Efficient visibility queries in simple polygons

  • Authors:

  • Prosenjit Bose;Anna Lubiw;J. Ian Munro

  • Affiliations:

  • School of Computer Science, Carleton University, 1125 Colonel By Drive, Ottawa, ON, Canada;Department of Computer Science, University of Waterloo, Waterloo, ON, Canada;Department of Computer Science, University of Waterloo, Waterloo, ON, Canada

  • Venue:
  • Computational Geometry: Theory and Applications
  • Year:
  • 2002

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Abstract

We present a method of decomposing a simple polygon that allows the preprocessing of the polygon to efficiently answer visibility queries of various forms in an output sensitive manner. Using O(n3 log n) preprocessing time and O(n3) space, we can, given a query point q inside or outside an n vertex polygon, recover the visibility polygon of q in O(log n + k) time, where k is the size of the visibility polygon, and recover the number of vertices visible from q in O(log n) time.The key notion behind the decomposition is the succinct representation of visibility regions, and tight bounds on the number of such regions. These techniques are extended to handle other types of queries, such as visibility of fixed points other than the polygon vertices, and for visibility from a line segment rather than a point. Some of these results have been obtained independently by Guibas, Motwani and Raghavan [18].