The Visibility-Voronoi Complex

Examples of a Visibility-Voronoi Diagrams

 Abstract

We introduce a new type of diagram called the 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 ∞. 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 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 VV(c) -diagram for that c-value.

The preprocessing time is O(n² 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 VV(c) -diagram in an exact manner for a given clearance value, and used it to plan natural-looking paths in various applications.

Illustration

C=0	C>0	C=inifinity

Links

• 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:
  
  • Proceedings of the 21st ACM Symposium on Computational Geometry (SoCG), pages 63-72, 2005
  • Proceedings of the European Workshop on Computational Geometry (EWCG), pages 151-154, 2005
• Ron Wein's presentation
  The Visibility–Voronoi Complex and Its Applications
  Presented at the Symposium of Computational Geometry (SoCG), June 2005 [ppt]
• Ron Wein
  The Integration of Exact Arrangements with Effective Motion Planning
  PhD Thesis, Tel Aviv University, March 2007 [pdf] [bibtex]
• Jur van den Berg
  Path Planning in Dynamic Environments.
  PhD Thesis, Utrecht University, The Netherlands, 2007
  

Contact

Ron Wein
Dan Halperin