Construction of Local-Shape-Controlled Quartic Generalized Said-Ball Model

Jiaoyue Zheng, Xiaomin Ji (), Zhaozhao Ma and Gang Hu ()
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Jiaoyue Zheng: School of Mechanical and Precision Instrument Engineering, Xi’an University of Technology, Xi’an 710048, China
Xiaomin Ji: School of Mechanical and Precision Instrument Engineering, Xi’an University of Technology, Xi’an 710048, China
Zhaozhao Ma: Department of Applied Mathematics, Xi’an University of Technology, Xi’an 710054, China
Gang Hu: Department of Applied Mathematics, Xi’an University of Technology, Xi’an 710054, China

Mathematics, 2023, vol. 11, issue 10, 1-21

Abstract: Said-Ball curves and surfaces are extensively applied in the realm of geometric modeling. Their appearance is only decided by the control points, which produces a great deal of inconvenience for the shape design of sophisticated products. To overcome this defect, we construct a novel kind of quartic generalized Said-Ball (QGS-Ball, for short) curves and surfaces, which contain multiple shape parameters, and the global and local shape can be easily modified via shape parameters. The specific research contents are as follows: Firstly, the QGS-Ball basis functions carrying multiple shape parameters are defined, and the correlative properties are proved. Secondly, the QGS-Ball curve is proposed according to the QGS-Ball basis functions, and the effect of shape parameters on the curve is discussed. Thirdly, in view of the constructed QGS-Ball curve, we further propose the combined quartic generalized Said-Ball (CQGS-Ball, for short) curves, and deduce the conditions of first-order and second-order geometric continuity (namely, G 1 and G 2 continuity). Finally, the QGS-Ball surface is defined by tensor product method, and the influence of shape parameters on the surface is analyzed. The main contribution of this article is to construct the QGS-Ball curve model, and deduce the G 1 and G 2 geometric joining conditions of QGS-Ball curves. Combined with some modeling examples, it further illustrates that the QGS-Ball curve as a new geometric model provides a powerful supplement for the geometric design of sophisticated form in computer-aided design (CAD) and computer-aided manufacturing (CAM) systems.

Keywords: quartic generalized Said-Ball curve and surface; three shape parameters; combined quartic generalized Said-Ball curves; G 1 and G 2 smooth joining (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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