{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,16]],"date-time":"2026-01-16T02:21:26Z","timestamp":1768530086302,"version":"3.49.0"},"reference-count":24,"publisher":"Association for Computing Machinery (ACM)","issue":"4","license":[{"start":{"date-parts":[[2019,12,9]],"date-time":"2019-12-09T00:00:00Z","timestamp":1575849600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"name":"Ministero dell\u00f0Istruzione, dell\u00f0Universit\u00e0 e della Ricerca","award":["CUP E83C18000100006"],"award-info":[{"award-number":["CUP E83C18000100006"]}]},{"DOI":"10.13039\/501100007642","name":"Universit\u00e0 degli Studi di Roma Tor Vergata","doi-asserted-by":"crossref","award":["CUP E81I18000060005"],"award-info":[{"award-number":["CUP E81I18000060005"]}],"id":[{"id":"10.13039\/501100007642","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Math. Softw."],"published-print":{"date-parts":[[2019,12,31]]},"abstract":"<jats:p>\n            Multi-degree splines are smooth piecewise-polynomial functions where the pieces can have different degrees. We describe a simple algorithmic construction of a set of basis functions for the space of multi-degree splines with similar properties to standard B-splines. These basis functions are called\n            <jats:italic>multi-degree B-splines<\/jats:italic>\n            (or\n            <jats:italic>MDB-splines<\/jats:italic>\n            ). The construction relies on an extraction operator that represents all MDB-splines as linear combinations of local B-splines of different degrees. This enables the use of existing efficient algorithms for B-spline evaluations and refinements in the context of multi-degree splines. A M\n            <jats:sc>ATLAB<\/jats:sc>\n            implementation is provided to illustrate the computation and use of MDB-splines.\n          <\/jats:p>","DOI":"10.1145\/3321514","type":"journal-article","created":{"date-parts":[[2019,12,9]],"date-time":"2019-12-09T13:35:27Z","timestamp":1575898527000},"page":"1-15","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":21,"title":["Algorithm 999"],"prefix":"10.1145","volume":"45","author":[{"given":"Hendrik","family":"Speleers","sequence":"first","affiliation":[{"name":"University of Rome Tor Vergata, Rome, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2019,12,9]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cagd.2017.10.003"},{"key":"e_1_2_2_2_1","doi-asserted-by":"publisher","DOI":"10.1002\/nme.2968"},{"key":"e_1_2_2_3_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0427(03)00533-8"},{"key":"e_1_2_2_4_1","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1007\/BF00348440","article-title":"Observations on the history of central B-splines","volume":"39","author":"Butzer P. L.","year":"1988","unstructured":"P. L. Butzer, M. Schmidt, and E. L. Stark. 1988. Observations on the history of central B-splines. Archive for History of Exact Sciences 39 (1988), 137--156.","journal-title":"Archive for History of Exact Sciences"},{"key":"e_1_2_2_5_1","volume-title":"Splines: An Introduction","author":"Cohen E.","year":"2001","unstructured":"E. Cohen, R. F. Riesenfeld, and G. Elber. 2001. Geometric Modeling with Splines: An Introduction. CRC Press, Boca Raton, FL."},{"key":"e_1_2_2_6_1","doi-asserted-by":"publisher","DOI":"10.1002\/9780470749081"},{"key":"e_1_2_2_7_1","doi-asserted-by":"publisher","DOI":"10.1093\/imamat\/10.2.134"},{"key":"e_1_2_2_8_1","first-page":"1114","article-title":"On spline distributions and their limits: The P\u00f3lya distribution functions","volume":"53","author":"Curry H. B.","year":"1947","unstructured":"H. B. Curry and I. J. Schoenberg. 1947. On spline distributions and their limits: The P\u00f3lya distribution functions. Bulletin of the American Mathematical Society 53 (1947), 1114.","journal-title":"Bulletin of the American Mathematical Society"},{"key":"e_1_2_2_9_1","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9045(72)90080-9"},{"key":"e_1_2_2_10_1","doi-asserted-by":"publisher","DOI":"10.1137\/0714026"},{"key":"e_1_2_2_11_1","volume-title":"Revised Edition","author":"de Boor C.","unstructured":"C. de Boor. 2001. A Practical Guide to Splines, Revised Edition. Springer-Verlag."},{"key":"e_1_2_2_12_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0021-9045(03)00117-5"},{"key":"e_1_2_2_13_1","doi-asserted-by":"publisher","DOI":"10.1007\/s11390-012-1268-2"},{"key":"e_1_2_2_14_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.compfluid.2016.03.020"},{"key":"e_1_2_2_15_1","doi-asserted-by":"publisher","DOI":"10.1137\/0515061"},{"key":"e_1_2_2_16_1","unstructured":"L. Piegl and W. Tiller. 2012. The NURBS Book. Springer-Verlag."},{"key":"e_1_2_2_17_1","doi-asserted-by":"publisher","DOI":"10.1002\/nme.5216"},{"key":"e_1_2_2_18_1","doi-asserted-by":"publisher","DOI":"10.1090\/qam\/15914"},{"key":"e_1_2_2_19_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-8396(03)00096-7"},{"key":"e_1_2_2_20_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cagd.2009.08.005"},{"key":"e_1_2_2_21_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2010.03.015"},{"key":"e_1_2_2_22_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02075454"},{"key":"e_1_2_2_23_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2016.11.009"},{"key":"e_1_2_2_24_1","volume-title":"Multi-degree B-splines: Algorithmic computation and properties. Computer Aided Geometric Design 76, Article 101792","author":"Toshniwal D.","year":"2020","unstructured":"D. Toshniwal, H. Speleers, R. R. Hiemstra, C. Manni, and T. J. R. Hughes. 2020. Multi-degree B-splines: Algorithmic computation and properties. Computer Aided Geometric Design 76, Article 101792 (2020). https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0167839619301013."}],"container-title":["ACM Transactions on Mathematical Software"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3321514","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3321514","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T23:54:38Z","timestamp":1750204478000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3321514"}},"subtitle":["Computation of Multi-Degree B-Splines"],"short-title":[],"issued":{"date-parts":[[2019,12,9]]},"references-count":24,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2019,12,31]]}},"alternative-id":["10.1145\/3321514"],"URL":"https:\/\/doi.org\/10.1145\/3321514","relation":{},"ISSN":["0098-3500","1557-7295"],"issn-type":[{"value":"0098-3500","type":"print"},{"value":"1557-7295","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,12,9]]},"assertion":[{"value":"2018-01-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2019-03-01","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2019-12-09","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}