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Blossoming stories

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Abstract

We review the main properties of blossoms along with their important repercussions in all aspects of geometric design. Not only are they an elegant and efficient tool to express all classical algorithms, but they are also a fundamental concept, as proven by the fact that their existence is equivalent to the existence of B-spline bases.

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References

  1. P.J. Barry, de Boor–Fix dual functionals and algorithms for Tchebycheffian B-splines curves, Construct. Approx. 12 (1996) 385–408.

    Google Scholar 

  2. J.-M. Carnicer and J.-M. Peña, Total positivity and optimal bases, in: Total Positivity and its Applications, eds. M. Gasca and C.A. Micchelli (Kluwer Academic, Dordrecht, 1996) pp. 133–155.

    Google Scholar 

  3. P. Costantini, Curve and surface construction using variable degree polynomial splines, Computer Aided Geom. Design 17 (2000) 419–446.

    Google Scholar 

  4. P. de Faget de Casteljau, Courbes et surfaces à pôles (Enveloppe 40040), Institut National de la Propriété Industrielle, Paris (1959).

  5. P. de Faget de Casteljau, Formes à Pôles (Hermes, 1985).

  6. N. Dyn and A. Ron, Recurrence relations for Tchebycheffian B-splines, J. Anal. Math. 51 (1988) 118–138.

    Google Scholar 

  7. N. Dyn and A. Ron, Cardinal translation invariant Tchebycheffian B-splines, Approx. Theory Appl. 6 (1990) 1–12.

    Google Scholar 

  8. H. Gonska and A. Lupaş, On an algorithm for Bernstein polynomials, in: Curve and Surface Design, Saint-Malo (2002), eds. T. Lyche, M.-L. Mazure and L.L. Schumaker (Nahsboro Press, Brentwood, TN, 2003) pp. 197–203.

    Google Scholar 

  9. T.N.T. Goodman Total positivity and the shape of curves, in: Total Positivity and its Applications, eds. M. Gasca and C.A. Micchelli (Kluwer Academic, Dordrecht, 1996) pp. 133–155.

    Google Scholar 

  10. T.N.T. Goodman and M.-L. Mazure, Blossoming beyond extended Chebyshev spaces, J. Approx. Theory 109 (2001) 48–81.

    Google Scholar 

  11. S. Karlin, Total Positivity (Stanford Univ. Press, Stanford, 1968).

    Google Scholar 

  12. S. Karlin and W.J. Studden, Tchebycheff Systems (Wiley Interscience, New York, 1966).

    Google Scholar 

  13. S. Karlin and Z. Ziegler, Chebyshevian spline functions, SIAM J. Numer. Anal. 3 (1966) 514–543.

    Google Scholar 

  14. P.E. Koch and T. Lyche, Construction of exponential tension B-splines of arbitrary order, in: Curves and Surfaces, eds. P.-J. Laurent, A. Le Méhauté and L.L. Schumaker (Academic Press, Boston, 1991) pp. 255–258.

    Google Scholar 

  15. E. Laguerre, Sur le rôle des émanants dans la théorie des équations numériques, Comptes Rendus des Séances de l’Académie des Sciences LXXVIII (1874); see also Oeuvres de Laguerre, Tome 1 (Chelsea, New York, 1972) pp. 48–50.

  16. T. Lyche, A recurrence relation for Chebyshevian B-splines, Construct. Approx. 1 (1985) 155–173.

    Google Scholar 

  17. T. Lyche, Trigonometric splines; a survey with new results, in: Shape Preserving Representations in Computer-Aided Geometric Design, ed. J.-M. Peña (Nova Science, 1999) pp. 201–227.

  18. T. Lyche and R. Winther, A stable recurrence relation for trigonometric B-splines, J. Approx. Theory 25 (1979) 266–279.

    Google Scholar 

  19. M. Marden, The Geometry of the Zeros of a Polynomial in a Complex Variable (Amer. Math. Soc., Providence, RI, 1949).

    Google Scholar 

  20. M.-L. Mazure, Blossoming of Chebyshev splines, in: Mathematical Methods for Curves and Surfaces, eds. T. Lyche and L.L. Schumaker (Vanderbilt Univ. Press, Nashville, TN, 1995) pp. 355–364.

    Google Scholar 

  21. M.-L. Mazure, Blossoming: A geometrical approach, Construct. Approx. 15 (1999) 33–68.

    Google Scholar 

  22. M.-L. Mazure, Chebyshev–Bernstein bases, Computer Aided Geom. Design 16 (1999) 649–669.

    Google Scholar 

  23. M.-L. Mazure, Blossoming and CAGD algorithms, in: Shape Preserving Representations in Computer-Aided Geometric Design, ed. J.-M. Peña (Nova Science, 1999) pp. 99–117.

