A note on Bernstein polynomials based on (p,q)-calculus

Agyuz, Erkan; Açıkgöz, Mehmet

doi:10.1063/1.5043690

Cited by 2 publications

(2 citation statements)

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“…Apart from their use in optimization theory, the shape operator properties of these surfaces are also important when we introduce basis-functions with shape-control parameters. Khan [25] introduced a new class of curves and surfaces recognized as (p, q)-Bernstein-Be ´zier curves and (p, q)-Bernstein-Be ´zier surfaces [26] and surfaces that are an extension of q-Bernstein-Be ´zier curves and q-Bernstein-Be ´zier surfaces respectively. With the help of the parameters p and q, curves and surfaces can be modified in shape without changing the position of control-points.…”

Section: Introductionmentioning

confidence: 99%

Exploring q-Bernstein-Bézier surfaces in Minkowski space: Analysis, modeling, and applications

Bashir,

Ahmad,

Ali

2024

PLoS ONE

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In this paper, we examine q-Bernstein-Bézier surfaces in Minkowski space-R 1 3 with q as the shape parameter. These surfaces, a generalization of Bézier surfaces, have applications in mathematics, computer-aided geometric design, and computer graphics for the surface formation and modeling. We analyze the timelike and spacelike cases of q-Bernstein-Bézier surfaces using known boundary control points. The mean curvature and Gaussian curvature of these q-Bernstein-Bézier surfaces are computed by finding the respective fundamental coefficients. We also investigate the shape operator dependency for timelike and spacelike q-Bernstein-Bézier surfaces in Minkowski space-R 1 3, and provide biquadratic and bicubic q-Bernstein-Bézier surfaces as illustrative examples for different values of the shape controlling parameter q.

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Section: Introductionmentioning

confidence: 99%

Exploring q-Bernstein-Bézier surfaces in Minkowski space: Analysis, modeling, and applications

Bashir,

Ahmad,

Ali

2024

PLoS ONE

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“…ey play a significant role for constructing the surfaces of different shapes and desired characteristics that depend on the choice of Bernstein polynomials, namely, the classical Bernstein polynomials, shifted knots Bernstein polynomials, and q-Bernstein polynomials, generally called the modified Bernstein polynomials. An extension of q-Bernstein polynomials in surface theory is (p, q)-Bernstein polynomials, basic properties, and generating functions for Bernstein polynomial-related results with the help of (p, q)-calculus, which can be seen in references [27,28]. Araci [29] and Jang et al [30] study the q-analogue of Euler numbers and polynomials naturally arising from the p-adic fermionic integrals on Z p and investigate some properties for these numbers and polynomials.…”

Section: Introductionmentioning

confidence: 99%

A Computational Model for q‐Bernstein Quasi‐Minimal Bézier Surface

Ahmad

Mahmood

Xin

et al. 2022

Journal of Mathematics

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A computational model is presented to find the q -Bernstein quasi-minimal Bézier surfaces as the extremal of Dirichlet functional, and the Bézier surfaces are used quite frequently in the literature of computer science for computer graphics and the related disciplines. The recent work [1–5] on q -Bernstein–Bézier surfaces leads the way to the new generalizations of q -Bernstein polynomial Bézier surfaces for the related Plateau–Bézier problem. The q -Bernstein polynomial-based Plateau–Bézier problem is the minimal area surface amongst all the q -Bernstein polynomial-based Bézier surfaces, spanned by the prescribed boundary. Instead of usual area functional that depends on square root of its integrand, we choose the Dirichlet functional. Related Euler–Lagrange equation is a partial differential equation, for which solutions are known for a few special cases to obtain the corresponding minimal surface. Instead of solving the partial differential equation, we can find the optimal conditions for which the surface is the extremal of the Dirichlet functional. We workout the minimal Bézier surface based on the q -Bernstein polynomials as the extremal of Dirichlet functional by determining the vanishing condition for the gradient of the Dirichlet functional for prescribed boundary. The vanishing condition is reduced to a system of algebraic constraints, which can then be solved for unknown control points in terms of known boundary control points. The resulting Bézier surface is q -Bernstein–Bézier minimal surface.

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A note on Bernstein polynomials based on (p,q)-calculus

Cited by 2 publications

References 10 publications

Exploring q-Bernstein-Bézier surfaces in Minkowski space: Analysis, modeling, and applications

Exploring q-Bernstein-Bézier surfaces in Minkowski space: Analysis, modeling, and applications

A Computational Model for q‐Bernstein Quasi‐Minimal Bézier Surface

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