In this chapter, recent near-shortest path-planning algorithms with O(nlog n) in the quadric plane based on the Delaunay triangulation, Ahuja-Dijkstra algorithm, and ridge points are reviewed. The shortest path planning in the general three-dimensional situation is an NP-hard problem. The optimal solution can be approached under the assumption that the number of Steiner points is infinite. The state-the-art method has at most 2.81% difference on the shortest path length, but the computation time is 4216 times faster. Compared to the other O(nlog n) time near-shortest path approach (Kanai and Suzuki, KS’s algorithm), the path length of the Delaunay triangulation method is 0.28% longer than the KS’s algorithm with three Steiner points, but the computation is about 31.71 times faster. This, however, has only a few path length differences, which promises a good result, but the best computing time. Notably, these methods based on Delaunay triangulation concept are ideal for being extended to solve the path-planning problem on the Quadric surface or even the cruise missile mission planning and Mars rover.
Part of the book: Advanced Path Planning for Mobile Entities