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Design with L-splines

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Abstract

We recently obtained a criterion to decide whether a given space of parametrically continuous piecewise Chebyshevian splines (i.e., splines with pieces taken from different Extended Chebyshev spaces) could be used for geometric design. One important field of application is the class of L-splines, that is, splines with pieces taken from the null space of some fixed real linear differential operator, generally investigated under the strong requirement that the null space should be an Extended Chebyshev space on the support of each possible B-spline. In the present work, we want to show the practical interest of the criterion in question for designing with L-splines. With this in view, we apply it to a specific class of linear differential operators with real constant coefficients and odd/even characteristic polynomials. We will thus establish necessary and sufficient conditions for the associated splines to be suitable for design. Because our criterion was achieved via a blossoming approach, shape preservation will be inherent in the obtained conditions. One specific advantage of the class of operators we consider is that hyperbolic and trigonometric functions can be mixed within the null space on which the splines are based. We show that this produces interesting shape effects.

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References

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Bister, D., Prautzsch, H.: A new approach to Tchebycheffian B-splines Curves and Surfaces with Applications in CAGD, pp. 35–41. Vanderbilt University Press, Nashville (1997)

  3. Brilleaud, M., Mazure, M.-L.: Mixed hyperbolic/trigonometric spaces for design. Comput. Math. Appl. 64, 2459–2477 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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

  5. Carnicer, J.-M., Mainar, E., Peña, J.-M.: Critical length for design purposes and extended Chebyshev spaces. Constr. Approx. 20, 55–71 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Goodman, T.N.T.: Total positivity and the shape of curves. In: Gasca, M., Micchelli, C.A. (eds.) Total Positivity and Its Applications, pp 157–186. Kluwer Academic Publication, Dordrecht (1996)

  7. Karlin, S.J., Studden, W.J.: Tchebycheff Systems: With Applications in Analysis and Statistics. Wiley Interscience, N.Y. (1966)

    MATH  Google Scholar 

  8. Koch, P.E., Lyche, T.: Exponential B-splines in tension. In: Chui, C.K., Schumaker, L.L., Ward, J.D. (eds.) Approximation Theory VI, p 361364. Academic Press, NY (1990)

    Google Scholar 

  9. Lü, Y., Wang, G., Yang, X.: Uniform hyperbolic polynomial B-spline curves. Comput. Aided Geom. Des. 19, 379–393 (2002)

    Article  Google Scholar 

  10. Lyche, T., Schumaker, L.L.: Total positivity properties of LB-splines. In: Gasca, M., Micchelli, C. (eds.) Total Positivity and Its Applications, pp. 35–46. Klüwer, Dordrecht (1996)

    Chapter  Google Scholar 

  11. Mainar, E., Peña, J.-M.: Quadratic-cycloidal curves. Adv. Comput. Math. 20, 161–175 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Mainar, E., Peña, J.-M.: A general class of, Bernstein-like bases. Comput. Math. Appl 53, 1686–1703 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Mainar, E., Peña, J.-M.: Optimal bases for a class of mixed spaces and their associated spline spaces. Comput. Math. Appl. 59, 1509–1523 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. Mazure, M.-L.: Chebyshev splines beyond total positivity. Adv. Comput. Math. 14, 129–156 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  15. Mazure, M.-L.: Blossoms and optimal bases. Adv. Comp. Math. 20, 177–203 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  17. Mazure, M.-L.: Chebyshev spaces and Bernstein bases. Constr. Approx. 22, 347–363 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. Mazure, M.-L.: Ready-to-blossom bases in Chebyshev spaces. In: Jetter, K., Buhmann, M., Haussmann, W., Schaback, R., Stoeckler, J. (eds.) Topics in Multivariate Approximation and Interpolation, pp. 109–148. Elsevier (2006)

  19. Mazure, M.-L.: Ready-to-blossom bases and the existence of geometrically continuous piecewise Chebyshevian B-splines. CRAS 347, 829–834 (2009)

    MATH  MathSciNet  Google Scholar 

  20. Mazure, M.-L.: On differentiation formulæ for Chebyshevian Bernstein and B-spline bases. Jaén J. Approx. 1, 111–143 (2009)

    MATH  MathSciNet  Google Scholar 

  21. Mazure, M.-L.: Finding all systems of weight functions associated with a given Extended Chebyshev space. J. Approx. Theory 163, 363–376 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  22. Mazure, M.-L.: How to build all Chebyshevian spline spaces good for Geometric Design. Numer. Math. 119, 517–556 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  23. Mazure, M.-L.: On a new criterion to decide whether a spline space can be used for design. BIT Num. Math. 52, 1009–1034 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  24. Mazure, M.-L.: From Taylor interpolation to Hermite interpolation via duality. Jaén J. Approx. 4(1), 15–45 (2012)

    Google Scholar 

  25. Peña, J.-M.: Bases with optimal shape preserving properties. In: Peña, J.M. (ed.) Shape Preserving Representations in Computer-Aided Geometric Design, pp. 63–84. Nova Sc. Pub. (1999)

  26. Pottmann, H.: The geometry of Tchebycheffian splines. Comput. Aided Geom. Des. 10, 181–210 (1993)

    Article  MATH  MathSciNet  Google Scholar 

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

  28. Schumaker, L.L.: Spline Functions. Wiley, N.Y. (1981)

    MATH  Google Scholar 

  29. Wang, G., Li, Y.: Optimal properties of the uniform algebraic trigonometric B-splines. Comput. Aided Geom. Des. 13, 226–238 (2006)

    Article  Google Scholar 

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Authors and Affiliations

  1. Laboratoire Jean Kuntzmann, Université Joseph Fourier, BP53, 38041, Grenoble Cedex 9, France

    Martine Brilleaud & Marie-Laurence Mazure

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  1. Martine Brilleaud
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  2. Marie-Laurence Mazure
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Correspondence to Marie-Laurence Mazure.

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Brilleaud, M., Mazure, ML. Design with L-splines. Numer Algor 65, 91–124 (2014). https://doi.org/10.1007/s11075-013-9697-8

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  • DOI: https://doi.org/10.1007/s11075-013-9697-8

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