De Casteljau’s geometric approach to geometric design still alive. (English) Zbl 07923123
Summary: With great enthusiasm and admiration we would like to pay tribute to Paul de Faget de Casteljau for his essential contribution to CAGD. Motivated by the development of automated human-computer collaboration for car industry, not only was he the very first pioneer in this field, but his initial geometric approach to creating shapes from poles was even undeniably the simplest and most remarkably effective. Two crucial points in this approach are to keep in mind: firstly, the idea of splitting one variable into several variables to facilitate the algorithmic construction of curves; secondly, the possibility of controlling shapes by means of osculating flats and corner-cutting algorithms. The present article is a partial survey on Chebyshevian blossoms intended to show that his ideas are still alive.
MSC:
| 65D17 | Computer-aided design (modeling of curves and surfaces) |
| 65D07 | Numerical computation using splines |
| 41A50 | Best approximation, Chebyshev systems |
| 41A20 | Approximation by rational functions |
| 47A58 | Linear operator approximation theory |
Keywords:
de Casteljau’s algorithm; free form curve design; polar forms and blossoms; extended Chebyshev spaces; shape parameters; dimension elevation; Bernstein-type operators; rational spacesReferences:
| [1] | Ait-Haddou, R., Dimension elevation in Müntz spaces: a new emergence of the Müntz condition, J. Approx. Theory, 181, 6-17, 2014 · Zbl 1288.65017 |
| [2] | Ait-Haddou, R.; Beccari, C.; Mazure, M.-L., Interpolation of \(G^1\) Hermite data by \(C^1\) cubic-like sparse Pythagorean hodograph splines, Comput. Aided Geom. Des., 79, Article 101838 pp., 2020 · Zbl 1505.65041 |
| [3] | Ait-Haddou, R.; Mazure, M.-L., Approximation by Müntz spaces on positive intervals, C. R. Acad. Sci. Paris, 351, 849-852, 2013 · Zbl 1323.41020 |
| [4] | Ait-Haddou, R.; Mazure, M.-L., Approximation by chebyshevian Bernstein operators versus convergence of dimension elevation, Constr. Approx., 43, 425-461, 2016 · Zbl 1350.41023 |
| [5] | Ait-Haddou, R.; Mazure, M.-L., Sparse Pythagorean hodograph curves, Comput. Aided Geom. Des., 55, 84-103, 2017 · Zbl 1375.65019 |
| [6] | Ait-Haddou, R.; Mazure, M.-L.; Render, H., Nested sequences of rational spaces: Bernstein approximation, dimension elevation, and Pólya-type theorems on positive polynomials, J. Approx. Theory, 234, 20-50, 2018 · Zbl 1396.41010 |
| [7] | Ait-Haddou, R.; Mazure, M-L.; Ruhland, H., A remarkable Wronskian with application to critical lengths of cycloidal spaces, Calcolo, 56, 2019 · Zbl 1431.41001 |
| [8] | Ait-Haddou, R.; Sakane, Y.; Nomura, T., Chebyshev blossoming in Müntz spaces: toward shaping with young diagrams, J. Comput. Appl. Math., 247, 172-208, 2013 · Zbl 1291.65051 |
| [9] | Ait-Haddou, R.; Sakane, Y.; Nomura, T., Gelfond-Bézier curves, Comput. Aided Geom. Des., 30, 199-225, 2013 · Zbl 1278.65012 |
| [10] | Ait-Haddou, R.; Sakane, Y.; Nomura, T., A Müntz type theorem for a family of corner cutting schemes, Comput. Aided Geom. Des., 30, 240-253, 2013 · Zbl 1263.65015 |
