Computing approximately shortest descending paths on convex terrains via multiple shooting

Given a polyhedral terrain and two points p,q on the terrain, a path from p to q on the terrain is descending if z-coordinate of a point v never increases while we move v along the path from p to q. The problem of finding shortest descending paths on polyhedral terrains was posed by de Berg and van Kreveld (Algorithmica 18:306-323, 1997). In this paper, the multiple shooting approach proposed by Hoai et al. (J Comp Appl Math 317:235-246, 2017) is applied for approximately computing shortest descending paths on convex polyhedral terrains. Three factors of the approach consisting of surface partition, straightness condition, and update of shooting points are presented. We also show that if the straightness condition is satisfied then a local shortest descending path is obtained. Proposed algorithm is implemented in C++. Numerical results indicate that once a local solution is obtained it is close to a global one.