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中文摘要：
基于Jacobi基的性质以及条件极值问题的求解,对4片相邻三角Bézier曲面进行了近似合并.首先利用Jacobi基的正交性及其与Bézier基之间的基转换矩阵,得到合并前后三角Bézier曲面距离函数的L2范数;为了保证合并前后三角Bézier曲面在边界C0连续以及角点处高阶连续,控制顶点必须满足一系列线性约束.为得到与原曲面距离最小的近似合并曲面,只需要利用Lagrange乘子法解决带线性约束的条件极值即可.合并三角Bézier曲面的控制顶点可用矩阵显式表达,且合并的逼近误差可由合并前后曲面距离函数的L2范数形式精确给出.通过提高合并三角Bézier曲面的次数,可减小合并误差、改善合并效果.数值实例表明,该方法计算简单、直接,适用性强,逼近效果佳.
英文摘要：
Based on the property of Jacobi polynomials and the solution of conditional extremum, the approximate merging of 4 neighbouring triangular Bezier surfaces with boundary constraints are discussed. With the help of the orthonormality of Jacobi polynomials and the Bernstein-Jaeobi transformation matrix, the distance function with respect to L2 norm between the surfaces before and after approximate merging is obtained. Meanwhile, based on the boundary constraints and the boundary continuity at three corners, control points are constrained by linear equations. With the help of Lagrange multipliers, we obtain the optimal approximate merging surface with minimal distance. Both the control points of approximate merging surfaces and the precise error of approximation are expressed explicitly in matrix form. The degree elevation could reduce the merging error. Finally, several numerical examples demonstrate the effectiveness and validity of the algorithm.
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Optimal constrained multi-degree reduction of B,zier curves with explicit expressions based on divid
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