| [11] | Aldaz, J. M.; Kounchev, O.; Render, H., Shape preserving properties of generalized Bernstein operators on extended Chebyshev spaces, Numer. Math., 114, 1-25, 2009 · Zbl 1184.41011 |
| [12] | Aldaz, J. M.; Kounchev, O.; Render, H., Bernstein operators for extended Chebyshev systems, Appl. Math. Comput., 217, 790-800, 2010 · Zbl 1204.41016 |
| [13] | Baker, B. M.; Handelman, D. E., Positive polynomials and time dependent integer-valued random variables, Can. J. Math., 44, 3-41, 1992 · Zbl 0774.60032 |
| [14] | Barry, P. J., De boor-fix dual functionals and algorithms for tchebycheffian B-splines curves, Constr. Approx., 12, 385-408, 1996 · Zbl 0854.41010 |
| [15] | Beccari, C. V.; Casciola, G.; Mazure, M-L., Piecewise extended Chebyshev spaces: a numerical test for design, Appl. Math. Comput., 296, 239-256, 2017 · Zbl 1411.65023 |
| [16] | Beccari, C. V.; Casciola, G.; Mazure, M-L., Critical length: an alternative approach, J. Appl. Math., 370, Article 112603 pp., 2020 · Zbl 1493.65024 |
| [17] | Beccari, C. V.; Casciola, G.; Mazure, M-L., Dimension elevation is not always corner-cutting, Appl. Math. Lett., 109, Article 106529 pp., 2020 · Zbl 1450.65011 |
| [18] | Beccari, C. V.; Casciola, G.; Mazure, M-L., Design or not design? A numerical characterisation for piecewise chebyshevian splines, Numer. Algorithms, 81, 1, 1-31, 2019 · Zbl 1415.65029 |
| [19] | Bernstein, S., Sur la représentation des polynômes positifs, Comm. Soc. Math. Kharkow, 14, 227-228, 1915 · JFM 48.1371.04 |
| [20] | Boehm, W.; Müller, A., On de Casteljau’s algorithm, Comput. Aided Geom. Des., 16, 587-605, 1999 · Zbl 0997.65017 |
| [21] | Bosner, T., Knot insertion algorithms for Chebyshev splines, 2006, Dept of Maths, University of Zagreb, PhD thesis |
| [22] | Borwein, P.; Erdélyi, T., Polynomials and Polynomial Inequalities, 1995, Wiley Interscience: Wiley Interscience N.Y. · Zbl 0840.26002 |
| [23] | Brilleaud, M.; Mazure, M.-L., Mixed hyperbolic/trigonometric spaces for design, Comput. Math. Appl., 64, 2459-2477, 2012 · Zbl 1268.65025 |
| [24] | Carnicer, J.-M.; Peña, J.-M., Total positivity and optimal bases, (Gasca, M.; Micchelli, C. A., Kluwer Ac. Kluwer Ac, Total Positivity and Its Applications, Math. Appl., vol. 359, 1996, Pub: Pub Dordrecht), 133-155 · Zbl 0892.15002 |
| [25] | Carnicer, J.-M.; Mainar, E.; Peña, J.-M., Critical length for design purposes and extended Chebyshev spaces, Constr. Approx., 20, 55-71, 2004 · Zbl 1061.41004 |
| [26] | Carnicer, J.-M.; Mainar, E.; Peña, J.-M., Critical length of cycloidal spaces are zeros of Bessel functions, Calcolo, 54, 1521-1531, 2017 · Zbl 1382.41001 |
| [27] | Costantini, P., Curve and surface construction using variable degree polynomial splines, Comput. Aided Geom. Des., 17, 41-446, 2000 · Zbl 0938.68128 |
| [28] | Costantini, P.; Lyche, T.; Manni, C., On a class of weak Tchebycheff systems, Numer. Math., 101, 333-354, 2004 · Zbl 1085.41002 |
| [29] | Erdélyi, T., The uniform closure of non-dense rational spaces on the unit interval, J. Approx. Theory, 131, 149-156, 2004 · Zbl 1057.41007 |
| [30] | de Faget de Casteljau, P., Outillage Méthodes Calcul, Enveloppe Soleau 40.040, 1959, Institute National de la Propriété Industrielle: Institute National de la Propriété Industrielle Paris |
