P2Seg: Distance query from point to segments

Abstract

Querying the nearest distance from a point to $n$ line segments in 2D is a textbook problem in computational geometry. This paper presents P2Seg, a novel algorithmic strategy that transforms the intricate problem into an accessible linear traversal. Our method precomputes a KD tree and a Voronoi diagram for the site collection $S$, where $S$ refers to the endpoints of all line segments. Obviously, for a query point $q$, the nearest site $s_i$ provides a crucial clue for pinpointing the nearest line segment, i.e., the pairing $(q,s_i)$ effectively reduces the search from $n$ line segments to a limited number, represented as $L(q,s_i)$. The key idea of this paper is driven by an insightful observation: if the ray $s_{i}q$ intersects with $s_i$’s Voronoi cell at a point, say $q^{\prime }$, then $L(q,s_i)$ is a subset of $L(q^{\prime },s_i)$. This suggests that preprocessing efforts can be substantially minimized by focusing solely on scenarios where the query point lies on the Voronoi edges, which are fundamentally one-dimensional. We further prove that the challenge of locating the nearest line segment from $L(q^{\prime },s_i)$ can be distilled down to a simple linear traversal. Testing on datasets of varying complexities shows that P2Seg significantly outperforms state-of-the-art techniques. For example, in scenarios involving 10K segments with an average length of 0.5, our method runs 2.2 times faster than P2M and 60 times faster than AABB, as illustrated in the teaser figure.