A de Casteljau Algorithm for Bernstein type Polynomials based on (p, q)-integers in CAGD

In this paper, a de Casteljau algorithm to compute (p, q)-Bernstein Bézier curves based on (p, q)integers is introduced. We study the nature of degree elevation and degree reduction for (p, q)-Bézier Bernstein functions. The new curves have some properties similar to q-Bézier curves. Moreover, we construct the corresponding tensor product surfaces over the rectangular domain (u, v) ∈ [0, 1]× [0, 1] depending on four parameters. We also study the de Casteljau algorithm and degree evaluation properties of the surfaces for these generalization over the rectangular domain. Furthermore, some fundamental properties for (p, q)-Bernstein Bézier curves are discussed. We get q-Bézier curves and surfaces for (u, v) ∈ [0, 1]× [0, 1] when we set the parameter p1 = p2 = 1.