<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.gif" display="inline" overflow="scroll"><mml:mi>q</mml:mi></mml:math>-Blossoming: A new approach to algorithms and identities for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si34.gif" display="inline" overflow="scroll"><mml:mi>q</mml:mi></mml:math>-Bernstein bases and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.gif" display="inline" overflow="scroll"><mml:mi>q</mml:mi></mml:math>-Bézier curv

Simeonov, Plamen; Zafiris, Vasilis; Goldman, Ron

doi:10.1016/j.jat.2011.09.006

Cited by 35 publications

(2 citation statements)

References 10 publications

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“…Based on the De Casteljau algorithm, Disibuyuk and Oruc [13] proposed rational Q-Bézier curves. Simeonov et al [14] deduced a recursive evaluation algorithm and an explicit subdivision procedure for Q-Bézier curves. Goldman et al [15] derived explicit formulas for the generating functions of the Q-Bernstein basis functions in terms of q-exponential functions and proved a variety of identities for the Q-Bernstein bases.…”

Section: Introductionmentioning

confidence: 99%

Degree Reduction of Q-Bézier Curves via Squirrel Search Algorithm

Liu

Abbas

et al. 2021

Mathematics

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Q-Bézier curves find extensive applications in shape design owing to their excellent geometric properties and good shape adjustability. In this article, a new method for the multiple-degree reduction of Q-Bézier curves by incorporating the swarm intelligence-based squirrel search algorithm (SSA) is proposed. We formulate the degree reduction as an optimization problem, in which the objective function is defined as the distance between the original curve and the approximate curve. By using the squirrel search algorithm, we search within a reasonable range for the optimal set of control points of the approximate curve to minimize the objective function. As a result, the optimal approximating Q-Bézier curve of lower degree can be found. The feasibility of the method is verified by several examples, which show that the method is easy to implement, and good degree reduction effect can be achieved using it.

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Section: Introductionmentioning

confidence: 99%

Degree Reduction of Q-Bézier Curves via Squirrel Search Algorithm

Liu

Abbas

et al. 2021

Mathematics

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“…Dişibüyük and Oruç [15, 16] defined the q generalization of rational Bernstein–Bézier curves and tensor product q -Bernstein–Bézier surfaces. Moreover, Simeonov et al [17] introduced a new variant of the blossom, the q blossom, which is specifically adapted to developing identities and algorithms for q -Bernstein bases and q -Bézier curves. In 2014, Han et al [18] proposed a generalization of q -analog Bézier curves with one shape parameter, and established degree evaluation and de Casteljau algorithms and some other properties.…”

Section: Introductionmentioning

confidence: 99%

Shape-preserving properties of a new family of generalized Bernstein operators

Cai

2018

J Inequal Appl

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In this paper, we introduce a new family of generalized Bernstein operators based on q integers, called -Bernstein operators, denoted by . We investigate a Kovovkin-type approximation theorem, and obtain the rate of convergence of to any continuous functions f. The main results are the identification of several shape-preserving properties of these operators, including their monotonicity- and convexity-preserving properties with respect to . We also obtain the monotonicity with n and q of .

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Optimal constrained polynomials approximation of hyperbolas based on Lupaş q‐Bézier curves

Han

Zhang

2018

Math Methods in App Sciences

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This paper presents an explicit optimal polynomial for approximating the quadratic Lupaş q‐Bézier curve. We first prove that the quadratic Lupaş q‐Bézier curve represents a hyperbola or a parabola. Then we research the approximation of quadratic Lupaş q‐Bézier curves by polynomials. Since the denominator of quadratic Lupaş q‐Bézier curves is a linear function, the explicit optimal constrained approximation is obtained. Finally, some numerical examples are presented to illustrate the effectiveness of the proposed method.

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q-Blossoming: A new approach to algorithms and identities for q-Bernstein bases and q-Bézier curv

Cited by 35 publications

References 10 publications

Degree Reduction of Q-Bézier Curves via Squirrel Search Algorithm

Degree Reduction of Q-Bézier Curves via Squirrel Search Algorithm

Shape-preserving properties of a new family of generalized Bernstein operators

Optimal constrained polynomials approximation of hyperbolas based on Lupaş q‐Bézier curves

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