Average and expected distortion of Voronoi paths and scapes
Edelsbrunner H, Nikitenko A. 2025. Average and expected distortion of Voronoi paths and scapes. Discrete and Computational Geometry. 73, 490–499.
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Abstract
The approximation of a circle with the edges of a fine square grid distorts the perimeter by a factor about 4/Pi. We prove that this factor is the same on average (in the ergodic sense) for approximations of any rectifiable curve by the edges of any non-exotic Delaunay mosaic (known as Voronoi path), and extend the results to all dimensions, generalizing Voronoi paths to Voronoi scapes.
Publishing Year
Date Published
2025-03-01
Journal Title
Discrete and Computational Geometry
Publisher
Springer Nature
Acknowledgement
The authors thank Ranita Biswas and Tatiana Ezubova for the collaboration on computational experiments that motivated the work reported in this paper. The authors also thank Daniel Bonnema for proofreading and noticing an issue with the original proof of Lemma 4.3.
Open access funding provided by Institute of Science and Technology (IST Austria).
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, Grant No. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), Grant No. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), Grant No. I 02979-N35.
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73
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490-499
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IST-REx-ID
Cite this
Edelsbrunner H, Nikitenko A. Average and expected distortion of Voronoi paths and scapes. Discrete and Computational Geometry. 2025;73:490-499. doi:10.1007/s00454-024-00660-y
Edelsbrunner, H., & Nikitenko, A. (2025). Average and expected distortion of Voronoi paths and scapes. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-024-00660-y
Edelsbrunner, Herbert, and Anton Nikitenko. “Average and Expected Distortion of Voronoi Paths and Scapes.” Discrete and Computational Geometry. Springer Nature, 2025. https://doi.org/10.1007/s00454-024-00660-y.
H. Edelsbrunner and A. Nikitenko, “Average and expected distortion of Voronoi paths and scapes,” Discrete and Computational Geometry, vol. 73. Springer Nature, pp. 490–499, 2025.
Edelsbrunner H, Nikitenko A. 2025. Average and expected distortion of Voronoi paths and scapes. Discrete and Computational Geometry. 73, 490–499.
Edelsbrunner, Herbert, and Anton Nikitenko. “Average and Expected Distortion of Voronoi Paths and Scapes.” Discrete and Computational Geometry, vol. 73, Springer Nature, 2025, pp. 490–99, doi:10.1007/s00454-024-00660-y.
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PMID: 39974750
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arXiv 2012.03350
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