{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T20:34:45Z","timestamp":1771533285334,"version":"3.50.1"},"reference-count":15,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2016,3,15]],"date-time":"2016-03-15T00:00:00Z","timestamp":1458000000000},"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>PN (point-normal) triangles are cubic B\u00e9zier triangles which meet at their edges to surface a triangular mesh, but this only achieves G0 continuity. We define blending regions that span the edges shared by adjacent pairs of triangular domains and blend the corresponding B\u00e9zier triangles using a univariate blending function formulated in terms of barycentric coordinates. This produces G2 continuity across boundaries while preserving G1 continuity at vertices. The sharpness of the blends can be controlled locally by varying the extent of these blending regions. We demonstrate the effectiveness of our technique by showing several modeling examples.<\/jats:p>","DOI":"10.3390\/sym8030013","type":"journal-article","created":{"date-parts":[[2016,3,15]],"date-time":"2016-03-15T11:10:18Z","timestamp":1458040218000},"page":"13","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["B\u00e9zier Triangles with G2 Continuity across Boundaries"],"prefix":"10.3390","volume":"8","author":[{"given":"Chang-Ki","family":"Lee","sequence":"first","affiliation":[{"name":"Department of Multimedia Engineering, Dongguk University"}]},{"given":"Hae-Do","family":"Hwang","sequence":"additional","affiliation":[{"name":"Department of Multimedia Engineering, Dongguk University"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0015-8305","authenticated-orcid":false,"given":"Seung-Hyun","family":"Yoon","sequence":"additional","affiliation":[{"name":"Department of Multimedia Engineering, Dongguk University"}]}],"member":"1968","published-online":{"date-parts":[[2016,3,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Hoppe, H., DeRose, T., Duchamp, T., Halstead, M., Jin, H., McDonald, J., Schweitzer, J., and Stuetzle, W. (1994, January 24\u201329). Piecewise Smooth Surface Reconstruction. Proceedings of the 21st Annual Conference on Computer Graphics and Interactive Techniques, Orlando, FL, USA.","DOI":"10.1145\/192161.192233"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Vlachos, A., Peters, J., Boyd, C., and Mitchell, J.L. (2001, January 19\u201321). Curved PN Triangles. Proceedings of the 2001 Symposium on Interactive 3D Graphics, Research Triangle Pk, NC, USA.","DOI":"10.1145\/364338.364387"},{"key":"ref_3","unstructured":"F\u00fcnfzig, C., M\u00fcller, K., Hansford, D., and Farin, G. PNG1 Triangles for Tangent Plane Continuous Surfaces on the GPU. Proceedings of the Graphics Interface 2008, Toronto, ON, Canada."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"801","DOI":"10.1016\/j.cagd.2008.08.003","article-title":"A Construction of Rational Manifold Surfaces of Arbitrary Topology and Smoothness from Triangular Meshes","volume":"29","author":"Vecchia","year":"2008","journal-title":"Comput. Aided Geometr. Des."},{"key":"ref_5","unstructured":"Vecchia, G.D., and J\u00fcttler, B. (2009, January 13\u201316). Piecewise Rational Manifold Surfaces with Sharp Features. Proceedings of the 13th IMA International Conference on Mathematics of Surfaces XIII, Blois, France."},{"key":"ref_6","unstructured":"Farin, G. (2002). Curves and Surfaces for CAGD: A Practical Guide, Morgan Kaufmann Publishers. [5th ed.]."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1016\/0010-4485(94)90023-X","article-title":"A survey of blending methods that use parametric surfaces","volume":"26","author":"Vida","year":"1994","journal-title":"Comput. Aided Des."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1016\/0010-4485(89)90046-8","article-title":"Constant-radius blending in surface modeling","volume":"21","author":"Choi","year":"1989","journal-title":"Comput. Aided Des."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1016\/S0167-8396(98)00006-5","article-title":"Differential geometry of G1 variable radius rolling ball blend surfaces","volume":"15","year":"1998","journal-title":"Comput. Aided Geometr. Des."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/PL00013398","article-title":"Parametric Gn Blending of curves and surfaces","volume":"17","author":"Hartmann","year":"2001","journal-title":"Vis. Comput."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"953","DOI":"10.1016\/j.cad.2007.06.009","article-title":"Generating Gn parametric blending surfaces based on partial reparameterization of base surfaces","volume":"39","author":"Song","year":"2007","journal-title":"Comput. Aided Des."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Grimm, C.M., and Hughes, J.F. (1995, January 6\u201311). Modeling Surfaces of Arbitrary Topology Using Manifolds. Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques, Los Angeles, CA, USA.","DOI":"10.1145\/218380.218475"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"643","DOI":"10.1016\/S0167-8396(00)00020-0","article-title":"Modeling surfaces from meshes of arbitrary topology","volume":"17","year":"2000","journal-title":"Comput. Aided Geometr. Des."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Cotrina-Navau, J., Pla-Garcia, N., and Vigo-Anglada, M. (2002, January 17\u201321). A Generic Approach to Free Form Surface Generation. Proceedings of the Seventh ACM Symposium on Solid Modeling and Applications, Saarbrucken, Germany.","DOI":"10.1145\/566282.566291"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1145\/1015706.1015714","article-title":"A Simple Manifold-based Construction of Surfaces of Arbitrary Smoothness","volume":"23","author":"Ying","year":"2004","journal-title":"ACM Trans. Graph."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/8\/3\/13\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T19:20:43Z","timestamp":1760210443000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/8\/3\/13"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,3,15]]},"references-count":15,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2016,3]]}},"alternative-id":["sym8030013"],"URL":"https:\/\/doi.org\/10.3390\/sym8030013","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,3,15]]}}}