Computation and complexity of visibility in geometric environments

Abstract

In this thesis, we study visibility problems, which are geometric in nature. This leads us to the field of computational geometry. The main topic of this thesis is visibility amidst objects in two and three dimensional environments. The objectives of visibility research in computational geometry are to unveil the computational hardness of a variety of visibility problems, and to develop, if possible, efficient algorithmic solutions to these problems. The motivation for studying these problems comes mainly from three different application areas: robotics, geographical information systems, and computer graphics. We consider a variety of visibility problems, both in two dimensions and in three-dimensional space. We prove new upper and lower bounds for several visibility computations and data structures. Moreover, we derive improved bounds for a special set of environments which satisfy certain model properties. Finally, we perform experiments to evaluate the parameters of the aforementioned model. We studied one problem in the plane. We tried to find a discretization of a simple polygon to aid computations for guarding problems. Unfortunately, we could not find such a discretization for all simple polygons, and thus we did not obtain the result that we hoped for. In three dimensions, we studied visibility in two different geometric environments: polyhedral terrains and general sets of triangles in three-space. First, we give three algorithms to compute whether two sets of triangles on a polyhedral terrain are mutually visible (under varying definitions). Next, we present upper and lower bounds on the complexity of visibility maps for two different view elements, namely line segments and triangles. We also develop a realistic input model for polyhedral terrains, and use this model to improve complexity bounds of the visibility map and two distance structures. Finally, we consider the experimental verification of that realistic input model. We perform experiments with real-world data to determine how realistic the assumptions of the input model are, by examining what the values of the model parameters are in practice. We provide evidence that our model is indeed realistic.