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

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1242
Author(s):  
Hu ◽  
Qiao ◽  
Qin ◽  
Wei

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.

2011 ◽  
Vol 48-49 ◽  
pp. 877-880
Author(s):  
Qing Xian Meng ◽  
Hui Li Liu

Regularity is one of important properties of curves in computer aided design. In this paper, we convert the problem of determining regularity of Bézier curves to that of detecting existence of zero points of polynomials. Based on the properties of algebraic equations and isolation theorem of roots, a simple and practical method is presented. Regularity of Bézier curves and number of singular points can be determined by easier computation.


Author(s):  
Yunbeom Park ◽  
U Jin Choi

AbstractThe error analysis of an algorithm for generating an approximation of degree n − 1 to an nth degree Bézier curve is presented. The algorithm is based on observations of the geometric properties of Bézier curves which allow the development of detailed error analysis. By combining subdivision with a degree reduction algorithm, a piecewise approximation can be generated, which is within some preset error tolerance of the original curve. The number of subdivisions required can be determined a priori and a piecewise approximation of degree m can be generated by iterating the scheme.


Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 505
Author(s):  
Xin Sun ◽  
Xiaomin Ji

The parametric method of product design is a pivotal and practical technique in computer-aided design and manufacturing (CAD/CAM) and used in many manufacturing sectors. In this paper, we presented a novel parametric method to design a kitchen product in the residential environment, a kitchen cabinet, by using cubic T-Bézier curves with constraints of geometric continuities. First, we introduced a class of cubic T-Bézier curves with two shape parameters and derived the G1 and G2 continuity conditions of the cubic T-Bézier curves. Then, we constructed shape-controlled complex contour curves of the kitchen cabinet by using closed composite cubic T-Bézier curves. The shapes of the contour curves can be adjusted intuitively and predictably by altering the values of the shape parameters. Finally, we studied shape optimization and representation of ellipses for the contour curves of the kitchen cabinet by finding optimal shape parameters and applicable control points respectively. The provided modeling examples showed that our method in this paper can improve the design and scheme adjustment effectively in the conceptual design stage of kitchen products.


Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2212
Author(s):  
Xiaomin Liu ◽  
Muhammad Abbas ◽  
Gang Hu ◽  
Samia BiBi

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.


Author(s):  
Pooja Arora ◽  
Anurag Dixit

Purpose The advancements in the cloud computing has gained the attention of several researchers to provide on-demand network access to users with shared resources. Cloud computing is important a research direction that can provide platforms and softwares to clients using internet. However, handling huge number of tasks in cloud infrastructure is a complicated task. Thus, it needs a load balancing (LB) method for allocating tasks to virtual machines (VMs) without influencing system performance. This paper aims to develop a technique for LB in cloud using optimization algorithms. Design/methodology/approach This paper proposes a hybrid optimization technique, named elephant herding-based grey wolf optimizer (EHGWO), in the cloud computing model for LB by determining the optimal VMs for executing the reallocated tasks. The proposed EHGWO is derived by incorporating elephant herding optimization (EHO) in grey wolf optimizer (GWO) such that the tasks are allocated to the VM by eliminating the tasks from overloaded VM by maintaining the system performance. Here, the load of physical machine (PM), capacity and load of VM is computed for deciding whether the LB has to be done or not. Moreover, two pick factors, namely, task pick factor (TPF) and VM pick factor (VPF), are considered for choosing the tasks for reallocating them from overloaded VM to underloaded VM. The proposed EHGWO decides the task to be allocated in the VM based on the newly derived fitness functions. Findings The minimum load and makespan obtained in the existing methods, constraint measure based LB (CMLB), fractional dragonfly based LB algorithm (FDLA), EHO, GWO and proposed EHGWO for the maximum number of VMs is illustrated. The proposed EHGWO attained minimum makespan with value 814,264 ns and minimum load with value 0.0221, respectively. Meanwhile, the makespan values attained by existing CMLB, FDLA, EHO, GWO, are 318,6896 ns, 230,9140 ns, 1,804,851 ns and 1,073,863 ns, respectively. The minimum load values computed by existing methods, CMLB, FDLA, EHO, GWO, are 0.0587, 0.026, 0.0248 and 0.0234. On the other hand, the proposed EHGWO with minimum load value is 0.0221. Hence, the proposed EHGWO attains maximum performance as compared to the existing technique. Originality/value This paper illustrates the proposed LB algorithm using EHGWO in a cloud computing model using two pitch factors, named TPF and VPF. For initiating LB, the tasks assigned to the overloaded VM are reallocated to under loaded VMs. Here, the proposed LB algorithm adapts capacity and loads for the reallocation. Based on TPF and VPF, the tasks are reallocated from VMs using the proposed EHGWO. The proposed EHGWO is developed by integrating EHO and GWO algorithm using a new fitness function formulated by load of VM, migration cost, load of VM, capacity of VM and makespan. The proposed EHGWO is analyzed based on load and makespan.


Author(s):  
Amolkumar Narayan Jadhav ◽  
Gomathi N.

The widespread application of clustering in various fields leads to the discovery of different clustering techniques in order to partition multidimensional data into separable clusters. Although there are various clustering approaches used in literature, optimized clustering techniques with multi-objective consideration are rare. This paper proposes a novel data clustering algorithm, Enhanced Kernel-based Exponential Grey Wolf Optimization (EKEGWO), handling two objectives. EKEGWO, which is the extension of KEGWO, adopts weight exponential functions to improve the searching process of clustering. Moreover, the fitness function of the algorithm includes intra-cluster distance and the inter-cluster distance as an objective to provide an optimum selection of cluster centroids. The performance of the proposed technique is evaluated by comparing with the existing approaches PSC, mPSC, GWO, and EGWO for two datasets: banknote authentication and iris. Four metrics, Mean Square Error (MSE), F-measure, rand and jaccord coefficient, estimates the clustering efficiency of the algorithm. The proposed EKEGWO algorithm can attain an MSE of 837, F-measure of 0.9657, rand coefficient of 0.8472, jaccord coefficient of 0.7812, for the banknote dataset.


Agriculture ◽  
2020 ◽  
Vol 10 (8) ◽  
pp. 313
Author(s):  
Hector A. Tinoco ◽  
Daniel R. Barco ◽  
Olga Ocampo ◽  
Jaime Buitrago-Osorio

The computer-aided design of fruits are used for different purposes, e.g., to determine mechanical properties by applying engineering simulations, to design postharvest equipment, and to study the natural changes related to the topology. This paper developed a methodology to model Valencia orange (Citrus sinensis), applying Bézier curves and an image-based CAD approach; the orange geometry was designed for different ripening stages. In the modeling process, a 3D construction was carried out using third-order Bézier curves, adjusted to the images taken in orthogonal planes. Four control points defined each profile to compose the geometric pattern of the orange, with geometric errors lower than 3%. Two prediction models were proposed to relate the orthogonal dimensions with a factor size; this means that two dimensions out of three can be predicted. The results showed that the shape ratios kept constant in any ripening stage; however, the radius of curvature evidenced differences in the analyzed shape profiles. The methodological framework presented in the paper might be used to draw other types of citrus fruits. This contribution is a tool to model fruits in 3D, instead of using expensive technological equipment, since it is only necessary to apply computer design tools.


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