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Visibility Maps of Segments and Triangles in 3D

  • Esther Moet
  • Christian Knauer
  • Marc van Kreveld
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3980)

Abstract

Let T be a set of n disjoint triangles in three-dimensional space, let s be a line segment, and let t be a triangle, both disjoint from T. We consider the visibility map of s with respect to T, i.e., the portions of T that are visible from s. The visibility map of t is defined analogously. We look at two different notions of visibility: strong (complete) visibility, and weak (partial) visibility. The trivial Ω(n 2) lower bound for the combinatorial complexity of the strong visibility map of both s and t is almost tight: we prove an O(n 2 log n) upper bound for both structures. Furthermore, we prove that the weak visibility map of s has complexity Θ(n 5), and the weak visibility map of t has complexity Θ(n 7). If T is a polyhedral terrain, the complexity of the weak visibility map is Ω(n 4) and O(n 5), both for a segment and a triangle. We also present efficient algorithms to compute all discussed structures.

Keywords

Algebraic Curf Quadratic Number Constant Degree Sweep Algorithm Aspect Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Esther Moet
    • 1
  • Christian Knauer
    • 2
  • Marc van Kreveld
    • 1
  1. 1.Department of Information and Computing SciencesUniversiteit UtrechtUtrechtThe Netherlands
  2. 2.Institute of Computer ScienceFreie Universität BerlinBerlinGermany

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