# Capturing the Shape of a Point Set with a Line Segment

arxiv（2024）

Abstract

Detecting location-correlated groups in point sets is an important task in a
wide variety of applications areas. In addition to merely detecting such
groups, the group's shape carries meaning as well. In this paper, we represent
a group's shape using a simple geometric object, a line segment. Specifically,
given a radius r, we say a line segment is representative of a point set P
if it is within distance r of each point p ∈ P. We aim to find the
shortest such line segment. This problem is equivalent to stabbing a set of
circles of radius r using the shortest line segment. We describe an algorithm
to find the shortest representative segment in O(n log h + h log^3 h) time.
Additionally, we show how to maintain a stable approximation of the shortest
representative segment when the points in P move.

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