Abstract
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.
Highlights
Parametric curves are an important research deimension in the fields of computer aided design (CAD)/CAM, but they play a crucial role in the shape design and geometric representation of various products
Based on the theory of grey wolf optimizer (GWO) algorithm [32,33,34,35], this paper considers the degree reduction of SG-Bézier curves under unrestricted condition and constraint condition of C0 and C1
The grey wolf optimizer algorithm is a new group intelligent optimizer algorithm proposed by Mirjalil et al [32] in 2014
Summary
Parametric curves are an important research deimension in the fields of CAD/CAM, but they play a crucial role in the shape design and geometric representation of various products. 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. This curve inherits many excellent characteristics of Bézier curves, and has flexible shape adjustability. Ahn [26] proposed an approximate degree reduction algorithm based on constrained Jacobi polynomial that maintains the curves reach Ck,k continuity at the endpoints. One method is to transform the problem of degree reduction into solving the optimization of objective function by using intelligent optimization algorithm; Ahn et al [29] showed that the constrained polynomial degree reduction in the L2 -norm equals best weighted Euclidean approximation of Bézier coefficients.
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