The visibility–Voronoi complex and its applications

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 ( 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 VV ( c ) -diagram in an exact manner for a given clearance value and used it to plan natural-looking paths in various applications.