{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/bounds-on-the-number-of-discontinuities-of","title":"Bounds on the number of discontinuities of Morton-type space-filling curves","arxiv_id":"1505.05055","date":"2015-03-24","proceeding":null,"authors":["Carsten Burstedde","Johannes Holke","Tobin Isaac"],"abstract":"The Morton- or z-curve is one example for a space filling curve: Given a level of refinement L, it maps the interval [0, 2**dL) one-to-one to a set of d-dimensional cubes of edge length 2**-L that form a subdivision of the unit cube. Similar curves have been proposed for triangular and tetrahedral unit domains. In contrast to the Hilbert curve that is continuous, the Morton-type curves produce jumps. We prove that any contiguous subinterval of the curve divides the domain into a bounded number of face-connected subdomains. For the hypercube case and arbitrary dimension, the subdomains are star-shaped and the bound is indeed two. For the simplicial case in dimensions 2 and 3, the bound is proportional to the depth of refinement L. We supplement the paper with theoretical and computational studies on the frequency of jumps for a quantitative assessment.","url_abs":"http://arxiv.org/abs/1505.05055v4","url_pdf":"http://arxiv.org/pdf/1505.05055v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"bounds-on-the-number-of-discontinuities-of","repo_url":"https://github.com/dlr-amr/t8code","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}