoi Cells of Lattices with Respect to Arbitrary Norms

Blömer, Johannes; Kohn, Kathlén

doi:10.1137/17m1132045

Cited by 3 publications

(3 citation statements)

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“…Voronoi diagrams and bisectors of sites under general convex distances have been extensively studied [Blömer and Kohn 2018;Corbalan et al 1996;Deza and Sikirić 2015;He et al 2011;Icking et al 2001;Ma 2000;Martini and Swanepoel 2004;Väisälä 2013]. The interested reader can find a global classification of distances for Voronoi diagrams in chapter 20 of the textbook by Deza and Deza [2013].…”

Section: Voronoi Diagrams Induced By Star-shaped Metricsmentioning

confidence: 99%

Star-shaped metrics for mechanical metamaterial design

et al. 2019

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We present a method for designing mechanical metamaterials based on the novel concept of Voronoi diagrams induced by star-shaped metrics. As one of its central advantages, our approach supports interpolation between arbitrary metrics. This capability opens up a rich space of structures with interesting aesthetics and a wide range of mechanical properties, including isotropic, tetragonal, orthotropic, as well as smoothly graded materials. We evaluate our method by creating large sets of example structures, provided as accompanying material. We validate the mechanical properties predicted by simulation through tensile tests on a set of physical prototypes.

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Section: Voronoi Diagrams Induced By Star-shaped Metricsmentioning

confidence: 99%

Star-shaped metrics for mechanical metamaterial design

et al. 2019

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“…Even with this information, the procedure might be time consuming, as there could be exponentially many Voronoi relevant vectors; cf. [9].…”

Section: Super-discrete Sets Of Sitesmentioning

confidence: 99%

Asymmetric tropical distances and power diagrams

Comăneci,

Joswig

2023

Algebraic Combinatorics

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We investigate Voronoi diagrams with respect to an asymmetric tropical distance function, in particular for infinite point sets. These Voronoi diagrams turn out to be much better behaved than those arising from the standard tropical distance, which is symmetric. In particular, we show that the asymmetric tropical Voronoi diagrams may be seen as tropicalizations of power diagrams over fields of real Puiseux series. Our results are then applied to rational lattices and Laurent monomial modules.

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“…Although the problem of finding a shortest basis is NP hard for general metrics [10], there is a single exponential time algorithm for the Euclidean space [11]. In non-Euclidean spaces, the number of Voronoi relevant vectors can be much larger [12].…”

Section: Choices Of Cellsmentioning

confidence: 99%

Note on minimal number of skewed unit cells for periodic distance calculation

Barthel

2018

Preprint

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How many copies of a parallelepiped are needed to ensure that for every point in the parallelepiped a copy of each other point exists, such that the distance between them equals the distance of the pair of points when the opposite sites of the parallelepiped are identified? This question is answered in Euclidean space by constructing the smallest domain that fulfills the above condition. We also describe how to obtain all primitive cells of a lattice (i.e., closures of fundamental domains) that realise the smallest number of copies needed and give them explicitly in 2D and 3D.

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Voronoi Cells of Lattices with Respect to Arbitrary Norms

Cited by 3 publications

References 20 publications

Star-shaped metrics for mechanical metamaterial design

Star-shaped metrics for mechanical metamaterial design

Asymmetric tropical distances and power diagrams

Note on minimal number of skewed unit cells for periodic distance calculation

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