Hierarchical Voronoi diagram-based path planning among polygonal obstacles for 3D virtual worlds

Path planning is one of important problems in virtual reality, computational geometry, robotic and GIS. This paper presents two methods of computing walkthrough paths in a polygon with h holes and a total of n vertices, for 3D virtual worlds such as virtual museums and games. For any two points s and t, one method is presented to compute the shortest Voronoi skeleton path SVSP(s,t) in O(k + h log h + log n) time, where k is the number of the Voronoi edges in SVSP(s,t) and O(k) ≤ O(n). An approximate shortest path from s to t can be computed in O(k + h log h + log n) time based on SVSP(s,t). The other is to compute the shortest path SP(s,t) in O(m + h log(n/h) + h² log h) time, where m is the number of the edges in SP(s,t) and m ≤ n. They are based on a hierarchical Voronoi diagram presented in this paper, which can be constructed in time O(n log n) with O(n) space in the preprocessing stage. This data structure also can be used to fast solve visibility information, collision detection and other problems in 3D virtual worlds. The methods are simpler and faster than other existed methods, with lower space.