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Graph."],"published-print":{"date-parts":[[2006,10]]},"abstract":"<jats:p>\n            In this article, we present regularly parametrized\n            <jats:italic>G<\/jats:italic>\n            <jats:sup>\n              <jats:italic>k<\/jats:italic>\n            <\/jats:sup>\n            free-form spline surfaces that extend box and half-box splines over regular triangular grids. The polynomial degree of these splines is max{4\n            <jats:italic>k<\/jats:italic>\n            + 1, \u23083k\/2 + 1\u2309r}, where\n            <jats:italic>r<\/jats:italic>\n            \u2208 \u2115 can be chosen arbitrarily and determines the flexibility at extraordinary points. The\n            <jats:italic>G<\/jats:italic>\n            <jats:sup>\n              <jats:italic>k<\/jats:italic>\n            <\/jats:sup>\n            splines presented in this article depend crucially on low-degree (re-)parametrizations of piecewise polynomial hole fillings. The explicit construction of such parametrizations forms the core of this work and we present two classes of singular and regular parametrizations. Also, we show how to build box and half-box spline surfaces of arbitrarily high smoothness with holes bounded by only\n            <jats:italic>n<\/jats:italic>\n            patches, in principle.\n          <\/jats:p>","DOI":"10.1145\/1183287.1183289","type":"journal-article","created":{"date-parts":[[2007,1,16]],"date-time":"2007-01-16T19:38:29Z","timestamp":1168976309000},"page":"1281-1293","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":3,"title":["Parametrizations for triangular\n            <i>G<\/i>\n            <sup>\n              <i>k<\/i>\n            <\/sup>\n            spline surfaces of low degree"],"prefix":"10.1145","volume":"25","author":[{"given":"Hartmut","family":"Prautzsch","sequence":"first","affiliation":[{"name":"University of Karlsruhe, Karlsruhe, Germany"}]},{"given":"Georg","family":"Umlauf","sequence":"additional","affiliation":[{"name":"University of Kaiserslautern, Kaiserslautern, Germany"}]}],"member":"320","published-online":{"date-parts":[[2006,10]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"crossref","first-page":"350","DOI":"10.1016\/0010-4485(78)90110-0","article-title":"Recursive generated b-spline surfaces on arbitrary topological meshes","volume":"10","author":"Catmull E.","year":"1978","unstructured":"Catmull , E. and Clark , J. 1978 . Recursive generated b-spline surfaces on arbitrary topological meshes . Comput. Aided Des. 10 , 6, 350 -- 355 . Catmull, E. and Clark, J. 1978. Recursive generated b-spline surfaces on arbitrary topological meshes. Comput. Aided Des. 10, 6, 350--355.","journal-title":"Comput. Aided Des."},{"key":"e_1_2_1_2_1","doi-asserted-by":"crossref","first-page":"356","DOI":"10.1016\/0010-4485(78)90111-2","article-title":"Behaviour of recursive division surfaces near extraordinary points","volume":"10","author":"Doo D.","year":"1978","unstructured":"Doo , D. and Sabin , M. 1978 . Behaviour of recursive division surfaces near extraordinary points . Comput. Aided Des. 10 , 6, 356 -- 360 . Doo, D. and Sabin, M. 1978. Behaviour of recursive division surfaces near extraordinary points. Comput. Aided Des. 10, 6, 356--360.","journal-title":"Comput. Aided Des."},{"doi-asserted-by":"publisher","key":"e_1_2_1_3_1","DOI":"10.1145\/78956.78958"},{"doi-asserted-by":"publisher","key":"e_1_2_1_4_1","DOI":"10.1016\/0167-8396(89)90006-X"},{"doi-asserted-by":"publisher","key":"e_1_2_1_5_1","DOI":"10.1016\/j.cagd.2004.04.005"},{"key":"e_1_2_1_6_1","volume-title":"Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques. 103--112","author":"Kobbelt L.","year":"2000","unstructured":"Kobbelt , L. 2000 . varun 3 subdivision . In Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques. 103--112 . 10.1145\/344779.344835 Kobbelt, L. 2000. varun 3 subdivision. In Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques. 103--112. 10.1145\/344779.344835"},{"key":"e_1_2_1_7_1","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1109\/TPAMI.1980.4766968","article-title":"A theoretical development for the computer generation and display of piecewise polynomial surfaces","volume":"2","author":"Lane J.","year":"1980","unstructured":"Lane , J. and Riesenfeld , R. 1980 . A theoretical development for the computer generation and display of piecewise polynomial surfaces . IEEE Trans. Pattern Anal. 2 , 1, 35 -- 46 . Lane, J. and Riesenfeld, R. 1980. A theoretical development for the computer generation and display of piecewise polynomial surfaces. IEEE Trans. Pattern Anal. 2, 1, 35--46.","journal-title":"IEEE Trans. Pattern Anal."