Abstract
We deal with the homographic and affine change of parameter of rational curves represented as BR-curves. Forcomputational purposes in CAGD and CAM we determine the new massic polygon through four different methods and we study the particular case of Bézier curves among the applications.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. Alt, Modélisation géométrique à l'aide de courbes et surfaces NURBS, PhD, University of Orsay, France (1991).
P. Bézier, Procédé de définition numérique des courbes et surfaces non mathématiques, système Unisurf, Automatisme 13 (1968) 189–196.
E. Cartan, Nombres complexes, in:Oeuvres Complètes, Partie II, Vol. I (Gauthier-Villars, Paris, 1953) pp. 107–246.
P. De Casteljau, Outillage, méthode de calcul, Rapport technique, André Citroën Automobile S.A., Paris (1959).
G. Farin, Algorithms for rational Bézier curves, Comput. Aided design 15 (1983) 73–77.
G. Farin, Rational curves and surfaces, in:Mathematical Methods in Computer Aided Geometric Design, eds. T. Lyche and L.L. Schumaker (Academic Press, New York, 1989) pp. 215–238.
G. Farin and A. Worsey, Reparameterization and degree elevation for rational Bézier curves, in:NURBS for Curve and Surface Design, ed. G. Farin (SIAM, Philadelphia, 1992) pp. 47–57.
G. Farin,Curves and Surfaces for Computer Aided Geometric Design (Academic Press, 1992).
I.E. Faux and M.J. Pratt,Computational Geometry for Design and Manufacture (Ellis Horwood, Chichester, 1979).
J.C. Fiorot and P. Jeannin, Courbes Bézier rationnelles,19ème Congrès National d'Analyse Numérique, Port-Barcarès, France (1986).
J.C. Fiorot and P. Jeannin, Nouvelle description et calcul des courbes rationnelles à l'aide de points et vecteurs de contrôle. C.R. Acad. Sci. Paris 305, Série I (1987) 435–440.
J.C. Fiorot and P. Jeannin, Surfaces rationnelles controlées par des points pondérés et des vecteurs, C.R. Acad. Sci. Paris 306, Série I (1988) 87–92.
J.C. Fiorot and P. Jeannin,Courbes et Surfaces Rationnelles. Applications à la CAO, RMA 12 (Masson, Paris, 1989).
J.C. Fiorot and P. Jeannin, Courbes splines rationnelles controlées par un polygone massique, C.R. Acad. Sci., Paris 310, Série I (1990) 839–844.
J.C. Fiorot and P. Jeannin, B-rational curves and surfaces, N-rational splines, in:Curves and Surfaces, eds. P.J. Laurent, A. Le Méhauté and L.L. Schumaker (Academic Press, New York, 1991) pp. 181–184.
J.C. Fiorot and P. Jeannin,Rational Curves and Surfaces. Applications to CAD (Wiley, 1992).
J.C. Fiorot and P. Jeannin,Courbes Splines Rationnelles. Applications à la CAO, RMA 24 (Masson, Paris, 1992).
J.C. Fiorot, P. Jeannin and S. Taleb, Homographic change of parameter for B-rational curves, LIMAV Research Report 10, Université de Valenciennes F-59304 (January 1993).
J.C. Fiorot and S. Taleb, Reparametrizations of polynomial and rational curves, in:Curves and Surfaces, eds. P.J. Laurent, A. Le Méhauté and L.L. Schumaker (Academic Press, New York, 1991) pp. 185–188.
J.C. Fiorot, P. Jeannin and S. Taleb, B-rational curves and reparametrization: the quadratic case, RAIRO, Math. Mod. Numer. Anal. 27 (1993) 289–311.
H. Grassmann,Die lineale Ausdehnungslehere, 1844, 1862, (Engel, Leipzig 1894, 1896).
E.T.Y. Lee, The rational Bézier representation for conics, in:Geometric Modelling: Algorithms and New Trends, ed. G. farin (SIAM, Philadelphia, 1987) pp. 3–19.
E.T.Y. Lee and M.L. Lucian, Möbius reparemetrizations of rational B-splines, Comp. Aided Geom. Design 8 (1991) 213–215.
M.L. Lucian, Linear fractional transformations of rational Bézier curves, in:NURBS for Curve and Surface Design, ed. G. Farin (SIAM, Philadelphia, 1992) pp. 131–139.
R.R. Patterson, Projective transformations of the parameter of a Bernstein-Bézier curve, J. ACM Trans. Graph. 4 (1985) 276–290.
L. Piegl, Infinite control points, a method for representing surfaces of revolution using boundary data, I.E.E.A. CGA (1987) 45–55.
L. Piegl, On the use of infinite control points in CAGD, Comp. Aided Geom. Design 4 (1987) 155–166.
S. Taleb, Incidence du paramétrage sur le contrôle des courbes rationnelles, PhD, University of Sciences and Technologies of Lille, France (1991).
K.J. Versprille, Computer aided design applications of the rational B-spline approximation form, PhD, Syracuse University (1975).
Author information
Authors and Affiliations
Additional information
Communicated by C. Brezinski
Rights and permissions
About this article
Cite this article
Fiorot, JC., Jeannin, P. & Taleb, S. New control massic polygon of a B-rational curve resulting from a homographic change of parameter. Numer Algor 6, 379–418 (1994). https://doi.org/10.1007/BF02142679
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02142679
Keywords
- Rational curves
- BR-curves
- homogeneous coordinates
- homographic change of parameter
- affine change of parameter
- massic polygons
- Bézier curves