Computing orthogonal polynomials on a triangle by degree raising

Shayne Waldron


We give an algorithm for computing orthogonal polynomials over triangular domains in Bernstein--B\'ezier 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.

Keywords: Bernstein--B\'ezier form, Hahn polynomials, Jacobi polynomials, surface smoothing

Math Review Classification: Primary 33C45, 65D17; Secondary 41A10

Length: 9 pages

Last Updated: 20 April 2006