AI 中文总结
研究光滑曲线最优分段线性逼近问题，通过确定二次抛物线及三次函数的最优划分与节点分布等，得到最小面积精确值等结果，可用于CAD系统基于最小偏差面积准则优化刀具路径。
AI 中文摘要
本文研究光滑曲线的最优分段线性逼近问题，即最小化原函数图像与构造的插值折线所围区域的面积。该问题出现在金属激光和等离子切割、铣削、增材制造等工艺过程以及CNC设备控制程序编制中。对于二次抛物线，确定了最小面积的精确值，并表明区间的最优划分是均匀的。对于三次函数，证明了最优插值节点非均匀分布；在有两个内部分割点的情况下，得到了它们的精确坐标和相应的最小面积。结果可应用于CAD系统，基于最小偏差面积准则优化刀具路径。
英文摘要
This paper investigates the problem of optimal piecewise linear approximation of a smooth curve, in which the area of the region enclosed between the graph of the original function and the constructed interpolating polyline is minimized. This problem arises in technological processes such as laser and plasma cutting of metals, milling, additive manufacturing, and in the preparation of control programs for CNC equipment. For a quadratic parabola, the exact value of the minimum area is established, and it is shown that the optimal partition of the interval is uniform. For a cubic function, it is demonstrated that the optimal interpolation nodes are distributed non-uniformly; in the case of two internal partition points, their exact coordinates and the corresponding minimum area are obtained. The results can be applied in CAD systems for optimizing tool paths based on the criterion of minimum deviation area.
