On kinetic Delaunay triangulations: A near-quadratic bound for unit speed motions

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Abstract

Let P be a collection of n points in the plane, each moving along some straight line at unit speed. We obtain an almost tight upper bound of O(n2+ε), for any ε > 0, on the maximum number of discrete changes that the Delaunay triangulation DT(P) of P experiences during this motion. Our analysis is cast in a purely topological setting, where we only assume that (i) any four points can be co-circular at most three times, and (ii) no triple of points can be collinear more than twice; these assumptions hold for unit speed motions.

Original languageEnglish
Article number25
JournalJournal of the ACM
Volume62
Issue number3
DOIs
StatePublished - 1 Jun 2015

Keywords

  • Algorithms
  • Combinatorial complexity
  • Computational geometry
  • Computing methodologies → computer graphics
  • Delaunay
  • Design
  • Discrete changes
  • Geometric arrangements
  • Kinetic data structures
  • Mathematics of computing → combinatorics; combinatorial algorithms
  • Moving points
  • Theory
  • Theory of computation → computational geometry
  • Triangulation
  • Voronoi diagram

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