目的 对B样条的改进方法大多从增加局部参数和在三角函数空间定义基两个角度出发，但仍存在缺陷，原因是通过模型的控制顶点对曲线进行编辑和处理，存在控制顶点给定时曲线较为固定的不足。为此，本文构造了一类基于全正基的非均匀三次加权λαβ-B样条基。方法 结合加权思想，首先证明三次有理基在相应空间上的全正性；其次对三次三角基和三次有理基同时进行扩展，得到新的λαβ-B样条基，新扩展基具有和经典B样条基相似的性质；最后对新扩展基进行线性组合，用得到的多项式构造非均匀三次加权λαβ-B样条基，并研究了曲线的定义及性质。结果 实验结果表明，新曲线保留传统B样条曲线基本性质的同时，还具有局部调整性，可以改善只通过调整控制顶点改变曲线形状的不足。结论 构造的新λαβ-B样条曲线可以有效克服传统方法在改进时的不足，适合曲线设计。

Objective The construction of B-spline basis functions has always been the focus of computer-aided design. The purpose of its research is mainly to solve the problem that the curve generated by the traditional method is fixed relative to the control vertices. The transformation form mainly incorporates the shape into the constructed basis function parameters to increase the flexibility of the curve, that is, to introduce free parameters to the expression of the classic Bernstein basis function or the extended Bernstein basis function, and adjusts the value of the parameter to adjust the shape of the curve. In recent years, researchers have proposed a large number of B-spline improvements, and they are mainly focused on two function spaces, namely, polynomial function space and trigonometric function space. The spline basis functions constructed in these two function spaces have their own advantages in addition to local adjustments to the corresponding curves. The spline curve constructed in the polynomial function space can be degenerated into a classic B-spline curve and has the advantages of simple calculation. Conversely, the basis function constructed in the trigonometric function space has the advantage of the derivation and cyclability of the trigonometric function. Both have high-order continuity, enabling the accurate representation of circle, ellipse, parabola, sine, cosine, cylindrical spiral, etc. The main purpose of this study is to combine the advantages of constructing spline basis functions in these two function spaces and use the weighting method to integrate the basis functions constructed in the two function spaces. The newly introduced weighting factors can be used as global parameters to make new extensions, and the flexibility of the curve is further enhanced. However, from the above two perspectives, some defects still exist. The reason is that the curve is edited and processed through the control vertices of the model, similar to the traditional method. When the control vertices are given, the curve is relatively fixed. Method First, construct a set of cubic rational basis functions in the polynomial function space and prove that they are fully positive. Then, use the weighting idea to weigh the newly constructed cubic rational basis functions and the cubic triangular basis functions constructed and new Bernstein basis and prove that it retains all the good properties of the classic Bernstein basis functions. Subsequently, the new extended basis is linearly combined to obtain the non-uniform cubic weighted B-spline basis, and its properties are studied. Finally, the definition and properties of the corresponding cubic spline curve are given, and the application of the new curve is given based on it. Result Experiments show that the curve constructed by the weighting method in this study retains the respective advantages of the polynomial function space and the trigonometric function space and also has local adjustment. At the same time, the introduced weighting factor also strengthens its global adjustment and further enhances the flexibility of the generated curve. It can improve the shortcomings of changing the curve shape only by adjusting the control vertices. Conclusion In this study, the new λαβ-B-spline curve constructed by the weighting method has a structure similar to the classic B-spline curve while retaining the same properties as the classic B-spline curve, such as convex hull, symmetry, geometric invariance, and change. In addition, the curve constructed by the weighting method in this paper has some unique advantages:first, it has two local shape parameters that can be adjusted locally; second, the weighting factor introduced can be used as a global parameter, and the generated λαβ-B-spline curve can perform global adjustment; third, the λαβ-B-spline curve constructed by the weighting method retains the respective advantages of the polynomial function space and the trigonometric function space. The results show that the shape parameter selection scheme of the constructed curve is correct and effective, which reflects the superiority of the method in this study over other similar methods in the literature. In addition, the construction method with parameter expansion base given in this study and the selection method of shape parameters are general, which can be extended to construct B-spline and triangle surfaces with shape parameters.