ALGORITHMS AND IDENTITIES FOR BIVARIATE $(h_1, h_2)$-BLOSSOMING
International Journal of Applied Mathematics
We extend the definition of h-blossoming for polynomials in one variable to the polynomials in two variables, and we use this bivariate (h 1 , h 2 )blossoming to study various properties, identities, and algorithms associated with (h 1 , h 2 )-Bézier surfaces. We construct a recursive (h 1 , h 2 )-midpoint subdivision algorithm for the (h 1 , h 2 )-Bézier surfaces and we prove its geometric rate of convergence. The quantum q-analogues of Bernstein basis functions were introduced and studied by
... ced and studied by , while the h-analogous were studied by Stancu in [16, 17] , Goldman in [1, 2] , and Goldman and Barry in  . The theory of quantum q-and h-Bézier curves, in the context of quantum q-and h-blossoming for polynomials in one variable, was introduced by Simeonov et al. in [14, 15] . The importance and usefulness of quantum q-and h-blossoming is the quantum blossoming representation of quantum q-and h-Bézier curves, surfaces, and splines, which gives efficient algorithms for recursive evaluation, degree evaluation, subdivision, and other properties (for example, see - by Simeonov and Goldman). Some of these properties, algorithms and identities were also derived using the standard mathematical induction and other elementary techniques in  by Jegdić, Larson, and Simeonov. Recently, Jegdić, Simeonov, and Zafiris used the tensor product and generalized concept of q-blossoming for polynomials in one variable introduced in  to define qblossoming for polynomials in two variables leading to the study of q-Bézier surfaces in  . The main goal of this paper is to extend the definition of h-blossoming for polynomials in one variable to (h 1 , h 2 )-blossoming for polynomials in two variables, and to use it to generalize identities and algorithms for h-Bernstein polynomials in one variable from  to the case of polynomials of two variables and to study (h 1 , h 2 )-Bézier surfaces.