An improved boundary-focused adaptive quadtree algorithm for circle-polygon intersection area approximation

TL;DR

This paper introduces an enhanced adaptive quadtree algorithm with curvature-guided sampling for efficient and accurate approximation of circle-polygon intersection areas, applicable in wireless sensor networks.

Contribution

It proposes a novel curvature-multiplicity-guided adaptive sampling strategy integrated with quadtree partitioning, improving computational efficiency and accuracy over classical methods.

Findings

Achieves O(1/ε^{3/2}) computational complexity with O(ε) error bound.

Outperforms five classical methods in relative error on complex polygons.

Demonstrates robustness and practical applicability in sensor network coverage estimation.

Abstract

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/{\epsilon}3/2) computational complexity while maintaining an O({\epsilon}) error bound, improving upon the O(1/{\epsilon}2) complexity of classical Monte Carlo and uniform grid methods for the same error tolerance {\epsilon}. 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.