In this paper, we present an improved numerical algorithm for computing the intersection area of multiple circles and a complex polygon efficiently. This geometric problem is fundamental to applications such as wireless sensor networks and base station deployment. The key idea is a curvature-multiplicity-guided adaptive sampling strategy that dynamically concentrates sampling points in geometrically complex boundary regions. The algorithm integrates three components: (i) adaptive quadtree partitioning, (ii) analytical integration via Green's theorem for cells intersecting a single circle, and (iii) curvature-multiplicity-guided Monte Carlo subsampling for cells intersecting multiple circles, where a minimum sample count and a constant factor are introduced into the sampling size. Theoretical analysis shows that the algorithm achieves O(1/ε3/2) computational complexity while maintaining an O(ε) error bound, improving upon the O(1/ε2) complexity of classical Monte Carlo and uniform grid methods for the same error tolerance ε. Numerical experiments on complex polygons, including synthetic data and real-world scenarios, demonstrate that our algorithm outperforms five classical methods in terms of relative error. Furthermore, parameter sensitivity analysis confirms that the algorithm is robust and could make it suited for practical applications such as wireless sensor network coverage estimation.

翻译：本文提出了一种改进的数值算法，用于高效计算多个圆与复杂多边形的相交面积。该几何问题在无线传感器网络和基站部署等应用中具有基础性意义。核心思想是一种曲率-多重度引导的自适应采样策略，该策略能够动态地将采样点集中在几何复杂的边界区域。算法整合了三个组成部分：（i）自适应四叉树分区；（ii）通过格林定理对与单个圆相交的单元进行解析积分；（iii）针对与多个圆相交的单元，采用曲率-多重度引导的蒙特卡洛子采样，其中在采样大小中引入了最小样本数和常数因子。理论分析表明，该算法在保持O(ε)误差界的同时实现了O(1/ε³/²)的计算复杂度，相比经典蒙特卡洛和均匀网格方法在相同容差ε下的O(1/ε²)复杂度有所改进。在包括合成数据和真实场景在内的复杂多边形上的数值实验表明，我们的算法在相对误差方面优于五种经典方法。此外，参数敏感性分析证实了该算法的鲁棒性，使其适用于无线传感器网络覆盖估计等实际应用。