{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,10]],"date-time":"2026-06-10T15:50:10Z","timestamp":1781106610328,"version":"3.54.1"},"reference-count":17,"publisher":"IGI Global Scientific Publishing","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2011,4,1]]},"abstract":"<p>This paper considers the conversion of the parametric B\u00e9zier surfaces, classically used in CAD-CAM, into patched of a class of non-spherical degree 4 algebraic surfaces called Dupin cyclides, and the definition of 3D triangle with circular edges on Dupin cyclides. Dupin cyclides was discovered by the French mathematician Pierre-Charles Dupin at the beginning of the 19th century. A Dupin cyclide has one parametric equation, two implicit equations, and a set of circular lines of curvature. The authors use the properties of these surfaces to prove that three families of circles (meridian arcs, parallel arcs, and Villarceau circles) can be computed on every Dupin cyclide. A geometric algorithm to compute these circles so that they define the edges of a 3D triangle on the Dupin cyclide is presented. Examples of conversions and 3D triangles are also presented to illustrate the proposed algorithms.<\/p>","DOI":"10.4018\/ijcvip.2011040104","type":"journal-article","created":{"date-parts":[[2011,10,19]],"date-time":"2011-10-19T12:09:04Z","timestamp":1319026144000},"page":"42-57","source":"Crossref","is-referenced-by-count":2,"title":["Construction of 3D Triangles on Dupin Cyclides"],"prefix":"10.4018","volume":"1","author":[{"given":"Bertrand","family":"Belbis","sequence":"first","affiliation":[{"name":"Universit\u00e9 de Bourgogne, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lionel","family":"Garnier","sequence":"additional","affiliation":[{"name":"Universit\u00e9 de Bourgogne, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sebti","family":"Foufou","sequence":"additional","affiliation":[{"name":"Universit\u00e9 de Bourgogne, France, and Qatar University, Qatar"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"2432","reference":[{"key":"ijcvip.2011040104-0","unstructured":"Albrecht, G., Belbis, B., & Garnier, L. (2009). Mod\u00e9lisation d'une surface de B\u00e9zier rationnelle biquadratique convertible en un carreau de supercyclide. GTMG, 77-85."},{"key":"ijcvip.2011040104-1","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-8396(97)00002-2"},{"key":"ijcvip.2011040104-2","author":"G.Darboux","year":"1917","journal-title":"Principes de g\u00e9om\u00e9trie analytique"},{"key":"ijcvip.2011040104-3","author":"C.Dupin","year":"1822","journal-title":"Application de g\u00e9om\u00e9trie et de m\u00e9chanique \u00e0 la marine, aux ponts et chauss\u00e9es, etc"},{"key":"ijcvip.2011040104-4","doi-asserted-by":"publisher","DOI":"10.1109\/38.180118"},{"key":"ijcvip.2011040104-5","author":"G.Farin","year":"1993","journal-title":"Curves and surfaces"},{"key":"ijcvip.2011040104-6","unstructured":"Farin, G. (1999). NURBS from projective geometry to practical use (2nd ed.). 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