Bezier curves based on Lupas (p,q)-analogue of Bernstein polynomials in CAGD

In this paper, we use the blending functions of Lupas type (rational) (p,q)-Bernstein operators based on (p,q)-integers for construction of Lupas (p,q)-Beezier curves (rational curves) and surfaces (rational surfaces) with two shape parameters. We study the nature of degree elevation and degree reduction for Lupas (p,q)-Bezier Bernstein functions. Parametric curves are represented using Lupas (p,q)-Bernstein basis and the concept of total positivity is applied to investigate the shape properties of the curve. We get q-B?ezier curve when we set the parameter p to the value 1: We also introduce a de Casteljau algorithm for Lupas type (p,q)-Bernstein Bezier curves. 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] \times [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 Lupas type (p,q)-Bernstein Bezier curves are discussed. We get q-Bezier curves and surfaces for (u,v) \in [0,1] \times [0,1] when we set the parameter p1 = p2 = 1. In Comparison to q-Bezier curves and surfaces based on Phillips q-Bernstein polynomials, our generalizations show more flexibility in choosing the value of p1; p2 and q1; q2 and superiority in shape control of curves and surfaces. The shape parameters provide more convenience for the curve and surface modeling.