Visibility queries in a polygonal region

  • Authors:

  • Rajasekhar Inkulu;Sanjiv Kapoor

  • Affiliations:

  • Institute for Computational Engineering and Sciences, The University of Texas at Austin, USA;Department of Computer Science, Illinois Institute of Technology, Chicago, USA

  • Venue:
  • Computational Geometry: Theory and Applications
  • Year:
  • 2009

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Abstract

In this paper, we consider the problem of computing the region visible to a query point located in a given polygonal domain. The polygonal domain is specified by a simple polygon with m holes and a total of n vertices. We provide two bounds on the complexity of this problem. One approach constructs a data structure with space complexity O(n^2) in time O(n^2lgn) and yields a query time of O((1+min(m,|V(q)|))lg^2n+m+|V(q)|). Here, V(q) represents the set of vertices of the visibility polygon of a query point q, and |E| denotes the number of edges in the visibility graph. The other approach provides a data structure with space complexity O(min(|E|,mn)+n) in time O(T+|E|+nlgn) with the query time of O(|V(q)|lgn+m). Here, T is the time to triangulate the given polygonal region (which is O(n+mlg^1^+^@em) for a small positive constant @e0). In both of these approaches, O(m) additive factor in the query time is eliminated with an additional O((min(|E|,mn))^2) space and an additional O(m(min(|E|,mn))^2) preprocessing time.