Abstract: Weighting idea and singular blending technology are used to extend the traditional quasi-Bézier curve, and a singular blending quasi-Bézier curve with shape parameters is constructed. Firstly, the singular blending function and the quasi-cubic Bézier basis function of the triangular polynomial space are combined to obtain the definition of the singular blending quasi-Bézier curve, and the singular blending quasi-Bézier basis function is deduced according to the definition of the singular blending quasi-Bézier curve. Secondly, we discuss the singular blending quasi-Bézier basis functions and the properties of their corresponding curves, and explore the influences of singular blending and parameters on them. Finally, an example of a singular blending quasi-Bézier curve and surface design is given. The experimental results show that the curve constructed in this paper has the flexible shape adjustability while having the practical properties of the traditional Bézier curve. The new curve can not only accurately represent conic curves such as elliptical arc, circular and parabola arc, but also achieve G1  and G2  continuity under certain conditions. Extending the curve to the surface using the tensor product method can also accurately represent the ellipsoid and the spherical surface. A large number of analysis and examples prove that the curves constructed in this paper are very effective in geometric design.

Key words: quasi-Bézier curve, singular blending, shape parameter, continuity