Geometric applications of bivariate q-bernstein and q-orthogonal polynomials

A q-version of a Bernstein basis of bivariate polynomials and a system of q-orthogonal bivariate polynomials over triangular domains are introduced to construct q-versions of the de Casteljau and the degree elevation algorithms. The polynomials are similar to the polynomials introduced recently by Farouki, Goodman, and Sauer [2]. The orthogonal polynomials are defined as products of scaled q-Legendre polynomials and Jacobi polynomials, a construction introduced by Xu [4]. The double integration is continuous in one variable and discrete in the other variable.