Chapter 7 A survey of computational geometry [chapter]

Joseph S.B. Mitchell, Subhash Suri
1995 Handbooks in Operations Research and Management Science  
1 2 J. Mitchell and S. Suri incidences between a set of lines and a set of points, and counting the number of lines that bisect a set of points. Both elds seem to have bene ted from each other: combinatorial bounds for certain structures have been obtained by analyzing an algorithm that enumerates them and, conversely, the analysis of algorithms often depends crucially on the combinatorial bound on some geometric objects. The eld of computational geometry has also bene ted from its interactions
more » ... with other disciplines within computer science such as VLSI, database theory, robotics, computer vision, computer graphics, pattern recognition and learning theory. These areas o er a rich variety of problems that are inherently geometrical. Due to its interconnections with many applications areas, the variety of problems studied in computational geometry is truly enormous. Our goal is this paper is quite modest: we survey state of the art in some selected areas of computational geometry, with a strong bias towards problems with an optimization component. In the process, we also hope to acquaint the reader with some of the fundamental techniques and structures in computational geometry. Our paper has seven main sections. The survey proper begins in Section 3, while Section 2 introduces some foundational material. In particular, we brie y describe ve key concepts and fundamental structures that permeate much of computational geometry, and therefore are somewhat essential to a proper understanding of the material in later sections. The structures covered are convex hulls, arrangements, geometric duality, Voronoi diagram, and point location data structures. The main body of our survey begins with Section 3, where we describe four popular geometric graphs: minimum and maximum spanning trees, relative neighborhood graphs, and Gabriel graphs. Section 4 is devoted to algorithms in path planning. The topic of path planning is a vast one, with problems ranging from nding shortest paths in a discrete graph to deciding the feasible motion of a complex robot in an environment full of complex obstacles. We brie y mention most of the major developments in path planning research over the last two decades, but to a large extent limit ourselves to issues related to shortest paths in a planar domain. In Section 5, we discuss the matching and the traveling salesman type problems in computational geometry. Section 6 describes results on a variety of problems related to shape analysis and pattern recognition. We close with some concluding remarks in Section 7. In each section, we also pose what in our opinion are the most important and interesting open problems on the topic. There are altogether twenty open problems in this survey.
doi:10.1016/s0927-0507(05)80124-0 fatcat:nogmdqmv6bflni3waox5dxrsiy