B样条曲面方向投影问题的几何计算方法

Geometric Method for the Directional Projection Problem of B-Spline Surfaces

B样条曲面方向投影问题可以通过求解方程组的方法来解决.由于方程组所有根中往往只有一个或甚至没有根与待求解的最近点对应,因而绝大多数的求根计算量是不必要的.为此讨论了B样条曲面的方向投影问题,提出一种简单且高效稳定的几何计算方法.该方法充分利用了B样条函数的凸包性,同时结合B样条函数稳定可靠的分裂算法给出了相应的几何剪枝方法.与传统的求解非线性方程组的计算方法相比,文中方法可以剪除绝大部分非线性方程组对应的根,且不需要Newton迭代,可以应用于平面/B样条曲面间的求交测试问题及B样条曲面包围盒的计算问题.实例结果表明,该方法具有比传统的相关方法更高的计算效率和更好的稳定性.

The directional projection problem of B roots of a non-linear equation system. Usually one -spline surfaces can be solved by computing the or none of the roots of the equation system is mapping to the closest point where the minimum distance occurs, and most of the computation on finding the roots of the equation system is unnecessary. A simple but efficient geometric pruning method is presented for the directional projection problem of B-spline surfaces. It utilizes the convex property of the B-spline basis functions and the robust subdivision algorithm of B-spline surface, and it is able to directly detect whether the minimum directional distance occurs at a corner point or at a boundary curve of the B-spline surface. Compared with conventional root-finding methods of a nonlinear equation system, it can exclude most of the roots and needs no numerical iterative method such as the Newton method. The proposed algorithm can be applied in the intersection testing problem between a plane and a B-spline surface, and the encapsulation box computation problem of B-spline surfaces. Examples are given to show both efficiency and robustness of the new method.