2015/05/07 by Khalid Khan, Khan, Khalid, D. K. Lobiyal +1 · 1 citation
Computer Science · Engineering · Mathematics · #41A10 #65D17 #Advanced Numerical Analysis Techniques #Approximation Theory and Sequence Spaces #FOS: Computer and information sciences #Graphics (cs.GR) #cs.GR #msc:41A10 #msc:65D17
paper · pdf · doi:10.48550/arxiv.1505.01810
24 pages, 9 figures, $(p,q)$-lupas operator and limit $(p,q)$-lupas operators and their property introduced, typo corrected
openalex publication_date 2015/05/07 · arxiv created 2016/06/11 · arxiv updated 2016/06/14 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
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\acuteezier 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\acuteezier 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\acuteezier curves. The new curves have some properties similar to q-B\acuteezier curves. Moreover, we construct the corresponding tensor product surfaces over the rectangular domain (u, v) ∈ [0, 1] × [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\acuteezier surfaces for (u, v) ∈ [0, 1] × [0, 1] when we set the parameter p1=p2=1. In comparison to q-B\acuteezier 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 Lnpn,qn(f,x) converges uniformly to f(x)∈ C[0,1] if and only if 0<qn<pn≤1 such that limn→∞ qn=1, limn→∞ pn=1 and limn→∞pnn=a, limn→∞qnn=b with 0<a,b≤1. On the other hand, for any p>0 fixed and p ≠ 1, the sequence Lnp,q(f,x) converges uniformly to f(x)~ ∈ C[0,1] if and only if f(x)=ax+b for some a, b ∈ ℝ.