多项式曲线包围面积的计算

ON COMPUTING THE AREA BOUNDED BY POLYNOMIAL CURVES

平面图形面积的计算具有较高的工程价值。本文首先给出幂形式多项式曲线所构成弓形的面积的一种高效求值算法，然后利用Bezier曲线、B-样条曲线与幂形式曲线的转换关系，将结果推广到不打结的平面拼接分段多项式曲线所包围的精确面积的高效求值的计算方法上来。这些多项式可以是CAD／CAM中常用多项式的任意一种，其边界线无形状限制且坐标原点无具体位置限制。该文还利用幂形式多项式的剖分方法，把结果推广到了一般的可打结的拼接曲线包围面积问题上，从而完整地给出了分段多项式曲线所包围面积的计算方法．

Computing the area of a planar graphs is valuable in engineering. A highly efficient algorithm for computing the area of an arch field of a power polynomial is presented at first. By means of conversions of Bezier curves, B-spline curves and uniform B-spline curves to power polynomial curves, and by means of the subdivision of power polynomial curves, an efficient computing method is presented for finding the exact area bounded by non-knotted planar composite piecewise polynomial curves, which may be any form of the four commonly used polynomials, mentioned above, in CAD/CAM. The shape of boundary is not restricted at all, and the origin of coordinate systems can be in any position. At last the results are generalized to the case for knotted composite curve boundary, therefore a complete resolution for computing the area bounded by composite polynomial curves is given.