目的 目前有很多研究B样条曲线的含参数扩展,给出的曲线都具备B样条曲线的局部形状控制性以及独立于控制顶点的形状可调性,但有些文献给出的参数是全局的,导致曲线不具备局部形状调整性,有些文献给出的调配函数不具有全正性,导致曲线不具备变差缩减性、保凸性。本文的出发点是构造同时具备保凸性、局部形状调整性、局部形状控制性的曲线。方法 首先运用拟扩展函数空间的理论框架证明了已有的3次Bézier曲线的扩展基,简称λμ-Bernstein基,恰好为所在空间中的规范B基。然后运用λμ-Bernstein基的线性组合来构造3次均匀B样条曲线的扩展基,根据预设的曲线性质反推出扩展基的性质,进而求出线性组合的系数,得出扩展基的表达式。扩展基可以表示成λμ-Bernstein基与一个转换矩阵的乘积,证明了转换矩阵的全正性,由扩展基定义了一种结构与3次B样条曲线相同的含一个局部形状参数的分段曲线。结果 转换矩阵的全正性决定了扩展基的全正性,扩展基的全正性决定了扩展曲线的变差缩减性、保凸性,形状参数的局部性决定了曲线的局部形状调整性,曲线的分段结构决定了曲线的局部形状控制性。结论 本文给出的构造具有全正性的B样条扩展基的方法具有一般性,与现有众多扩展曲线相比,本文方法构造的曲线因为具有变差缩减性和保凸性,从而为保形设计提供了一种有效方法。

Objective Many studies have been conducted on the extension of the B-spline curve with parameters. These extended curves have similar local shape controlability as the B-spline curve and have shape adjustability that is independent of the control points. However, previous studies on this topic used global parameters, thus preventing the shape of the curves to be locally adjusted. The blending functions in several studies do not have total positivity, thus removing the variation-diminishing properties and convex-preserving properties of the curves. The purpose of this paper is to construct a curve with convexity-preserving property, local shape adjustment, and local shape control. Method By using the theoretical framework of quasi-extended function space, we first prove that the existing extended basis of the cubic Bézier curve, called λμ-Bernstein basis, is exactly the normalized B-basis of the corresponding space. Thereafter, we use the linear combination of the λμ-Bernstein basis to express the extended basis of the cubic uniform B-spline curve. According to the preset properties of the curve, we deduce the properties of the extended basis and then determine the coefficients of the linear combination and the expression of the basis. The extended basis can be represented as the product of the λμ-Bernstein basis and a conversion matrix. We prove the totally positive property of the matrix. By using this basis, we define a piecewise curve with one local shape parameter with the same structure as the cubic B-spline curve. Result The total positivity of the conversion matrix determines the total positivity of the extended basis. The total positivity of the basis determines that the extended curve has a variation-diminishing property and convex-preserving property. The locality of the shape parameter determines that the shape of the curve can be adjusted locally. The piecewise structure indicates that the curve has local shape control ability. Conclusion The method for constructing the extended B-spline basis with total positivity has generality. Compared with most extended curves in literature, the curve given in this paper has variation diminishing property and convex-preserving property; thus, providing an efficient method for conformal design.