基于广义逆矩阵的张量积Bézier曲面合并逼近

Approximate merging of tensor product Bézier surfaces based on generalized inverse matrix

为了对CAD系统中的几何数据进行压缩,研究2张相邻张量积Bézier曲面合并逼近的问题.为了更好地进行曲面合并逼近,利用张量积Bézier曲面细分后的矩阵表示给出相邻张量积曲面可精确合并的充要条件,在此基础上通过广义逆矩阵的方法求解出在L2范数下合并逼近后的张量积Bézier曲面,得到其控制顶点的显示表达式.同时给出带角点插值条件的曲面合并逼近的结果.利用广义逆矩阵可以方便地求得最小二乘解,得到能够显示表示、算法执行时间最短且逼近效果好的合并逼近算法.数值实例显示了算法的有效性.

Approximate merging of two adjacent tensor product Bézier surfaces was investigated to guarantee the compression of geometric data in CAD system. The sufficient and necessary condition for precise merging of adjacent tensor product surfaces was obtained by using the matrix representation of subdivided Bézier surface. Then the merged tensor product Bézier surface was solved by the generalized inverse matrix in L2 norm based on precise merging condition,and the explicit representation of the merged surface＇s control points was also obtained. Meanwhile,the results of approximate merging with corner interpolation were shown. Since the minimal least squares solution can be directly obtained by the generalized inverse matrix,the algorithm possesses explicit formula,less time consumption and good approximation results. Numerical results demonstrated the effectiveness of the algorithm.