Given a set of $n$ circular arcs of the same radius in the plane, we consider the problem of computing the number of intersections among the arcs. The problem was studied before and the previously best algorithm solves the problem in $O(n^{4/3+ε})$ time [Agarwal, Pellegrini, and Sharir, SIAM J. Comput., 1993], for any constant $ε>0$. No progress has been made on the problem for more than 30 years. We present a new algorithm of $O(n^{4/3}\log^{16/3}n)$ time and improve it to $O(n^{1+ε}+K^{1/3}n^{2/3}(\frac{n^2}{n+K})^ε\log^{16/3}n)$ time for small $K$, where $K$ is the number of intersections of all arcs.

翻译：给定平面上 $n$ 条半径相同的圆弧，我们研究计算这些圆弧之间交点数量的问题。该问题先前已有研究，此前最佳算法可在 $O(n^{4/3+ε})$ 时间内解决问题 [Agarwal, Pellegrini, and Sharir, SIAM J. Comput., 1993]，其中 $ε>0$ 为任意常数。三十余年来该问题未取得进展。我们提出一种 $O(n^{4/3}\log^{16/3}n)$ 时间的新算法，并针对较小的 $K$ 将其改进为 $O(n^{1+ε}+K^{1/3}n^{2/3}(\frac{n^2}{n+K})^ε\log^{16/3}n)$ 时间，其中 $K$ 表示所有圆弧的交点总数。