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

In this paper, we use the blending functions of Lupa\c{s} type (rational) $(p,q)$-Bernstein operators based on $(p,q)$-integers for construction of Lupa\c{s} $(p,q)$-B$\acute{e}$zier curves (rational curves) and surfaces (rational surfaces) with shape parameters. We study the nature of degree elevation and degree reduction for Lupa\c{s} $(p,q)$-B$\acute{e}$zier Bernstein functions. Parametric curves are represented using Lupa\c{s} $(p,q)$-Bernstein basis. We introduce affine de Casteljau algorithm for Lupa\c{s} type $(p,q)$-Bernstein B$\acute{e}$zier curves. The new curves have some properties similar to $q$-B$\acute{e}$zier 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 Lupa\c{s} type $(p,q)$-Bernstein B$\acute{e}$zier curves and surfaces are discussed. We get $q$-B$\acute{e}$zier surfaces for $(u, v) \in [0, 1] \times [0, 1] $ when we set the parameter $p_1=p_2=1.$ In comparison to $q$-B$\acute{e}$zier curves and surfaces based on Lupa\c{s} $q$-Bernstein polynomials, our generalization gives us more flexibility in controlling the shapes of curves and surfaces. We also show that the $(p,q)$-analogue of Lupa\c{s} Bernstein operator sequence $L^{n}_{p_n,q_n}(f,x)$ converges uniformly to $f(x)\in C[0,1]$ if and only if $0<q_n<p_n\leq1$ such that $\lim\limits_{n\to\infty} q_n=1$ and $\lim\limits_{n\to\infty} p_n=1.$ On the other hand, for any $p>0$ fixed and $p \neq 1,$ the sequence $L^{n}_{p,q}(f,x)$ converges uniformly to $f(x)~ \in C[0,1]$ if and only if $f(x)=ax+b$ for some $a, b \in \mathbb{R}.$