Close Weighted Shortest Paths on 3D Terrain Surfaces

Abstract

This paper proposes an efficient method for the weighted region problem (WRP) on the surface of three-dimensional terrains. WRP is a classical path planning problem, asking for the minimum cost path between two given points crossing different regions in which each region is assigned a traversal cost per unit distance. Although WRP has been studied for decades, the exact solution for WRP, even in a two-dimensional environment, is unknown. Thus, the existing solutions for WRP are all approximations with decomposition-based and heuristic methods being the most widely-used in practice. However, when a very-close to optimal path is required, especially on real terrains with many regions, these approaches are not guaranteed or cannot return a satisfactory result in reasonable time. In this paper, we first present a new algorithm of finding a very-close optimal path, based on a user-defined parameter , between two points, crossing the surface of a sequence of regions in 3D, using Snell's law of physical refraction. We then show how to combine this algorithm with one existing decomposition-based method to compute a close optimal path over the whole terrain. In addition to a theoretical analysis, with an extensive set of test cases, the practicality and feasibility of our method are confirmed by that, our method always runs faster and returns closer to optimal paths in comparison with the existing ones.