一种 G2 连续组合曲线的表示

针对 Bézier 曲线以及现有众多含形状参数的扩展 Bézier 曲线的 G2 拼接条件均对控制顶点有严 格要求的问题，拟提出一种 G2 连续组合曲线，其能综合 Bézier 与 B 样条方法的优点，其基函数具有显式表达 式，既具有 B 样条方法的自动光滑性，又能轻松拥有 Bézier 曲线的端点几何特征。为此，构造了一组含 6 个 参数的基函数，按照 3 次 Bézier 曲线的定义方式由之构造了基于 4 个控制顶点的曲线段，根据曲线段的拼接条 件，按照 3 次 B 样条曲线的定义方式构造了基于 4 点分段的组合曲线。基函数具有全正性，其同时包含 3 次 Bernstein 基函数和所有由内部节点重复度均为 1 的节点向量所确定的 3 次 B 样条基函数作为特例。曲线段具 有保凸性、端点位置以及形状可调性，其同时包含 3 次 Bézier 曲线和 3 次 B 样条曲线段作为特例。组合曲线 的定义方式自动保证了其整体 G2 连续，将部分参数取特定值，即可使其端点插值、端边相切，此时其中依然 存在用于调整内部形状的独立参数。按一定规则选取组合曲线中的参数，即可重构 C2 连续的 3 次 B 样条曲线。

Representation of a kind of G2 continuous composite curve

To meet the strict requirements for the control points made by the G2 continuity conditions of the Bézier curve and many existing extended Bézier curves with shape parameter, a G2 continuous composite curve representation method was proposed. The method could synthesize the advantages of the Bézier method and B-spline method, and its basis function had explicit expression. It was of the automatic smoothness as that of the B-spline method, easily possessing the end-point geometric characteristic of the Bézier curve. To this end, a set of basis function with six parameters was constructed. On this basis, a curve segment based on four control points was constructed according to the definition mode of the cubic Bézier curve. According to the -continuity conditions between the curve segments, a kind of composite curve on four-point piecewise scheme was constructed according to the definition mode of the cubic B-spline curve. The basis function was of total positivity, and contained the cubic Bernstein basis functions and the cubic B-spline basis functions that were determined by the node vector with the repetition degree of all internal nodes being one. The curve segment had the feature of convexity-preserving, endpoint position, and adjustable shape, and contained the cubic Bézier curve and the cubic B-spline curve segment as special cases. The definition of the composite curve could automatically ensure its G2 continuity at each junction. The composite curve could have end-point interpolation and end-edge tangency by setting some of its parameters as specific values. At this point, the composite curve still contained independent parameters used to adjust its internal shape. As long as the parameters of the composite curve were selected according to certain rules, the C2 continuous cubic B-spline curve could be reconstructed.