Bezier curves based on Lupas (p,q)-analogue of Bernstein polynomials in CAGD
Abstract
In this paper, we use the blending functions of Lupas type (rational) (p,q)-Bernstein operators based on (p,q)-integers for construction of Lupas (p,q)-Bezier curves (rational curves) and surfaces (rational surfaces) with shape parameters. We study the nature of degree elevation and degree reduction for Lupas (p,q)-Bezier Bernstein functions. Parametric curves are represented using Lupas (p,q)-Bernstein basis. We introduce affine de Casteljau algorithm for Lupas type (p,q)-Bernstein Bezier curves. The new curves have some properties similar to q-Bezier 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-Bezier surfaces for (u, v) ∈ [0, 1] × [0, 1] when we set the parameter p1=p2=1. In comparison to q-Bezier curves and surfaces based on Lupas 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 Lupas 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 n∞ qn=1, n∞ pn=1 and n∞pnn=a, n∞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.
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