| [31] | de Faget Casteljau, P., Courbes et Surfaces à Pôles, Microfiche P 4147-1, André Citroën Automobile, 1-58, 1963, SA: SA Paris |
| [32] | de Faget Casteljau, P., Formes à Pôles, Mathématiques et CAO, vol. 2, 1985, Editions Hermès: Editions Hermès Paris |
| [33] | Farin, G., Algorithms for rational Bézier curves, Comput. Aided Des., 15, 73-77, 1983 |
| [34] | Farin, G., Curves and Surfaces for Computer-Aided Geometric Design: A Practical Guide, 1988, Ac. Press · Zbl 0694.68004 |
| [35] | Fiorot, J.-C.; Jeannin, P., Rational Curves and Surfaces: Applications to CAD, 1992, John Wiley & Sons, Inc.: John Wiley & Sons, Inc. New York, NY · Zbl 0809.68114 |
| [36] | Farouki, R., The Bernstein polynomial basis: a centennial retrospective, Comput. Aided Geom. Des., 29, 379-419, 2012 · Zbl 1252.65039 |
| [37] | Goodman, T. N.T., Total positivity and the shape of curves, (Gasca, M.; Micchelli, C. A., Kluwer Ac. Kluwer Ac, Total Positivity and Its Applications, Math. Appl., vol. 359, 1996, Pub: Pub Dordrecht), 157-186 · Zbl 0894.68159 |
| [38] | Goodman, T. N.T.; Micchelli, C. A., Corner cutting algorithms for the Bézier representation of free form curves, Linear Algebra Appl., 99, 225-252, 1988 · Zbl 0652.41003 |
| [39] | Goodman, T. N.T.; Mazure, M.-L., Blossoming beyond extended Chebyshev spaces, J. Approx. Theory, 109, 48-81, 2001 · Zbl 0996.41005 |
| [40] | Kaklis, P. D.; Pandelis, D. G., Convexity preserving polynomial splines of non-uniform degree, IMA J. Numer. Anal., 10, 223-234, 1990 · Zbl 0699.65007 |
| [41] | Karlin, S. J.; Studden, W. J., Tchebycheff Systems: with Applications in Analysis and Statistics, 1966, Wiley Interscience: Wiley Interscience N.Y. · Zbl 0153.38902 |
| [42] | Koch, P. E.; Lyche, T., Construction of exponential tension B-splines of arbitrary order, (Laurent, P.-J.; Le Méhauté, A.; Schumaker, L. L., Curves and Surfaces, 1991, Acad. Press: Acad. Press Boston), 255-258 · Zbl 0736.41013 |
| [43] | Korovkin, P. P., Linear Operators and Approximation Theory, 1960, Hindustan Publ. Corp. |
| [44] | Laurent, P.-J.; Mazure, M.-L., Nested sequences of Chebyshev spaces, Math. Model. Numer. Anal., 32, 773-788, 1998 · Zbl 0922.65010 |
| [45] | Laurent, P.-J.; Mazure, M.-L., Piecewise smooth spaces in duality: application to blossoming, J. Approx. Theory, 98, 316-353, 1999 · Zbl 0952.41010 |
| [46] | Lyche, T., A recurrence relation for Chebyshevian B-splines, Constr. Approx., 1, 155-173, 1985 · Zbl 0583.41011 |
| [47] | Mazure, M.-L., Vandermonde type determinants and blossoming, Adv. Comput. Math., 8, 291-315, 1998 · Zbl 0906.65014 |
| [48] | Mazure, M.-L., Blossoming: a geometrical approach, Constr. Approx., 15, 33-68, 1999 · Zbl 0924.65010 |
| [49] | Mazure, M.-L., Chebyshev-Bernstein bases, Comput. Aided Geom. Des., 16, 649-669, 1999 · Zbl 0997.65022 |
| [50] | Mazure, M.-L., Chebyshev spaces with polynomial blossoms, Adv. Comput. Math., 10, 219-238, 1999 · Zbl 0933.65016 |
| [51] | Mazure, M.-L., Bernstein bases in Müntz spaces, Numer. Algorithms, 22, 285-304, 1999 · Zbl 0952.65024 |
| [52] | Mazure, M.-L., Four properties to characterize Chebyshev blossoms, Constr. Approx., 17, 319-333, 2001 · Zbl 0987.65017 |
