{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T08:14:24Z","timestamp":1768292064038,"version":"3.49.0"},"reference-count":18,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,1,1]],"date-time":"2020-01-01T00:00:00Z","timestamp":1577836800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Subdivision schemes are extensively used in scientific and practical applications to produce continuous geometrical shapes in an iterative manner. We construct a numerical algorithm to estimate subdivision depth between the limit curves\/surfaces and their control polygons after k-fold subdivisions. In this paper, the proposed numerical algorithm for subdivision depths of binary subdivision curves and surfaces are obtained after some modification of the results given by Mustafa et al in 2006. This algorithm is very useful for implementation of the parametrization.<\/jats:p>","DOI":"10.3390\/sym12010066","type":"journal-article","created":{"date-parts":[[2020,1,3]],"date-time":"2020-01-03T11:55:07Z","timestamp":1578052507000},"page":"66","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["A Novel Numerical Algorithm to Estimate the Subdivision Depth of Binary Subdivision Schemes"],"prefix":"10.3390","volume":"12","author":[{"given":"Aamir","family":"Shahzad","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan"}]},{"given":"Faheem","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5994-8440","authenticated-orcid":false,"given":"Abdul","family":"Ghaffar","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, BUITEMS, Quetta 87300, Pakistan"}]},{"given":"Ghulam","family":"Mustafa","sequence":"additional","affiliation":[{"name":"Department of Mathematics, The Islamia University of bahawalpur, Punjab 63100, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5769-4320","authenticated-orcid":false,"given":"Kottakkaran Sooppy","family":"Nisar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Arts and Sciences, Prince Sattam bin Abdulaziz University, Wadi Aldawaser 11991, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0286-7244","authenticated-orcid":false,"given":"Dumitru","family":"Baleanu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Cankaya University, Ankara 06530, Turkey"},{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40447, Taiwan"},{"name":"Institute of Space Sciences, 077125 Magurele-Bucharest, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2020,1,1]]},"reference":[{"key":"ref_1","unstructured":"Barnhill, R.E., and Riesenfeld, R.F. (1974). Computer Aided Geometric Design, Academic Press."},{"key":"ref_2","unstructured":"Faux, I., and Pratt, M. (1979). Computational Geometry for Design and Manufacture, Ellis Horwood."},{"key":"ref_3","first-page":"697","article-title":"Error estimation for Doo-Sabin surfaces","volume":"12","author":"Huawei","year":"2002","journal-title":"Prog. Nat. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"596","DOI":"10.1016\/j.cam.2005.06.030","article-title":"Estimating error bounds for binary subdivision curves\/surfaces","volume":"193","author":"Mustafa","year":"2006","journal-title":"J. Comput. Appl. Math."},{"key":"ref_5","first-page":"179","article-title":"Estimating error bounds for non-stationary binary subdivision curves\/surfaces","volume":"2","author":"Mustafa","year":"2007","journal-title":"J. Inf. Comput. 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[Master\u2019s Thesis, Tel Aviv University]."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/1\/66\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T13:28:31Z","timestamp":1760362111000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/1\/66"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1,1]]},"references-count":18,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,1]]}},"alternative-id":["sym12010066"],"URL":"https:\/\/doi.org\/10.3390\/sym12010066","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,1,1]]}}}