Continuity conditions for tensor product Q-Bézier surfaces of degree ($$m,\, n$$m,n)

Hu, Gang; Bo, Cuicui; Qin, Xinqiang

doi:10.1007/s40314-017-0568-0

Cited by 7 publications

(7 citation statements)

References 17 publications

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“…As previously mentioned, the L 2 -norm-based error function in (19) can be used to measure the approximation effect between the degree reduced Q-Bézier curves and the original Q-Bézier curves. Nevertheless, the error calculated by the function in Equation ( 19) is an absolute error, and its value is affected by the coordinate value of the control points of the curves, which brings inconvenience to the objective evaluation of the approximation effect.…”

Section: Results Discussionmentioning

confidence: 99%

“…Liu et al [18] provided some remarks on the Q-Bézier curves constructed by [17] and revisited the rational quadratic case of the curves. Hu et al [19] derived G 1 and G 2 continuity conditions of adjacent (m, n)-degree Q-Bézier surfaces with shape parameters. Jegdić et al [20] defined a (q, h)-blossom operator of bivariate Bernstein polynomials, and presented the corresponding rectangular tensor product (q, h)-Bézier surfaces.…”

Section: Introductionmentioning

confidence: 99%

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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: Results Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Degree Reduction of Q-Bézier Curves via Squirrel Search Algorithm

Liu

Abbas

et al. 2021

Mathematics

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“…Similarly, if two surfaces need to satisfy G 1 continuity, they should not only have a common boundary curve, but also have a common tangent plane at the common joint. In order to solve the problem mentioned above, many international scholars have done many studies on the related algorithms of geometric continuity conditions for different geometric models [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30].…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, then the conditions of G 1 and G 2 geometric continuity between two adjacent SG-Bézier curves and surfaces are derived [21,22]. Subsequently, the geometric continuity conditions of Q-Bézier curves and surfaces are studied in [23] and [24], respectively. Furthermore, Qian and Tang [25] investigated the sufficient and necessary conditions for continuity of curvature and tangency between H-Bézier curves of degree 4.…”

Section: Introductionmentioning

confidence: 99%

Explicit Continuity Conditions for G1 Connection of S-λ Curves and Surfaces

Abbas

et al. 2020

Mathematics

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The S-λ model is one of the most useful tools for shape designs and geometric representations in computer-aided geometric design (CAGD), which is due to its good geometric properties such as symmetry, shape adjustable property. With the aim to solve the problem that complex S-λ curves and surfaces cannot be constructed by a single curve and surface, the explicit continuity conditions for G1 connection of S-λ curves and surfaces are investigated in this paper. On the basis of linear independence and terminal properties of S-λ basis functions, the conditions of G1 geometric continuity between two adjacent S-λ curves and surfaces are proposed, respectively. Modeling examples imply that the continuity conditions proposed in this paper are easy and effective, which indicate that the S-λ curves and surfaces can be used as a powerful supplement of complex curves and surfaces design in computer aided design/computer aided manufacturing (CAD/CAM) system.

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“…To maintain the advantages of the Bézier method and increase the ability to adjust its shape, as well as to improve its approximation to real curves, scholars have constructed many non-rational Bézier curves with shape parameters [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Based on a new set of generalized Bernstein basis functions of explicit expressions, Hu et al [20] proposed a generalized Bézier (SG-Bézier) curve with various shape parameters.…”

Section: Introductionmentioning

confidence: 99%

Approximate Multi-Degree Reduction of SG-Bézier Curves Using the Grey Wolf Optimizer Algorithm

Qiao

Qin

et al. 2019

Symmetry

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SG-Bézier curves have become a useful tool for shape design and geometric representation in computer aided design (CAD), owed to their good geometric properties, e.g., symmetry and convex hull property. Aiming at the problem of approximate degree reduction of SG-Bézier curves, a method is proposed to reduce the n-th SG-Bézier curves to m-th (m < n) SG-Bézier curves. Starting from the idea of grey wolf optimizer (GWO) and combining the geometric properties of SG-Bézier curves, this method converts the problem of multi-degree reduction of SG-Bézier curves into solving an optimization problem. By choosing the fitness function, the approximate multi-degree reduction of SG-Bézier curves with adjustable shape parameters is realized under unrestricted and corner interpolation constraints. At the same time, some concrete examples of degree reduction and its errors are given. The results show that this method not only achieves good degree reduction effect, but is also easy to implement and has high accuracy.

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Continuity conditions for tensor product Q-Bézier surfaces of degree ($$m,\, n$$m,n)

Cited by 7 publications

References 17 publications

Degree Reduction of Q-Bézier Curves via Squirrel Search Algorithm

Degree Reduction of Q-Bézier Curves via Squirrel Search Algorithm

Explicit Continuity Conditions for G1 Connection of S-λ Curves and Surfaces

Approximate Multi-Degree Reduction of SG-Bézier Curves Using the Grey Wolf Optimizer Algorithm

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