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Capturing the Shape of a Point Set with a Line Segment

Authors Nathan van Beusekom , Marc van Kreveld , Max van Mulken , Marcel Roeloffzen , Bettina Speckmann , Jules Wulms

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Nathan van Beusekom
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Marc van Kreveld
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands
Max van Mulken
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Marcel Roeloffzen
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Bettina Speckmann
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Jules Wulms
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands

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Nathan van Beusekom, Marc van Kreveld, Max van Mulken, Marcel Roeloffzen, Bettina Speckmann, and Jules Wulms. Capturing the Shape of a Point Set with a Line Segment. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 26:1-26:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.MFCS.2024.26

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 of n points 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 log³h) time, where h is the size of the convex hull of P. Additionally, we show how to maintain a stable approximation of the shortest representative segment when the points in P move.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Shape descriptor
  • Stabbing
  • Rotating calipers

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