{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:45:05Z","timestamp":1787319905658,"version":"build-2736575974"},"reference-count":17,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","funder":[{"DOI":"10.13039\/100010663","name":"H2020 European Research Council","doi-asserted-by":"publisher","award":["StG 757609"],"award-info":[{"award-number":["StG 757609"]}],"id":[{"id":"10.13039\/100010663","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100009024","name":"Exploratory Research for Advanced Technology","doi-asserted-by":"publisher","award":["JPMJER1201"],"award-info":[{"award-number":["JPMJER1201"]}],"id":[{"id":"10.13039\/501100009024","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001700","name":"Ministry of Education, Culture, Sports, Science and Technology","doi-asserted-by":"publisher","award":["17K12635"],"award-info":[{"award-number":["17K12635"]}],"id":[{"id":"10.13039\/501100001700","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CCF-1422311"],"award-info":[{"award-number":["CCF-1422311"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>We consider the problem of digitalizing Euclidean line segments from $\\mathbb{R}^d$ to $\\mathbb{Z}^d$. Christ, P\u00e1lv\u00f6lgyi, and Stojakovi\u0107, [ Discrete Comput. Geom., 47 (2012), pp. 691--710] showed how to construct a set of consistent digital segments (CDS) for $d=2$: a collection of segments connecting any two points in $\\mathbb{Z}^2$ that satisfies the natural extension of the Euclidean axioms to $\\mathbb{Z}^d$. In this paper we study the construction of CDSs in higher dimensions. We extend some of their results to higher dimensions. Specifically we show that any total order can be used to create a set of consistent digital rays CDR in $\\mathbb{Z}^d$ (a set of rays emanating from a fixed point $p$ that satisfies the extension of the Euclidean axioms). Then we use the same approach to create a CDS in $\\mathbb{Z}^d$ and observe that it only works in some cases. We fully characterize for which total orders the construction is consistent (and thus gives a CDS). In particular, this positively answers the question posed by Christ and co-authors.<\/jats:p>","DOI":"10.1137\/17m1136572","type":"journal-article","created":{"date-parts":[[2018,11,15]],"date-time":"2018-11-15T13:23:22Z","timestamp":1542288202000},"page":"2566-2590","source":"Crossref","is-referenced-by-count":1,"title":["High Dimensional Consistent Digital Segments"],"prefix":"10.1137","volume":"32","author":[{"given":"Man-Kwun","family":"Chiu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Matias","family":"Korman","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,11,15]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1142\/S1793830910000863"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1142\/S0218195901000420"},{"key":"atypb3","unstructured":"M. Chiu and M. Korman,\n                      High dimensional consistent digital segments\n                      , in Proceedings of the 33rd Annual Symposium on Computational Geometry, 2017, pp. 31:1-31:15."},{"key":"atypb4","first-page":"337","author":"Chowdhury I.","year":"2015","journal-title":"Berlin"},{"key":"atypb5","first-page":"263","volume":"9644","author":"Chowdhury I.","year":"2016","journal-title":"American Theoretical Informatics Symposium"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-012-9411-y"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1016\/j.cviu.2012.03.003"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-009-9166-2"},{"key":"atypb9","doi-asserted-by":"crossref","unstructured":"M. Gibson, K. Varadarajan, and X. Wu,\n                      Computing regions decomposable into m stars\n                      , in Proceedings of the 22nd Annual European Symposium on Algorithms, Vol. 8737, 2014, pp. 480-491.","DOI":"10.1007\/978-3-662-44777-2_40"},{"key":"atypb10","doi-asserted-by":"crossref","unstructured":"M. T. Goodrich, L. J. Guibas, J. Hershberger, and P. J. Tanenbaum,\n                      Snap rounding line segments efficiently in two and three dimensions\n                      , in Proceedings of the 13th Annual Symposium on Computational Geometry, 1997, pp. 284-293.","DOI":"10.1145\/262839.262985"},{"key":"atypb11","doi-asserted-by":"crossref","unstructured":"D. H. Greene and F. F. Yao,\n                      Finite-resolution computational geometry\n                      , in Proceedings of the 27th Annual Symposium on Foundations of Computer Science, 1986, pp. 143-152.","DOI":"10.1109\/SFCS.1986.19"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1007\/s11263-005-6299-0"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.2004.1262177"},{"key":"atypb14","first-page":"397","author":"Luby M. G.","year":"1987","journal-title":"Proceedings of the NATO Conference on Graphics\/CAD"},{"key":"atypb15","doi-asserted-by":"crossref","unstructured":"K. Sugihara,\n                      Robust geometric computation based on topological consistency\n                      , in Proceedings of the 9th International Conference on Computational Science, 2001, pp. 12-26.","DOI":"10.1007\/3-540-45545-0_10"},{"key":"atypb16","unstructured":"J. van der Corput,\n                      Verteilungsfunktionen I (in German)\n                      , Proc. Koninklijke Nederlandse Akademie van Wetenschappen, 38 (1935), pp. 813-820."},{"key":"atypb17","unstructured":"J. van der Corput,\n                      Verteilungsfunktionen II (in German)\n                      , Proc. Koninklijke Nederlandse Akademie van Wetenschappen, 38 (1935), pp. 1058-1066."}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/17M1136572","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T12:53:55Z","timestamp":1787316835000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/17M1136572"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1]]},"references-count":17,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2018,1]]}},"alternative-id":["10.1137\/17M1136572"],"URL":"https:\/\/doi.org\/10.1137\/17m1136572","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1]]}}}