Robust Geometric Computing and Optimal Visibility Coverage

1 2 Preface This book presents work in two of the most popular areas of geometric computation: Robust Geometric Computing and Optimal Visibility Coverage. Both areas have been extensively researched in recent years and attracted the attention of many researchers and developers in both the academic and the industrial worlds. The work we present here aims to contribute to this effort by providing efficient tools that can be used to solve important problems and challenges that arise in the areas specified above. We focus on both the theoretic and practical aspects of the problems we solve; we theoretically justify their efficiency and correctness and give experiential evidence, using applications that were developed for that purpose, to show their practicality and usefulness. Implementation of geometric algorithms is generally difficult because one must deal with both precision problems and degenerate input. While these issues are usually ignored when describing geometric algorithms in theory, overlooking them in practice often result in program crashes and incorrect results. Chapters 1, 2 and 3 are devoted to three algorithms that approximate arrangements of line segments in in order to make them more robust for further manipulation and computation. We describe the algorithms in detail, prove important properties that they hold and present extensive experimental results obtained with their implementation. In Chapter 1 we present an algorithm, Iterated Snap Rounding [52], to convert an arrangement of line segments into a more robust representation for algorithms that further manipulate the arrangement. More specifically, the algorithm converts each line segment into a polygonal chain of segments such that the vertices of the new arrangement have integer coordinates. The main goal of the algorithm is to provide an efficient alternative to the well known Snap Rounding algorithm: While in a snap-rounded arrangement, the distance between a vertex and a non-incident edge can be extremely small compared with the width of a pixel in the grid used for rounding, Iterated Snap Rounding rounds the arrangement such that each vertex is at least half-the-width-of-a-pixel away from any non-incident edge. By doing so, Iterated Snap Rounding produces more robust results that are safer to further manipulate. We note that in Iterated Snap Rounding the deviation of the output arrangement from the input arrangement may be huge. In Chapter 2 we present another Snap Rounding variant, Iterated Snap Rounding with Bounded Drift [73, 75, 76], that comes to improve the output quality of Iterated …