{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,22]],"date-time":"2023-10-22T07:34:16Z","timestamp":1697960056570},"reference-count":11,"publisher":"Association for Computing Machinery (ACM)","issue":"3","content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Graph."],"published-print":{"date-parts":[[1989,7]]},"abstract":"\n In this paper, we analyze planar parametric cubic curves to determine conditions for loops, cusps, or inflection points. By expressing the curve to be analyzed as a linear combination of control points, it can be transformed such that three of the control points are mapped to specific locations on the plane. We call this image curve the\n canonical curve<\/jats:italic>\n . Affine maps do not affect inflection points, cusps, or loops, so the analysis can be applied to the canonical curve instead of the original one. Since the first three points are fixed, the canonical curve is completely characterized by the position of its fourth point. The analysis therefore reduces to observing which region of the canonical plane the fourth point occupies. We demonstrate that for all parametric cubes expressed in this form, the boundaries of these regions are tonics and straight lines. Special cases include B\u00e9zier curves, B-splines, and Beta-splines. Such a characterization forms the basis for an easy and efficient solution to this problem.\n <\/jats:p>","DOI":"10.1145\/77055.77056","type":"journal-article","created":{"date-parts":[[2002,7,27]],"date-time":"2002-07-27T11:29:00Z","timestamp":1027769340000},"page":"147-163","update-policy":"http:\/\/dx.doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":64,"title":["A geometric characterization of parametric cubic curves"],"prefix":"10.1145","volume":"8","author":[{"given":"Maureen C.","family":"Stone","sequence":"first","affiliation":[{"name":"Xerox PARC, Palo Alto, CA"}]},{"given":"Tony D.","family":"DeRose","sequence":"additional","affiliation":[{"name":"Univ. of Washington, Seattle"}]}],"member":"320","published-online":{"date-parts":[[1989,7]]},"reference":[{"key":"e_1_2_1_1_2","volume-title":"Palo Alto, Calif.","author":"BARTELS R. H.","year":"1987","unstructured":"BARTELS , R. H. , BEATTY , ,J. C., AND BARSKY , B. A. An Introduction to Splines For Use in Computer Graphics and Geometric Modeling. Morgan-Kaufmann , Palo Alto, Calif. 1987 . BARTELS, R. H., BEATTY, ,J. C., AND BARSKY, B. A. An Introduction to Splines For Use in Computer Graphics and Geometric Modeling. Morgan-Kaufmann, Palo Alto, Calif. 1987."},{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1145\/15886.15912"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/0167-8396(84)90003-7"},{"key":"e_1_2_1_4_2","volume-title":"Curves and Surfuces for Computer Aided Geometric Design","author":"FARIN G.","year":"1987","unstructured":"FARIN , G. Curves and Surfuces for Computer Aided Geometric Design . Academic Press , Orlando, Fla . 1987 . FARIN, G. Curves and Surfuces for Computer Aided Geometric Design. Academic Press, Orlando, Fla. 1987."},{"key":"e_1_2_1_5_2","volume-title":"University of Cambridge","author":"FORREST A. R.","year":"1970","unstructured":"FORREST , A. R. Shape classification of the non-rational twisted cubic curve in terms of Bezier polygons. CAD Group Document No. 52 , University of Cambridge , Cambridge, England , Dec. 1970 . FORREST, A. R. Shape classification of the non-rational twisted cubic curve in terms of Bezier polygons. CAD Group Document No. 52, University of Cambridge, Cambridge, England, Dec. 1970."},{"key":"e_1_2_1_6_2","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1016\/0010-4485(80)90149-9","volume":"12","author":"FORREST A. R.","year":"1980","unstructured":"FORREST , A. R. The twisted cubic curve: A computer-aided geometric design approach. Comput. Aided Des. 12 , 4 ( July 1980 ), 165-172. FORREST, A. R. The twisted cubic curve: A computer-aided geometric design approach. Comput. Aided Des. 12, 4 (July 1980), 165-172.","journal-title":"Comput. Aided Des."},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/0167-8396(88)90028-3"},{"key":"e_1_2_1_8_2","volume-title":"Projective Geometry and Its Applications to Computer Graphics","author":"PENNA M. A.","year":"1986","unstructured":"PENNA , M. A. , AND PATTERSON , R. R. Projective Geometry and Its Applications to Computer Graphics . Prentice-Hall , Englewood Cliffs, NJ , 1986 . PENNA, M. A., AND PATTERSON, R. R. Projective Geometry and Its Applications to Computer Graphics. Prentice-Hall, Englewood Cliffs, NJ, 1986."},{"key":"e_1_2_1_9_2","volume-title":"Proceedings of the International Conference on Electronic Publishing, Document Manipulation-Typography (EP88)","author":"PIER K.","year":"1988","unstructured":"PIER , K. , BIER , E. , AND STONE , M. An Introduction to Gargoyle: An interactive illustration tool. In Document Manipulation-Typography , Proceedings of the International Conference on Electronic Publishing, Document Manipulation-Typography (EP88) (Nice, France , Apr. 20-22, 1988 ). Cambridge University Press, 1988, pp. 223-238. PIER, K., BIER, E., AND STONE, M. An Introduction to Gargoyle: An interactive illustration tool. In Document Manipulation-Typography, Proceedings of the International Conference on Electronic Publishing, Document Manipulation-Typography (EP88) (Nice, France, Apr. 20-22, 1988). Cambridge University Press, 1988, pp. 223-238."},{"key":"e_1_2_1_11_2","first-page":"3","volume":"24","author":"SU B.","year":"1983","unstructured":"SU , B. , AND LIU , D. An affine invarient and its application in computational geometry. Scientia Sinica (Series A) 24 , 3 ( Mar. 1983 ), 259-267. SU, B., AND LIU, D. An affine invarient and its application in computational geometry. Scientia Sinica (Series A) 24, 3 (Mar. 1983), 259-267.","journal-title":"Scientia Sinica (Series A)"},{"key":"e_1_2_1_12_2","first-page":"4","volume":"13","author":"WANG C. Y.","year":"1981","unstructured":"WANG , C. Y. Shape classification of the parametric cubic curve and parametric B-spline cubic curve. Comput. Aided Des. 13 , 4 ( 1981 ), 199-206. WANG, C. Y. Shape classification of the parametric cubic curve and parametric B-spline cubic curve. Comput. Aided Des. 13, 4 (1981), 199-206.","journal-title":"Comput. Aided Des."}],"container-title":["ACM Transactions on Graphics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/77055.77056","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,3]],"date-time":"2023-01-03T07:27:51Z","timestamp":1672730871000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/77055.77056"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1989,7]]},"references-count":11,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1989,7]]}},"alternative-id":["10.1145\/77055.77056"],"URL":"http:\/\/dx.doi.org\/10.1145\/77055.77056","relation":{},"ISSN":["0730-0301","1557-7368"],"issn-type":[{"value":"0730-0301","type":"print"},{"value":"1557-7368","type":"electronic"}],"subject":["Computer Graphics and Computer-Aided Design"],"published":{"date-parts":[[1989,7]]},"assertion":[{"value":"1989-07-01","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}