Computational geometry

Computational geometry [Edelsbrunner 1987; Preparata and Shamos 1985] evolves from the classical discipline of design and analysis of algorithms. It is concerned with the computational complexity of geometric problems that arise in various disciplines such as pattern recognition, computer graphics, computer vision, robotics, VLSI layout, operations research, statistics, and so on. In contrast with the classical approach to proving mathematical theorems about geometry-related problems, this discipline emphasizes the computational aspect of these problems and attempts to exploit the underlying geometric properties possible, for example, the metric space, to derive efficient algorithmic solutions. An objective of this discipline in the theoretical context is to study the intrinsic difficulty of geometric computation under a certain computation model and to devise provably efficient algorithms for the problems at hand. The measure of intrinsic difficulty of a geometric problem or efficiency of an algorithm is in terms of time and space complexities when the problem size tends to `. This is the so-called asymptotic computational complexity. Traditionally there are two types of complexity analysis of the asymptotic behavior of an algorithm: worst-case analysis and average-case analysis. In the former, the behavior of the algorithm with respect to the worst possible case of the input instance is considered, whereas in the latter, the expected behavior is considered when the input is randomly drawn from a given distribution. Recently the notion of randomized algorithms has emerged as an important area of study within algorithm design and analysis. Similar to average-case analysis, the analysis of randomized algorithms is also probabilistic with the exception that in the latter the input instance is fixed whereas the steps taken by the algorithm are random and governed by the outcomes of some probabilistic event, for example, the toss of a coin. Due to its applications to various science and engineering related disciplines, researchers in computational geometry have begun to address the efficacy of the algorithms; the issues concerning robustness and numerical stability [Fortune 1993] and the actual running times of their implementations have now become centers of attention as well. We concentrate here on the theoretical development of this field in the context of sequential computation. We adopt the real RAM (random-access machine) model of computation in which all arithmetic operations, comparisons, kth-root, exponential, or logarithmic functions take unit time. For more details refer to [Preparata and Shamos 1985]. Those who are interested in the implementations or would like more information about available software may consult the Proceedings of the Annual ACM Symposium on Computational Geometry, which has a video session, or the WWW page on Geometry in Action