ALGORITHMS AND IDENTITIES FOR BIVARIATE $(h_1, h_2)$-BLOSSOMING

I. Jegdi\'c
2017 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
more » ... ced and studied by , while the h-analogous were studied by Stancu in [16, 17] , Goldman in [1, 2] , and Goldman and Barry in [3] . 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 [11]-[13] by Simeonov and Goldman). Some of these properties, algorithms and identities were also derived using the standard mathematical induction and other elementary techniques in [4] 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 [15] to define qblossoming for polynomials in two variables leading to the study of q-Bézier surfaces in [5] . 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 [14] to the case of polynomials of two variables and to study (h 1 , h 2 )-Bézier surfaces.
doi:10.12732/ijam.v30i4.5 fatcat:z42avurjhre2phowrbbpp3weni