Bèzier curves based on Lupaş (p,q)-analogue of Bernstein functions in CAGD

摘要

This paper deals with the extension of rational Lupaş Bernstein functions, Lupaş Bèzier curves and surfaces involving (p,q)-integers as shape parameters for all p>0 and q>0. Two different techniques such as de-Casteljau’s algorithm and Korovkin’s type approximation based on (p,q)-integers are used. A two parameter family for Lupaş (p,q)-Bernstein functions is constructed and their degree elevation and reduction properties have been studied. For Lupaş (p,q)-Bèzier curves, some of their basic properties as well as degree elevation and de Casteljau algorithm have been discussed. The new curves have some properties similar to Lupaş q-Bèzier curves. Similarly, the corresponding tensor product for Lupaş Bèzier surfaces over the rectangular domain (u,v)∈[0,1]×[0,1] depending on four parameters is constructed. The de Casteljau algorithm and degree evaluation properties of the surfaces for these generalizations and some fundamental properties are discussed. We get Lupaş q-Bèzier surfaces for (u,v)∈[0,1]×[0,1] when we set the parameter p1=p2=1.With the help of rational Lupaş Bernstein functions, (p,q)-Lupaş Bernstein operators are constructed. Based on Korovkin’s type approximation, it has been shown that the sequence of (p,q)-analogue of Lupaş Bernstein operators Lpn,qnn(f,x) converges uniformly to f(x)∈C[0,1] if and only if 00 fixed and p≠1, the sequence Lp,qn(f,x) converges uniformly to f(x)∈C[0,1] if and only if f(x)=ax+b for some a,b∈R.Furthermore, in comparison to q-Bèzier curves and surfaces based on Lupaş q-Bernstein rational functions, our generalization gives us more flexibility in controlling the shapes of curves and surfaces.