一种G^(2)连续组合曲线的表示

Representation of a kind of G^(2) continuous composite curve

针对Bézier曲线以及现有众多含形状参数的扩展Bézier曲线的G^(2)拼接条件均对控制顶点有严格要求的问题,拟提出一种G^(2)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G^(2)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C~2连续的3次B样条曲线。

To meet the strict requirements for the control points made by the G^(2) continuity conditions of the Bézier curve and many existing extended Bézier curves with shape parameter, a G^(2) 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 G^(2) 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 C~2 continuous cubic B-spline curve could be reconstructed.