# Computing orthogonal polynomials on a triangle by degree raising

## Shayne Waldron

## Abstract:

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

## Availability: