Computational geometry in two and a half dimensions

In this thesis, we study computational geometry in two and a half dimensions. These so-called polyhedral terrains have many applications in practice: computer graphics, navigation and motion planning, CAD/CAM, military surveillance, forest fire monitoring, etc.We investigate a series of fundamental problems regarding polyhedral terrains and present efficient algorithms to solve them. We propose an O(n) time algorithm to decide whether or not a geometric object is a terrain and an O(n log n) time algorithm to compute the shortest watchtower of a polyhedral terrain. We study the terrain guarding problem, obtain tight bounds for vertex and edge guards and O(n) algorithms to place these guards. We study the tetrahedralization of certain simple and non-simple polyhedra (which include some special classes of solid terrains) and present efficient algorithms to tetrahedralize them. We also investigate the problem of computing the $alpha$-hull of a terrain. Finally, we present efficient algorithms for the intersection detection and computation of Manhattan terrains