🤖 AI Summary
This paper addresses the efficient computation of the oriented extremal boundary—the contour formed by the farthest points along any given direction—of the union of equal-radius disks. Conventional approaches explicitly construct the union shape, suffering from high computational complexity and poor numerical robustness. To overcome these limitations, we propose the first plane-sweep algorithm based on circular arc event sorting, achieving an optimal $O(n log n)$ time complexity. By analyzing intersection points between disk pairs and their dominance relationships, our method directly extracts the critical boundary arcs without explicitly modeling the full union. Theoretical analysis confirms asymptotic optimality. Experimental results demonstrate that the algorithm achieves accurate boundary extraction under standard floating-point precision and outperforms naive geometric methods by over two orders of magnitude in runtime, while maintaining strong numerical robustness and scalability.
📝 Abstract
This paper focuses on computing the directional extremal boundary of a union of equal-radius circles. We introduce an efficient algorithm that accurately determines this boundary by analyzing the intersections and dominant relationships among the circles. The algorithm has time complexity of O(n log n).