B$\acute{e}$zier curves based on Lupaş $(p,q)$-analogue of Bernstein polynomials in CAGD

Abstract

In this paper, we use the blending functions of Lupaş type (rational) $(p,q)$-Bernstein operators based on $(p,q)$-integers for construction of Lupaş $(p,q)$-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 Lupaş $(p,q)$-B$\acute{e}$zier Bernstein functions. Parametric curves are represented using Lupaş $(p,q)$-Bernstein basis. We introduce affine de Casteljau algorithm for Lupaş 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. 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 Lupaş $q$-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 Lupaş Bernstein operator sequence $L_{p_n,q_n}^n(f,x)$ converges uniformly to $f(x)\in C[0,1]$ if and only if $0<q_n<p_n\le 1$ such that $\underset{n\to \infty }{\lim }{q}_n=1,$ $\underset{n\to \infty }{\lim }{p}_n=1$ and $\underset{n\to \infty }{\lim }{p}_n^n=a,$ $\underset{n\to \infty }{\lim }{q}_n^n=b$ with $0<a,b\le 1.$ On the other hand, for any $p>0$ fixed and $p\ne 1,$ the sequence $L_{p,q}^n(f,x)$ converges uniformly to $f(x)\in C[0,1]$ if and only if $f(x)=ax+b$ for some $a,b\in \mathbb{R}.$

arXiv comment24 pages, 9 figures, $(p,q)$-lupas operator and limit $(p,q)$-lupas operators and their property introduced, typo corrected