  24. M.-L. Mazure, Chebyhev spaces with polynomial blossoms, Adv. Comput. Math. 10 (1999) 219–238.

    Google Scholar 

  25. M.-L. Mazure, Four properties to characterize Chebyshev blossoms, Construct. Approx. 17 (2001) 319–333.

    Google Scholar 

  26. M.-L. Mazure, Quasi-Chebyshev splines with connexion matrices. Application to variable degree polynomial splines, Computer Aided Geom. Design 18 (2001) 287–298.

    Google Scholar 

  27. M.-L. Mazure, B-spline bases and osculating flats: One result of H.-P. Seidel revisited, Math. Modelling Numer. Anal. 36 (2002) 1177–1186.

    Google Scholar 

  28. M.-L. Mazure, Blossoms and optimal bases, Adv. Comput. Math. 20 (2004) 177–203.

    Google Scholar 

  29. M.-L. Mazure, On the equivalence between existence of B-spline bases and existence of blossoms, Construct. Approx. 20 (2004) 603–624.

    Google Scholar 

  30. M.-L. Mazure and P.J. Laurent, Affine and non-affine blossoms, in: Computational Geometry, eds. A. Conte, V. Demichelis, F. Fontanella and I. Galligani (World Scientific, Singapore, 1993) pp. 201–230.

    Google Scholar 

  31. M.-L. Mazure and P.J. Laurent, Marsden identities, blossoming and de Boor–Fix formula, in: Advanced Topics in Multivariate Approximation (World Scientific, Singapore, 1996) pp. 227–242.

    Google Scholar 

  32. M.-L. Mazure and P.J. Laurent, Piecewise smooth spaces in duality: Application to blossoming, J. Approx. Theory 98 (1999) 316–353.

    Google Scholar 

  33. M.-L. Mazure and P.-J. Laurent, Polynomial Chebyshev splines, Computer Aided Geom. Design 16 (1999) 317–343.

    Google Scholar 

  34. M.-L. Mazure and H. Pottmann, Tchebycheff curves, in: Total Positivity and its Applications, eds. M. Gasca and C.A. Micchelli (Kluwer Academic, Dordrecht, 1996) pp. 187–218.

    Google Scholar 

  35. C.A. Micchelli, Cardinal ℒ-splines, in: Studies in Spline Functions and Approximation Theory (Academic Press, New York, 1976) pp. 203–250.

    Google Scholar 

  36. J.-M. Peña, Shape preserving representations for trigonometric polynomial curves, Computer Aided Geom. Design 14 (1997) 5–11.

    Google Scholar 

  37. G. Pólya and G. Szegö, Aufgaben und Lehrsätze aus der Analysis, Vol. II (Berlin, 1925).

  38. T. Popoviciu, Despre cea mai bun \v{a} aproximaţie a funcţiilor continue prin polinoame, Institutul de Arte Grafice Ardealul, Cluj (1937).

  39. H. Pottmann, The geometry of Tchebycheffian splines, Computer Aided Geom. Design 10 (1993) 181–210.

    Google Scholar 

  40. H. Pottmann, A geometric approach to variation diminishing free-form curve schemes, in: Shape Preserving Representations in Computer-Aided Geometric Design, ed. J.-M. Peña (Nova Science, 1999) pp. 119–131.

  41. H. Pottmann and M.G. Wagner, Helix splines as an example of affine Tchebycheffian splines, Adv. Comput. Math. 2 (1994) 123–142.

    Google Scholar 

  42. L. Ramshaw, Blossoming: A connect-the-dots approach to splines, Research Report 19, Compaq Systems Research Center, Palo Alto, CA (June 1987).

  43. L. Ramshaw, Blossoms are polar forms, Computer Aided Geom. Design 6 (1989) 323–358.

    Google Scholar 

  44. I.J. Schoenberg, On trigonometric spline interpolation, J. Math. Mech. 13 (1964) 795–826.

    Google Scholar 

  45. L.L. Schumaker, Spline Functions (Wiley Interscience, New York, 1981).

    Google Scholar 

  46. D. Schweikert, An interpolation curve using a spline in tension, J. Math. Phys. 45 (1966) 312–317.

    Google Scholar 

  47. H.-P. Seidel, New algorithms and techniques for computing with geometrically continuous spline curves of arbitrary degree, Math. Modelling Numer. Anal. 26 (1992) 149–176.

    Google Scholar 

  48. H.-P. Seidel, Polar forms for geometrically continuous spline curves of arbitrary degree, ACM Trans. Graphics 12 (1993) 1–34.

    Google Scholar 

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Correspondence to Marie-Laurence Mazure.

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Mazure, ML. Blossoming stories. Numer Algor 39, 257–288 (2005). https://doi.org/10.1007/s11075-004-3642-9

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  • DOI: https://doi.org/10.1007/s11075-004-3642-9

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