{"abstract":[{"lang":"eng","text":"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."}],"file":[{"date_updated":"2025-04-23T07:31:32Z","checksum":"ffb0c818222138f9f113f4bbea41e834","file_name":"2025_DiscreteComputGeom_EdelsbrunnerHe.pdf","file_id":"19610","success":1,"date_created":"2025-04-23T07:31:32Z","content_type":"application/pdf","access_level":"open_access","relation":"main_file","creator":"dernst","file_size":283443}],"quality_controlled":"1","language":[{"iso":"eng"}],"volume":73,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","scopus_import":"1","doi":"10.1007/s00454-024-00660-y","month":"03","status":"public","OA_type":"hybrid","publication_status":"published","corr_author":"1","day":"01","publisher":"Springer Nature","article_processing_charge":"Yes (via OA deal)","oa_version":"Published Version","arxiv":1,"file_date_updated":"2025-04-23T07:31:32Z","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"isi":1,"year":"2025","date_updated":"2026-02-16T12:18:50Z","ec_funded":1,"pmid":1,"ddc":["510"],"intvolume":"        73","article_type":"original","date_created":"2024-06-16T22:01:07Z","oa":1,"page":"490-499","type":"journal_article","has_accepted_license":"1","external_id":{"pmid":["39974750"],"arxiv":["2012.03350"],"isi":["001238566200004"]},"tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"_id":"17149","project":[{"call_identifier":"H2020","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended"},{"name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342","call_identifier":"FWF"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","call_identifier":"FWF"}],"author":[{"full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner"},{"last_name":"Nikitenko","orcid":"0000-0002-0659-3201","first_name":"Anton","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","full_name":"Nikitenko, Anton"}],"title":"Average and expected distortion of Voronoi paths and scapes","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.\r\nOpen access funding provided by Institute of Science and Technology (IST Austria).\r\nThis 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.","publication":"Discrete & Computational Geometry","date_published":"2025-03-01T00:00:00Z","department":[{"_id":"HeEd"}],"citation":{"ieee":"H. Edelsbrunner and A. Nikitenko, “Average and expected distortion of Voronoi paths and scapes,” <i>Discrete &#38; Computational Geometry</i>, vol. 73. Springer Nature, pp. 490–499, 2025.","mla":"Edelsbrunner, Herbert, and Anton Nikitenko. “Average and Expected Distortion of Voronoi Paths and Scapes.” <i>Discrete &#38; Computational Geometry</i>, vol. 73, Springer Nature, 2025, pp. 490–99, doi:<a href=\"https://doi.org/10.1007/s00454-024-00660-y\">10.1007/s00454-024-00660-y</a>.","apa":"Edelsbrunner, H., &#38; Nikitenko, A. (2025). Average and expected distortion of Voronoi paths and scapes. <i>Discrete &#38; Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-024-00660-y\">https://doi.org/10.1007/s00454-024-00660-y</a>","ista":"Edelsbrunner H, Nikitenko A. 2025. Average and expected distortion of Voronoi paths and scapes. Discrete &#38; Computational Geometry. 73, 490–499.","short":"H. Edelsbrunner, A. Nikitenko, Discrete &#38; Computational Geometry 73 (2025) 490–499.","ama":"Edelsbrunner H, Nikitenko A. Average and expected distortion of Voronoi paths and scapes. <i>Discrete &#38; Computational Geometry</i>. 2025;73:490-499. doi:<a href=\"https://doi.org/10.1007/s00454-024-00660-y\">10.1007/s00454-024-00660-y</a>","chicago":"Edelsbrunner, Herbert, and Anton Nikitenko. “Average and Expected Distortion of Voronoi Paths and Scapes.” <i>Discrete &#38; Computational Geometry</i>. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/s00454-024-00660-y\">https://doi.org/10.1007/s00454-024-00660-y</a>."},"OA_place":"publisher"}