| [53] | Mazure, M.-L., Blossoms and optimal bases, Adv. Comput. Math., 20, 177-203, 2004 · Zbl 1042.65016 |
| [54] | Mazure, M.-L., Chebyshev spaces and Bernstein bases, Constr. Approx., 22, 347-363, 2005 · Zbl 1116.41006 |
| [55] | Mazure, M.-L., Ready-to-blossom bases in Chebyshev spaces, (Jetter, K.; Buhmann, M.; Haussmann, W.; Schaback, R.; Stoeckler, J., Topics in Multivariate Approximation and Interpolation, 2006, Elsevier), 109-148 · Zbl 1205.65095 |
| [56] | Mazure, M.-L., Extended Chebyshev Piecewise spaces characterised via weight functions, J. Approx. Theory, 145, 33-54, 2007 · Zbl 1110.41002 |
| [57] | Mazure, M.-L., Bernstein-type operators in Chebyshev spaces, Numer. Algorithms, 52, 93-128, 2009 · Zbl 1200.41023 |
| [58] | Mazure, M.-L., Dimension elevation formula for Chebyshevian blossoms, Comput. Aided Geom. Des., 26, 1016-1027, 2009 · Zbl 1205.41030 |
| [59] | Mazure, M.-L., On a general new class of quasi-Chebyshevian splines, Numer. Algorithms, 58, 3, 399-438, 2011 · Zbl 1232.41008 |
| [60] | Mazure, M.-L., Finding all systems of weight functions associated with a given extended Chebyshev space, J. Approx. Theory, 163, 363-376, 2011 · Zbl 1217.41030 |
| [61] | Mazure, M.-L., How to build all Chebyshevian spline spaces good for geometric design?, Numer. Math., 119, 517-556, 2011 · Zbl 1230.65023 |
| [62] | Mazure, M.-L., Extended Chebyshev spaces in rationality, BIT Numer. Math., 53, 1013-1045, 2013 · Zbl 1287.65010 |
| [63] | Mazure, M.-L., Polynomial spaces revisited via weight functions, Jaen J. Approx., 6, 167-198, 2014 · Zbl 1379.41028 |
| [64] | Mazure, M.-L., Design with Quasi extended Chebyshev Piecewise spaces, Comput. Aided Geom. Des., 47, 3-28, 2016 · Zbl 1418.41009 |
| [65] | Melkémi, K., Orthogonalité des B-splines de Chebyshev Cardinales dans Un Espace de Sobolev Pondéré, 1999, Doctorat de Luniversité Joseph Fourier: Doctorat de Luniversité Joseph Fourier Grenoble |
| [66] | Mühlbach, G., A recurrence formula for generalized divided differences and some applications, J. Approx. Theory, 9, 165-172, 1973 · Zbl 0273.41026 |
| [67] | Piegl, L., A geometric investigation of the rational Bézier scheme of computer aided design, Comput. Ind., 7, 401-410, 1986 |
| [68] | Pinkus, A., Density in approximation theory, Surv. Approx. Theory, 1, 1-45, 2005 · Zbl 1067.41003 |
| [69] | Pólya, G., Über Positive Darstellung Von Polynomen, Vierteljschr. Naturforsch. Ges. Zürich, vol. 73, 141-145, 1928, MIT Press, pp. 309-313. In Collected Papers 2 (1974) · JFM 54.0138.01 |
| [70] | Pottmann, H., The geometry of Tchebycheffian splines, Comput. Aided Geom. Des., 10, 181-210, 1993 · Zbl 0777.41016 |
| [71] | Ramshaw, L., Blossoming: A Connect-the-dots Approach to Splines Digital Systems Research, 1987, Center |
| [72] | Ramshaw, L., Blossoms are polar forms, Comput. Aided Geom. Des., 6, 323-358, 1989 · Zbl 0705.65008 |
| [73] | Róth, A., Algorithm 992: an OpenGL and C++ based function library for curve and surface modeling in a large class of extended Chebyshev spaces, ACM Trans. Math. Softw., 2019 · Zbl 1471.65015 |
| [74] | Schumaker, L. L., Spline Functions, 1981, Wiley Interscience: Wiley Interscience N.Y. · Zbl 0449.41004 |
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