Advertisement

Tensor Product q −Bernstein Bézier Patches

  • Çetin Dişibüyük
  • Halil Oruç
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5434)

Abstract

In this work we define a new de Casteljau type algorithm, which is in barycentric form, for the q −Bernstein Bézier curves. We express the intermediate points of the algorithm explicitly in two ways. Furthermore we define tensor product patches, based on this algorithm, depending on two parameters. Degree elevation procedure for the tensor product patch is studied. Finally, the matrix representation of tensor product patch is given and we find the transformation matrix between classical tensor product Bézier patch and tensor product q −Bernstein Bézier patch.

Keywords

Tensor Product Transformation Matrix Point Interpolation Bernstein Polynomial Intermediate Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Andrews, G.E.: The theory of partitions. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1998); Reprint of the 1976 originalzbMATHGoogle Scholar
  2. 2.
    Dişibüyük, Ç., Oruç, H.: A generalization of rational Bernstein-Bézier curves. BIT 47(2), 313–323 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Farin, G.: Curves and surfaces for computer-aided geometric design. In: Computer Science and Scientific Computing, 5th edn. Academic Press Inc., San Diego (2002)Google Scholar
  4. 4.
    Lewanowicz, S., Woźny, P.: Generalized Bernstein polynomials. BIT 44(1), 63–78 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Oruç, H., Phillips, G.M.: q-Bernstein polynomials and Bézier curves. J. Comput. Appl. Math. 151(1), 1–12 (2003)CrossRefzbMATHGoogle Scholar
  6. 6.
    Oruç, H., Tuncer, N.: On the convergence and iterates of q-Bernstein polynomials. J. Approx. Theory 117(2), 301–313 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Phillips, G.M.: Bernstein polynomials based on the q-integers. Ann. Numer. Math. 4(1-4), 511–518 (1997); The heritage of P. L. Chebyshev: a Festschrift in honor of the 70th birthday of T. J. RivlinMathSciNetzbMATHGoogle Scholar
  8. 8.
    Phillips, G.M.: A de Casteljau algorithm for generalized Bernstein polynomials. BIT 37(1), 232–236 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Çetin Dişibüyük
    • 1
  • Halil Oruç
    • 1
  1. 1.Department of MathematicsDokuz Eylül UniversityİzmirTurkey

Personalised recommendations