{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T09:38:24Z","timestamp":1778751504940,"version":"3.51.4"},"reference-count":30,"publisher":"Association for Computing Machinery (ACM)","issue":"3","license":[{"start":{"date-parts":[[2007,7,29]],"date-time":"2007-07-29T00:00:00Z","timestamp":1185667200000},"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":[[2007,7,29]]},"abstract":"<jats:p>Designing rotational symmetries on surfaces is a necessary task for a wide variety of graphics applications, such as surface parameterization and remeshing, painterly rendering and pen-and-ink sketching, and texture synthesis. In these applications, the<jats:italic>topology<\/jats:italic>of a rotational symmetry field such as<jats:italic>singularities<\/jats:italic>and<jats:italic>separatrices<\/jats:italic>can have a direct impact on the quality of the results. In this paper, we present a design system that provides control over the topology of rotational symmetry fields on surfaces.<\/jats:p><jats:p>As the foundation of our system, we provide comprehensive analysis for rotational symmetry fields on surfaces and present efficient algorithms to identify singularities and separatrices. We also describe design operations that allow a rotational symmetry field to be created and modified in an intuitive fashion by using the idea of basis fields and relaxation. In particular, we provide control over the topology of a rotational symmetry field by allowing the user to remove singularities from the field or to move them to more desirable locations.<\/jats:p><jats:p>At the core of our analysis and design implementations is the observations that<jats:italic>N<\/jats:italic>-way rotational symmetries can be described by symmetric<jats:italic>N<\/jats:italic>-th order tensors, which allows an efficient vector-based representation that not only supports coherent definitions of arithmetic operations on rotational symmetries but also enables many analysis and design operations for vector fields to be adapted to rotational symmetry fields.<\/jats:p><jats:p>To demonstrate the effectiveness of our approach, we apply our design system to pen-and-ink sketching and geometry remeshing.<\/jats:p>","DOI":"10.1145\/1276377.1276446","type":"journal-article","created":{"date-parts":[[2007,9,14]],"date-time":"2007-09-14T13:44:55Z","timestamp":1189777495000},"page":"55","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":151,"title":["Rotational symmetry field design on surfaces"],"prefix":"10.1145","volume":"26","author":[{"given":"Jonathan","family":"Palacios","sequence":"first","affiliation":[{"name":"Oregon State University"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Eugene","family":"Zhang","sequence":"additional","affiliation":[{"name":"Oregon State University"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2007,7,29]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"publisher","DOI":"10.1145\/882262.882296"},{"key":"e_1_2_2_2_1","doi-asserted-by":"publisher","DOI":"10.1109\/TVCG.2007.1021"},{"key":"e_1_2_2_3_1","doi-asserted-by":"publisher","DOI":"10.1145\/777792.777839"},{"key":"e_1_2_2_4_1","doi-asserted-by":"crossref","unstructured":"Delmarcelle T. and Hesselink L. 1994. The topology of symmetric second-order tensor fields. IEEE Computer Graphics and Applications 140--147. Delmarcelle T. and Hesselink L. 1994. The topology of symmetric second-order tensor fields. IEEE Computer Graphics and Applications 140--147.","DOI":"10.1109\/VISUAL.1994.346326"},{"key":"e_1_2_2_5_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cagd.2005.04.004"},{"key":"e_1_2_2_6_1","doi-asserted-by":"publisher","DOI":"10.1145\/1141911.1141993"},{"key":"e_1_2_2_7_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-8396(02)00002-5"},{"key":"e_1_2_2_8_1","doi-asserted-by":"publisher","DOI":"10.1145\/340916.340922"},{"key":"e_1_2_2_9_1","doi-asserted-by":"publisher","DOI":"10.1109\/38.79452"},{"key":"e_1_2_2_10_1","doi-asserted-by":"publisher","DOI":"10.1145\/344779.345074"},{"key":"e_1_2_2_11_1","volume-title":"12th Pacific Conference on (PG'04)","author":"Marinov M.","unstructured":"Marinov , M. , and Kobbelt , L . 2004. Direct anisotropic quad-dominant remeshing. Computer Graphics and Applications , 12th Pacific Conference on (PG'04) , 207--216. Marinov, M., and Kobbelt, L. 2004. Direct anisotropic quad-dominant remeshing. Computer Graphics and Applications, 12th Pacific Conference on (PG'04), 207--216."},{"key":"e_1_2_2_12_1","doi-asserted-by":"crossref","unstructured":"Meyer M. Desbrun M. Schr\u00f6der P. and Barr A. H. 2002. Discrete differential-geometry operators for triangulated 2-manifolds. VisMath. Meyer M. Desbrun M. Schr\u00f6der P. and Barr A. H. 2002. Discrete differential-geometry operators for triangulated 2-manifolds. VisMath .","DOI":"10.1007\/978-3-662-05105-4_2"},{"key":"e_1_2_2_13_1","first-page":"393","article-title":"Conley index. Hand-book of Dynamic Systems","volume":"2","author":"Mischaikow K.","year":"2002","unstructured":"Mischaikow , K. , and Mrozek , M. 2002 . Conley index. Hand-book of Dynamic Systems , North-Holland 2 , 393 -- 460 . Mischaikow, K., and Mrozek, M. 2002. Conley index. Hand-book of Dynamic Systems, North-Holland 2, 393--460.","journal-title":"North-Holland"},{"key":"e_1_2_2_14_1","doi-asserted-by":"publisher","DOI":"10.1145\/344779.344987"},{"key":"e_1_2_2_15_1","doi-asserted-by":"publisher","DOI":"10.1145\/1183287.1183297"},{"key":"e_1_2_2_16_1","unstructured":"Ray N. Vallet B. Li W.-C. and Levy B. 2006. N-symmetry direction fields on surfaces of arbitrary genus. In Tech Report. Ray N. Vallet B. Li W.-C. and Levy B. 2006. N-symmetry direction fields on surfaces of arbitrary genus. In Tech Report ."},{"key":"e_1_2_2_17_1","doi-asserted-by":"publisher","DOI":"10.1109\/3DPVT.2004.54"},{"key":"e_1_2_2_18_1","doi-asserted-by":"publisher","DOI":"10.1109\/2945.694953"},{"key":"e_1_2_2_19_1","doi-asserted-by":"publisher","DOI":"10.1145\/882262.882338"},{"key":"e_1_2_2_20_1","volume-title":"Computer Graphics Forum (Proceedings Eurographics","volume":"21","author":"Theisel H.","year":"2002","unstructured":"Theisel , H. 2002 . Designing 2d vector fields of arbitrary topology . In Computer Graphics Forum (Proceedings Eurographics 2002), vol. 21 , 595--604. Theisel, H. 2002. Designing 2d vector fields of arbitrary topology. In Computer Graphics Forum (Proceedings Eurographics 2002), vol. 21, 595--604."},{"key":"e_1_2_2_21_1","doi-asserted-by":"publisher","DOI":"10.1145\/882262.882290"},{"key":"e_1_2_2_22_1","volume-title":"ACM\/Eurographics Symposium on Geometry Processing, 201--210","author":"Tong Y.","unstructured":"Tong , Y. , Alliez , P. , Cohen-Steiner , D. , and Desbrun , M . 2006. Designing quadrangulations with discrete harmonic forms . ACM\/Eurographics Symposium on Geometry Processing, 201--210 . Tong, Y., Alliez, P., Cohen-Steiner, D., and Desbrun, M. 2006. Designing quadrangulations with discrete harmonic forms. 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