Bézier curves based on Lupaş (p,q)-analogue of Bernstein polynomials in CAGD
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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)-Bézier curves (rational curves) and surfaces (rational surfaces) with shape parameters. We study the nature of degree elevation and degree reduction for Lupaş (p,q)-Bézier Bernstein functions. Parametric curves are represented using Lupaş (p,q)-Bernstein basis. We introduce affine de Casteljau algorithm for Lupaş type (p,q)-Bernstein Bézier curves. The new curves have some properties similar to q-Bézier 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-Bézier surfaces for (u, v) ∈ [0, 1] × [0, 1] when we set the parameter p_1=p_2=1. In comparison to q-Bézier 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^n_p_n,q_n(f,x) converges uniformly to f(x)∈ C[0,1] if and only if 0<q_n<p_n≤1 such that _n→∞ q_n=1, _n→∞ p_n=1 and _n→∞p_n^n=a, _n→∞q_n^n=b with 0<a,b≤1. On the other hand, for any p>0 fixed and p ≠ 1, the sequence L^n_p,q(f,x) converges uniformly to f(x) ∈ C[0,1] if and only if f(x)=ax+b for some a, b ∈R.
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