Numerical Algorithms

, Volume 42, Issue 2, pp 171–179 | Cite as

Computing orthogonal polynomials on a triangle by degree raising

  • Shayne Waldron


We give an algorithm for computing orthogonal polynomials over triangular domains in Bernstein–Bézier form which uses only the operator of degree raising and its adjoint. This completely avoids the need to choose an orthogonal basis (or tight frame) for the orthogonal polynomials of a given degree, and hence the difficulties inherent in that approach. The results are valid for Jacobi polynomials on a simplex, and show the close relationship between the Bernstein form of Jacobi polynomials, Hahn polynomials and degree raising.


Bernstein–Bézier form Hahn polynomials Jacobi polynomials surface smoothing 

AMS (MOS) Subject Classifications

primary 33C45 65D17 secondary 41A10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    de Boor, C.: \(B\)-form basics. In: Farin, G.E. (ed.) Geometric Modeling: Algorithms and New Trends, pp. 131–148. SIAM, Philadelphia, Pennsylvania (1987)Google Scholar
  2. [2]
    Dunkl, C.F., Xu, Y.: Orthogonal Polynomials of Several Variables. Cambridge University Press, Cambridge (2001)MATHGoogle Scholar
  3. [3]
    Farouki, R.T.: Legendre–Bernstein basis transformations. J. Assoc. Comput. Mach. 119(1–2), 145–160 (2000)MATHMathSciNetGoogle Scholar
  4. [4]
    Farouki, R.T., Goodman, T.N.T., Sauer, T.: Construction of orthogonal bases for polynomials in Bernstein form on triangular and simplex domains. Comput. Aided Geom. Design 20(4), 209–230 (2003)MATHMathSciNetGoogle Scholar
  5. [5]
    Kim, H.J., Ahn, Y.J.: Good degree reduction of Bézier curves using Jacobi polynomials. Comput. Math. Appl. 40, 1205–1215 (2000)MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Waldron, S.: On the Bernstein–Bézier form of Jacobi polynomials on a simplex. J. Approx. Theory 140(1), 86–99 (2006)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of AucklandAucklandNew Zealand

Personalised recommendations