G^(1)连续组合曲线曲面的构造

Construction of G^(1) Continuous Composite Curves and Surfaces

为了在保留B样条方法自动光滑性的基础上突破其要求各部分曲线曲面的次数必须相等的限制,同时赋予曲线曲面独立于控制顶点的形状可调性,提出一种G^(1)连续组合曲线曲面的构造方法.首先构造一组含2个自由参数的n(n≥2)次基函数并分析其性质,基于该基函数,定义了结构与n次Bézier曲线曲面相同的新曲线曲面,其包含n次Bézier曲线曲面为特例;然后分析新曲线曲面的G^(1)光滑拼接条件,根据拼接条件,采用与B样条方法相同的组合思想但是不同的组合方式,定义基于新曲线曲面的分段组合曲线与分片组合曲面,其包含2次均匀和2次准均匀B样条为特例.实例结果表明,定义方式自动保证了组合曲线曲面在连接处的G^(1)连续性,组合曲线曲面的端点与角点位置以及内部形状都可以通过改变自由参数的取值来进行调整,且调整方式既可以是全局的又可以是局部的;所提方法为复杂曲线曲面的造型设计提供了便利.

In order to break through the limitation that all parts of the B-spline curve and surface must have exactly the same degree while retaining the automatic smoothness of B-spline method,and to have shape adjustability independent of the control points,this paper proposes a method for constructing G^(1) continuous composite curves and surfaces.Firstly,a group of basis functions of degree n(n≥2)which contain two free parameters is constructed,and the properties are analyzed.Based on these basis functions,a new kind of degree n curves and surfaces which have the same structure as the Bézier curves and surfaces is defined.They contain the degree n Bézier curve and surface as special cases.Next,the G^(1) smooth join conditions of the new curves and surfaces are analyzed.According to the join conditions,the piecewise combination curves and surfaces based on the new curves and surfaces are defined.They contain the quadratic uniform and quasi-uniform B-spline curve and surface as special cases.The combination idea is the same as the B-spline method,but the combination method is different.The definition automatically ensures the G^(1) continuity of the composite curves and surfaces at the joints.The position of the end(corner)points and the internal shape of the composite curves and surfaces can be adjusted by changing the free parameters,and the adjustment can be both global and local.This method facilitates the design of complex curve and surface.