B-样条曲线升阶的几何收敛性

B-样条曲线的升阶算法是CAD系统相互沟通必不可少的手段之一。B-样条曲线的控制多边形经过不断升阶以后，和Bézier曲线一样都会收敛到初始B-样条曲线。根据双次数B-样条的升阶算法，得到了B-样条曲线升阶的收敛性证明。与以往升阶算法不同的是，双次数B-样条的升阶算法具有割角的性质，这就使B-样条曲线升阶有了鲜明的几何意义。得到的结论可以使B-样条曲线像Bézier曲线一样，通过几何割角法生成。

Geometric Convergence of Degree Elevation of B-Spline Curves

Degree elevation of B-spline curves is an essential measure for communication between CAD systems. The sequence of B-spline’s control polygon convergences to initial B-spline curve is similar to the Bézier curve. The convergence proof of B-spline curve is obtained based on the degree elevation algorithm by the bi-degree B-spline. In contrast to traditional methods, degree elevation algorithm by bi-degree B-spline can be interpreted as corner cutting process, so degree elevation of B-spline curve has obvious geometric meaning. The result makes B-spline curve obtained by geometric corner cutting algorithm as Bézier curve.