[NURBS ou pas ?]
On peut assez naturellement considérer que la plus grande classe d'espaces de splines utiles pour le design est celle des espaces de splines à sections dans différents espaces « quasi-Chebyshev généralisés » reliées entre elles par des matrices de connexion, espaces qui, de plus, contiennent les constantes et possédent des floraisons. Nous avons récemment donné une construction itérative étonnamment simple de cette classe très difficile. Une application adéquate d'une étape de cette itération peut être interprétée comme la construction de splines rationnelles (donc de NURBS) associées, tout se passant à l'intérieur de la même classe.
The view adopted here is that the largest class of splines which are useful for design is composed of all spaces of geometrically continuous piecewise quasi-Chebyshevian splines that contain constants and possess blossoms. We recently described an iterative construction of all spaces of this class. This note announces the possibility of building associated rational spline spaces (and therefore, associated NURBS) while remaining in the same class. This is obtained when applying in an appropriate way one step of this construction.
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@article{CRMATH_2016__354_7_747_0,
author = {Marie-Laurence Mazure},
title = {NURBS or not {NURBS?}},
journal = {Comptes Rendus. Math\'ematique},
pages = {747--750},
publisher = {Elsevier},
volume = {354},
number = {7},
year = {2016},
doi = {10.1016/j.crma.2016.01.027},
language = {en},
}
Marie-Laurence Mazure. NURBS or not NURBS?. Comptes Rendus. Mathématique, Volume 354 (2016) no. 7, pp. 747-750. doi : 10.1016/j.crma.2016.01.027. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.01.027/
[1] Knot insertion algorithms for Chebyshev splines, Dept. of Maths, University of Zagreb, 2006 (PhD thesis)
[2] Variable degree polynomial splines are Chebyshev splines, Adv. Comput. Math., Volume 38 (2013), pp. 383-400
[3] Curve and surface construction using variable degree polynomial splines, Comput. Aided Geom. Des., Volume 17 (2000), pp. 419-446
[4] On a class of weak Tchebycheff systems, Numer. Math., Volume 101 (2005), pp. 333-354
[5] NURBS: from Projective Geometry to Practical Use, A.K. Peters, Natick, MA, USA, 1999
[6] Rational Curves and Surfaces: Applications to CAD, John Wiley & Sons, NY, 1992
[7] Total positivity and the shape of curves (M. Gasca; C.A. Micchelli, eds.), Total Positivity and Its Applications, Kluwer Academic Publishers, 1996, pp. 157-186
[8] Blossoming beyond extended Chebyshev spaces, J. Approx. Theory, Volume 109 (2001), pp. 48-81
[9] Convexity preserving polynomial splines of non-uniform degree, IMA J. Numer. Anal., Volume 10 (1990), pp. 223-234
[10] Quasi-Chebyshev splines with connection matrices. Application to variable degree polynomial splines, Comput. Aided Geom. Des., Volume 18 (2001), pp. 287-298
[11] Blossoms and optimal bases, Adv. Comput. Math., Volume 20 (2004), pp. 177-203
[12] Which spaces for design, Numer. Math., Volume 110 (2008), pp. 357-392
[13] Ready-to-blossom bases and the existence of geometrically continuous piecewise Chebyshevian B-splines, C. R. Acad. Sci. Paris, Ser. I, Volume 347 (2009) no. 13–14, pp. 829-834
[14] Quasi-extended Chebyshev spaces and weight functions, Numer. Math., Volume 118 (2011), pp. 79-108
[15] On a general new class of quasi-Chebyshevian splines, Numer. Algorithms, Volume 58 (2011) no. 3, pp. 399-438
[16] How to build all Chebyshevian spline spaces good for geometric design, Numer. Math., Volume 119 (2011), pp. 517-556
[17] Piecewise Chebyshev–Schoenberg operators: shape preservation, approximation and space embedding, J. Approx. Theory, Volume 166 (2013), pp. 106-135
[18] Which spline spaces for design?, C. R. Acad. Sci. Paris, Ser. I, Volume 353 (2015) no. 8, pp. 761-765 | DOI
[19] Spline Functions, Wiley–Interscience, New York, 1981
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Commentaires - Politique
Chebyshevian splines: interpolation and blossoms
Alexander Kayumov; Marie-Laurence Mazure
C. R. Math (2007)
Marie-Laurence Mazure
C. R. Math (2009)