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BIT Numerical Mathematics

, Volume 47, Issue 2, pp 313–323 | Cite as

A generalization of rational Bernstein–Bézier curves

  • Çetin Dişibüyük
  • Halil Oruç
Article

Abstract

This paper is concerned with a generalization of Bernstein–Bézier curves. A one parameter family of rational Bernstein–Bézier curves is introduced based on a de Casteljau type algorithm. A subdivision procedure is discussed, and matrix representation and degree elevation formulas are obtained. We also represent conic sections using rational q-Bernstein–Bézier curves.

Key words

q-Bernstein polynomials rational Bézier curves de Casteljau algorithm subdivision degree elevation 

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References

  1. 1.
    G. E. Andrews, The Theory of Partitions, Cambridge University Press, Cambridge, 1998.zbMATHGoogle Scholar
  2. 2.
    G. Farin, Curves and Surfaces for CAGD, a Practical Guide, 5th edn., Academic Press, San Diego USA, 2002.Google Scholar
  3. 3.
    T. N. T. Goodman, Total positivity and shape of curves, in Total Positivity and its Applications, M. Gasca and C. A. Micchelli eds., pp. 157–186, Kluwer Academic Publishers, Dordrecht, 1996.Google Scholar
  4. 4.
    T. N. T. Goodman, H. Oruç, and G. M. Phillips, Convexity and generalized Bernstein polynomials, Proc. Edinb. Math. Soc., 42 (1999), pp. 179–190.zbMATHCrossRefGoogle Scholar
  5. 5.
    J. Hoschek and D. Lasser, Fundamentals of Computer Aided Geometric Design, A.K. Peters, Wellesley, Mass., 1993.zbMATHGoogle Scholar
  6. 6.
    S. Lewanowicz and P. Woźny, Generalized Bernstein polynomials, BIT, 44 (2004), pp. 63–78.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    H. Oruç, LU factorization of the Vandermonde matrix and its applications, Appl. Math. Lett., to appear.Google Scholar
  8. 8.
    H. Oruç and H. K. Akmaz, Symmetric functions and the Vandermonde matrix, J. Comput. Appl. Math., 172 (2004), pp. 49–64.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    H. Oruç and G. M. Phillips, q-Bernstein polynomials and Bézier curves, J. Comput. Appl. Math., 151 (2003), pp. 1–12.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    H. Oruç and G. M. Phillips, A generalization of the Bernstein polynomials, Proc. Edinb. Math. Soc., 42 (1999), pp. 403–413.zbMATHGoogle Scholar
  11. 11.
    G. M. Phillips, A de Casteljau algorithm for generalized Bernstein polynomials, BIT, 36 (1996), pp. 232–236.Google Scholar
  12. 12.
    G. M. Phillips, Bernstein polynomials based on the q-integers, Ann. Numer. Math., 4 (1997), pp. 511–518.zbMATHMathSciNetGoogle Scholar
  13. 13.
    G. M. Phillips, Interpolation and Approximation by Polynomials, Springer, New York, 2003.zbMATHGoogle Scholar
  14. 14.
    H. Prautzsch and L. Kobbelt, Convergence of subdivision and degree elevation, Adv. Comput. Math., 2 (1994), pp. 143–154.CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of Mathematics, Fen Edebiyat FakültesiDokuz Eylül UniversityBuca, İzmirTurkey

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