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

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, G1 and G2 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 G1 and G2 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.