The visibility-Voronoi complex and its applications

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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( n2logn), 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. © 2006 Elsevier B.V. All rights reserved.

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Wein, R., Van Den Berg, J. P., & Halperin, D. (2007). The visibility-Voronoi complex and its applications. In Computational Geometry: Theory and Applications (Vol. 36, pp. 66–87). https://doi.org/10.1016/j.comgeo.2005.11.007

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