},{"doi-asserted-by":"publisher","key":"e_1_2_1_9_1","DOI":"10.1016\/S0167-8396(01)00081-4"},{"doi-asserted-by":"publisher","key":"e_1_2_1_10_1","DOI":"10.1016\/j.cagd.2004.04.06"},{"doi-asserted-by":"publisher","key":"e_1_2_1_12_1","DOI":"10.1016\/S0167-8396(96)00029-5"},{"key":"e_1_2_1_13_1","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1023\/A:1018945708536","article-title":"Smoothness of subdivision surfces at extraordinary points","volume":"9","author":"Prautzsch H.","year":"1998","unstructured":"Prautzsch , H. 1998 . Smoothness of subdivision surfces at extraordinary points . Adv. Comput. Math. 9 , 377 -- 389 . Prautzsch, H. 1998. Smoothness of subdivision surfces at extraordinary points. Adv. Comput. Math. 9, 377--389.","journal-title":"Adv. Comput. Math."},{"doi-asserted-by":"crossref","unstructured":"Prautzsch H. and Boehm W. 2002. Box splines. In Handbook of CAGD G. Farin et al. Eds. Elsevier 255--282.  Prautzsch H. and Boehm W. 2002. Box splines. In Handbook of CAGD G. Farin et al. Eds. Elsevier 255--282.","key":"e_1_2_1_14_1","DOI":"10.1016\/B978-044451104-1\/50011-3"},{"doi-asserted-by":"crossref","unstructured":"Prautzsch H. Boehm W. and Paluszny M. 2002. B\u00e9zier and B-Spline Techniques. Springer-Verlag New York.   Prautzsch H. Boehm W. and Paluszny M. 2002. B\u00e9zier and B-Spline Techniques. Springer-Verlag New York.","key":"e_1_2_1_15_1","DOI":"10.1007\/978-3-662-04919-8"},{"key":"e_1_2_1_16_1","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1023\/A:1018922530826","article-title":"Degree estimates for Ck-piecewise polynomial subdivision surfaces","volume":"10","author":"Prautzsch H.","year":"1999","unstructured":"Prautzsch , H. and Reif , U. 1999 . Degree estimates for Ck-piecewise polynomial subdivision surfaces . Adv. Comput. Math. 10 , 2, 209 -- 217 . Prautzsch, H. and Reif, U. 1999. Degree estimates for Ck-piecewise polynomial subdivision surfaces. Adv. Comput. Math. 10, 2, 209--217.","journal-title":"Adv. Comput. Math."},{"unstructured":"Prautzsch H. and Umlauf G. 2000. Triangular G2-splines. In Curve and Surface Design---Saint-Malo 1999. P.-L. Laurent et al. Eds. Vanderbilt University Nashville TN. 335--342.  Prautzsch H. and Umlauf G. 2000. Triangular G2-splines. In Curve and Surface Design---Saint-Malo 1999. P.-L. Laurent et al. Eds. Vanderbilt University Nashville TN. 335--342.","key":"e_1_2_1_17_1"},{"key":"e_1_2_1_19_1","doi-asserted-by":"crossref","first-page":"2167","DOI":"10.1090\/S0002-9939-96-03366-7","article-title":"A degree estimate for subdivision surfaces of higher regularity","volume":"124","author":"Reif U.","year":"1996","unstructured":"Reif , U. 1996 . A degree estimate for subdivision surfaces of higher regularity . Proc. Amer. Math. Soc. 124 , 7, 2167 -- 2174 . Reif, U. 1996. A degree estimate for subdivision surfaces of higher regularity. Proc. Amer. Math. Soc. 124, 7, 2167--2174.","journal-title":"Proc. Amer. Math. Soc."},{"key":"e_1_2_1_20_1","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1007\/s003659900063","article-title":"TURBS---Topologically unrestricted rational B-splines","volume":"14","author":"Reif U.","year":"1998","unstructured":"Reif , U. 1998 . TURBS---Topologically unrestricted rational B-splines . Constr. Approx. 14 , 1, 57 -- 78 . Reif, U. 1998. TURBS---Topologically unrestricted rational B-splines. Constr. Approx. 14, 1, 57--78.","journal-title":"Constr. Approx."},{"key":"e_1_2_1_21_1","volume-title":"Geometric Modeling and Computing---Seattle","author":"Umlauf G.","year":"2003","unstructured":"Umlauf , G. 2004. A technique for verifying the smoothness of subdivison schemes . In Geometric Modeling and Computing---Seattle 2003 . M. Lucian and M. Neamtu, Eds . Nashboro Press , 513--521. Umlauf, G. 2004. A technique for verifying the smoothness of subdivison schemes. In Geometric Modeling and Computing---Seattle 2003. M. Lucian and M. Neamtu, Eds. Nashboro Press, 513--521."},{"doi-asserted-by":"publisher","key":"e_1_2_1_22_1","DOI":"10.1016\/S0167-8396(01)00040-1"}],"container-title":["ACM Transactions on Graphics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/1183287.1183289","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/1183287.1183289","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T15:06:37Z","timestamp":1750259197000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/1183287.1183289"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,10]]},"references-count":19,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2006,10]]}},"alternative-id":["10.1145\/1183287.1183289"],"URL":"https:\/\/doi.org\/10.1145\/1183287.1183289","relation":{},"ISSN":["0730-0301","1557-7368"],"issn-type":[{"type":"print","value":"0730-0301"},{"type":"electronic","value":"1557-7368"}],"subject":[],"published":{"date-parts":[[2006,10]]},"assertion":[{"value":"2006-10-01","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}