Subdivisions of Ring Dupin Cyclides Using Bézier Curves with Mass Points

Author:

Garnier Lionel1,Druoton Lucie2,Bécar Jean-Paul3,Fuchs Laurent4,Morin Géraldine5

Affiliation:

1. L.I.B., University of Burgundy, B.P. 47870, 21078 Dijon Cedex, France

2. I.U.T. of Dijon, University of Burgundy-Franche-Comté, B.P. 47870, 21078 Dijon Cedex, France

3. U.P.H.F. - Campus Mont Houy - 59313 Valenciennes Cedex 9, France

4. X.L.I.M., U.M.R. 7252, University of Poitiers, France

5. Laboratoire IRIT, U.M.R. 5505, University Paul Sabatier, 31 000 Toulouse, France

Abstract

The paper deals in the Computer-Aided Design or Computer-Aided Manufacturing domain with the Dupin cyclides as well as the Bézier curves. It shows that the same algorithms can be used either for subdivisions of ring Dupin cyclides or Bézier curves. The Bézier curves are described with mass points here. The Dupin cyclides are considered in the Minkowski-Lorentz space. This makes a Dupin cyclide as the union of two conics on the unit pseudo-hypersphere, called the space of spheres. And the conics are quadratic Bézier curves modelled by mass points. The subdivision of any Dupin cyclide, is equivalent to subdivide two curves of degree 2, independently, whereas in the 3D Euclidean space, the same work implies the subdivision of a rational quadratic Bézier surface and resolutions of systems of three linear equations. The first part of this work is to consider ring Dupin cyclides because the conics are bounded circles which look like ellipses.

Publisher

World Scientific and Engineering Academy and Society (WSEAS)

Subject

General Mathematics

Reference41 articles.

1. G. Albrecht and W. Degen, Construction of Bézier rectangles and triangles on the symmetric Dupin horn cyclide by means of inversion, Computer Aided Geometric Design 14 (1996), pp. 349–375.

2. J.P. Bécar, Forme (BR) des coniques et de leurs faisceaux, Ph.D. thesis, Université de Valenciennes et de Hainaut-Cambrésis, LIMAV (1997).

3. J.P. Bécar, L. Fuchs, and L. Garnier, Courbe d’une fraction rationnelle et courbes de Bffzier ff points massiques, in , 20-21 mars, Toulouse, France, 2019.

4. J.P. Bécar and L. Garnier, Points massiques, courbes de Bézier quadratiques et coniques : un état de l’art, in G.T.M.G. 2014, 26 au 27 mars, Lyon, 2014.

5. M. Berger, M. Cole, and S. Levy, Geometry II, no. vol. 2 in Universitext (1979), Springer (2009), URL http://books.google. fr/books?id=iER421ZjkqcC.

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