Construction of orthogonal bases for polynomials in Bernstein form on triangular and simplex domains

A scheme for constructing orthogonal systems of bivariate polynomials in the Bernstein–Bézier form over triangular domains is formulated. The orthogonal basis functions have a hierarchical ordering by degree, facilitating computation of least-squares approximations of increasing degree (with permanence of coefficients) until the approximation error is subdued below a prescribed tolerance. The orthogonal polynomials reduce to the usual Legendre polynomials along one edge of the domain triangle, and within each fixed degree are characterized by vanishing Bernstein coefficients on successive rows parallel to that edge. Closed-form expressions and recursive algorithms for computing the Bernstein coefficients of these orthogonal bivariate polynomials are derived, and their application to surface smoothing problems is sketched. Finally, an extension of the scheme to the construction of orthogonal bases for polynomials over higher-dimensional simplexes is also presented.