Work

Counting Unit Circular Arc Intersections

Year: 2026

Type: preprint

Abstract: 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.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2602.01074

Author: Wang, Haitao

Topic: Computational Geometry and Mesh Generation

Subfield: Computer Graphics and Computer-Aided Design

Field: Computer Science

Domain: Physical Sciences

Open Access status: green

OpenAlex ID: W7127405692

Cites:

Cited by:

Related to: