Capturing the Shape of a Point Set with a Line Segment

TL;DR

The paper studies the problem of representing a point set $P$ by the shortest line segment $q_1q_2$ with all points within distance $r$, effectively a radius-$r$ stabbing problem that also encodes the shape of the set. It first solves the static problem via a fixed-orientation then full-rotation approach based on rotating calipers, reducing to convex-hull geometry and maintaining two convex envelopes $S_1,S_2$ to describe feasible endpoints. The main contributions are an $O(n \log h + h \log^3 h)$ static algorithm and a kinetic framework that yields a stable, approximately optimal segment for moving points, with endpoint speeds bounded by a linear function of $r$ and a constant-factor radius relaxation. These results advance previous subquadratic-time approaches and provide practical, stable shape descriptors suitable for visualization and dynamic data analysis in motion-rich settings.

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 $p \in P$. 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.

Figures (15)

• Figure 1: The line segment (blue) must hit every circle of radius $r$, centered at the points in $P$.
• Figure 2: Two extremal tangents $\tau_1$ and $\tau_2$ for horizontal orientation $\alpha$. The shortest line segment of orientation $\alpha$ that intersects all circles, ends at the boundary of the gray regions.
• Figure 4: Two convex sequences between $\tau_1$ and $\tau_2$. There are multiple points on the left convex sequence that have the same tangent as the right yellow vertex. Still, there is only one line segment in horizontal orientation for which the tangents of its endpoints are equal (blue).
• Figure 5: When $q_1$/$q_2$ is at a vertex of $S_1$/$S_2$, it stops moving.
• Figure 6: When the defining circle of $\tau_1$/$\tau_2$ changes, $\tau_1$/$\tau_2$ is parallel to a convex hull edge.

Theorems & Definitions (25)

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