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The Visibility-Voronoi Complex


The Visibility-Voronoi Complex
Examples of a Visibility-Voronoi Diagrams

We introduce a new type of diagram called the \(\text{VV}(c)\)-diagram (the Visibility–Voronoi diagram for clearance \(c\)), which is a hybrid between the visibility graph and the Voronoi diagram of polygons in the plane. It evolves from the visibility graph to the Voronoi diagram as the parameter c grows from 0 to \(\infty\). This diagram can be used for planning natural-looking paths for a robot translating amidst polygonal obstacles in the plane. A natural-looking path is short, smooth, and keeps—where possible—an amount of clearance \(c\) from the obstacles. The \(\text{VV}(c)\)-diagram contains such paths.

We also propose an algorithm that is capable of preprocessing a scene of configuration-space polygonal obstacles and constructs a data structure called the VV-complex. The VV-complex can be used to efficiently plan motion paths for any start and goal configuration and any clearance value \(c\), without having to explicitly construct the \(\text{VV}(c)\)-diagram for that \(c\)-value.

The Visibility-Voronoi Complex

The preprocessing time is \(O(n^2 \log n)\), where \(n\) is the total number of obstacle vertices, and the data structure can be queried directly for any c-value by merely performing a Dijkstra search. We have implemented a CGAL-based software package for computing the \(\text{VV}(c)\)-diagram in an exact manner for a given clearance value, and used it to plan natural-looking paths in various applications.

Visibility-Voronoi Diagrams with different clearance values





  • Ron Wein, Jur van den Berg and Dan Halperin
    The Visibility–Voronoi Complex and Its Applications

    Computational Geometry: Theory and Applications, vol. 36(1): 66-87, 2007 [link] [bibtex]
    Preliminary versions appeared in:

    • In Proceedings of the 21st ACM Symposium on Computational Geometry (SoCG), pages 63-72, 2005
    • In Proceedings of the European Workshop on Computational Geometry (EWCG), pages 151-154, 2005
  • Ron Wein
    The Integration of Exact Arrangements with Effective Motion Planning
    Ph.D. Thesis, Tel Aviv University, March 2007 [pdf] [bibtex]
  • Jur van den Berg
    Path Planning in Dynamic Environments.
    Ph.D. Thesis, Utrecht University, The Netherlands, 2007


Ron Wein
Dan Halperin

Yair Oz - Webcreator


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