{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T18:14:40Z","timestamp":1758824080123,"version":"3.41.0"},"reference-count":21,"publisher":"Association for Computing Machinery (ACM)","issue":"2","license":[{"start":{"date-parts":[[2006,4,1]],"date-time":"2006-04-01T00:00:00Z","timestamp":1143849600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Graph."],"published-print":{"date-parts":[[2006,4]]},"abstract":"<jats:p>In this article, we propose a new subdivision scheme based on uniform Powell-Sabin spline subdivision. It belongs to the class of vector subdivision schemes; for each vertex, we have three control points that form a control triangle tangent to the surface instead of one control point. The main advantage of this scheme is that we can choose the values of the normals in the initial vertices which results in more design possibilities. At first sight, it is an approximating scheme because the control points change each iteration. However, the point where the control triangle is tangent to the surface remains the same. Therefore, it is an interpolating scheme. In the regular regions, we use the uniform Powell-Sabin rules, and we develop additional subdivision rules for the new vertices in the neighborhood of extraordinary vertices. The scheme yields<jats:italic>C<\/jats:italic><jats:sup>1<\/jats:sup>continuous surfaces. We also do the convergence analysis based on the eigenproperties of the subdivision matrix and the properties of the characteristic map.<\/jats:p>","DOI":"10.1145\/1138450.1138458","type":"journal-article","created":{"date-parts":[[2006,7,25]],"date-time":"2006-07-25T14:14:26Z","timestamp":1153836866000},"page":"340-355","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":4,"title":["A tangent subdivision scheme"],"prefix":"10.1145","volume":"25","author":[{"given":"Evelyne","family":"Vanraes","sequence":"first","affiliation":[{"name":"Katholieke Universiteit Leuven, Leuven, Belgium"}]},{"given":"Adhemar","family":"Bultheel","sequence":"additional","affiliation":[{"name":"Katholieke Universiteit Leuven, Leuven, Belgium"}]}],"member":"320","published-online":{"date-parts":[[2006,4]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1145\/42458.42459"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1145\/344779.344841"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1145\/280811.280992"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-8396(97)81785-2"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1145\/280811.280991"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1145\/78956.78958"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1145\/166117.166121"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1111\/1467-8659.1530409"},{"key":"e_1_2_1_10_1","doi-asserted-by":"publisher","DOI":"10.1016\/0167-8396(91)90051-C"},{"key":"e_1_2_1_11_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142996304346"},{"key":"e_1_2_1_12_1","doi-asserted-by":"publisher","DOI":"10.1145\/355759.355761"},{"key":"e_1_2_1_13_1","unstructured":"Prautzsch H. and Reif U. 1997. Necessary conditions for subdivision surfaces. Tech. rep. 97\/04 Universitat Stuttgart. Prautzsch H. and Reif U. 1997. Necessary conditions for subdivision surfaces. Tech. rep. 97\/04 Universitat Stuttgart."},{"key":"e_1_2_1_14_1","doi-asserted-by":"publisher","DOI":"10.1023\/A:1018922530826"},{"key":"e_1_2_1_15_1","series-title":"Series in Approximations and Decompositions 2. World Scientific","volume-title":"Some new results on subdivision algorithms for meshes of arbitrary topology","author":"Reif U.","unstructured":"Reif , U. 1995a. Some new results on subdivision algorithms for meshes of arbitrary topology . In Wavelets and MultiLevel Approximation, C. Chui and L. Schumaker, Eds. Series in Approximations and Decompositions 2. World Scientific , Singapore , 367--374. Reif, U. 1995a. Some new results on subdivision algorithms for meshes of arbitrary topology. In Wavelets and MultiLevel Approximation, C. Chui and L. Schumaker, Eds. Series in Approximations and Decompositions 2. World Scientific, Singapore, 367--374."},{"key":"e_1_2_1_16_1","doi-asserted-by":"publisher","DOI":"10.1016\/0167-8396(94)00007-F"},{"key":"e_1_2_1_17_1","unstructured":"Reif U. and Schr\u00f6der P. 2000. Curvature smoothness of subdivision surfaces. Tech. rep. TR-00-03 California Institute of Technology. Reif U. and Schr\u00f6der P. 2000. Curvature smoothness of subdivision surfaces. Tech. rep. TR-00-03 California Institute of Technology."},{"key":"e_1_2_1_18_1","unstructured":"Sabin M. A. and Barthe L. 2002. Artifacts in recursive subdivision schemes. In Curve and Surface Fitting: Saint-Malo 2002 A. Cohen J.-L. Merrien and L. Schumaker Eds. Nashboro Press 353--362. Sabin M. A. and Barthe L. 2002. Artifacts in recursive subdivision schemes. In Curve and Surface Fitting: Saint-Malo 2002 A. Cohen J.-L. Merrien and L. Schumaker Eds. Nashboro Press 353--362."},{"key":"e_1_2_1_19_1","doi-asserted-by":"crossref","unstructured":"Salomon G. Leclercq A. Akkouche S. and Galin E. 2002. Normal control using n-adic subdivision schemes. Shape Modeling International. 21--28. Salomon G. Leclercq A. Akkouche S. and Galin E. 2002. Normal control using n-adic subdivision schemes. Shape Modeling International. 21--28.","DOI":"10.1109\/SMI.2002.1003524"},{"key":"e_1_2_1_20_1","doi-asserted-by":"publisher","DOI":"10.1007\/s003659910006"},{"key":"e_1_2_1_21_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-8396(99)00002-3"},{"key":"e_1_2_1_22_1","unstructured":"Zheludev V. A. Averbuch A. Z. and Gruzd M. 2003. Interpolatory subdivision schemes generated by splines. In Curve and Surface Fitting: Saint-Malo 2002 A. Cohen J.-L. Merrien and L. Schumaker Eds. Nashboro Press 393--402. Zheludev V. A. Averbuch A. Z. and Gruzd M. 2003. Interpolatory subdivision schemes generated by splines. In Curve and Surface Fitting: Saint-Malo 2002 A. Cohen J.-L. Merrien and L. Schumaker Eds. 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