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中文摘要：
针对由于几何图形最高阶数不同而引起的NURBS曲面降阶逼近问题，基于NURBS曲面的显式矩阵表示，结合Chebyshev多项式逼近理论，提出了一种NURBS曲面降阶新方法．分别对一小片NURBS曲面和整张NURBS曲面进行降多阶，并导出了误差界计算公式．当对整张曲面降阶时先分别对各小片操作，再对各片降阶逼近曲面的控制顶点，集中其下标相重的部分做加权平均得到最终的整张降阶逼近曲面．提出的算法可以一次降多阶，所得NURBS降阶逼近曲面具有显式表达式，实现了NURBS曲面降阶的最佳或近似最佳一致逼近．
英文摘要：
Aiming at the degree reduction problem of non-uniform rational B-spline （NURBS） surfaces aroused by different requirements and limitations of the degree of graphics in different systems, a new degree reduction method of NURBS surfaces was presented based on the explicit matrix representation of NURBS surfaces and combing with the theory of Chebyshev polynomials approximation. The degree reduction of a NURBS surface on each knot span region and the degree reduction of a whole NURBS surface were carried out respectively, and the error bounds were also provided. Multi-degree reduction of a whole NURBS suface includes multi-degree reduction to each surface segments respectively and calculating the weighted average of the obtained control points, of which the subscripts are the same. The new algorithm can do multi-degree reduction each time and the degree reduced NURBS surfaces have explicit representations. The optimal or nearly optimal uniform approximation can be obtained for degree reduction of NURBS surfaces by using the method.
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