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Visibility Maps of Segments and Triangles in 3D

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Computational Science and Its Applications - ICCSA 2006 (ICCSA 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3980))

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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.

This research was initiated during a visit of the first author to Freie Universität Berlin, which was supported by a Marie-Curie scholarship.

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Moet, E., Knauer, C., van Kreveld, M. (2006). Visibility Maps of Segments and Triangles in 3D. In: Gavrilova, M., et al. Computational Science and Its Applications - ICCSA 2006. ICCSA 2006. Lecture Notes in Computer Science, vol 3980. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11751540_3

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  • DOI: https://doi.org/10.1007/11751540_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-34070-6

  • Online ISBN: 978-3-540-34071-3

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