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

In this paper, a de Casteljau algorithm to compute (p; q)-Bernstein Bezier curves based on (p; q)- integers are introduced. The nature of degree elevation and degree reduction for (p; q)-Bezier Bernstein functions are studied. The new curves have some properties similar to q-Bezier curves. Moreover, we construct the corresponding tensor product surfaces over the rectangular domain (u; v) in [0; 1] x [0; 1] depending on four parameters. De Casteljau algorithm and degree evaluation properties of the surfaces for these generalization over the rectangular domain are investigated. Furthermore, some fundamental properties for (p; q)-Bernstein Bezier curves are discussed.We get q-Bezier curves and surfaces for (u; v) in [0; 1] x [0; 1] when we set the parameter p1 = p2 = 1: In comparison to q-Bezier curves based on q-Bernstein polynomials, this generalization gives us more flexibility in controlling the shapes of curves