%0 Journal Article
%T Parabolic Spiral Search Plan for a Randomly Located Target in the Plane
%A Mohamed Beltagy
%A Mohamed Abd Allah El-Hadidy
%J ISRN Mathematical Analysis
%D 2013
%R 10.1155/2013/151598
%X This paper addresses the problem of searching for a located target in the plane by using one searcher starting its motion from the point . The searcher moves along parabolic spiral curve. The position of the target has a known distribution. We show that the distance between the target position and the searcher starting point depends on the number of revolutions, where the complete revolution is done when . Furthermore, we study this technique in the one-dimensional case (i.e., when the searcher moves with linear search technique). It is desired to get the expected value of the time for detecting the target. Illustrative examples are given to demonstrate the applicability of this technique assuming circular normal distributed estimates of the target position. 1. Introduction Search problem dates back to World War II, and the works of Koopman [1¨C3] and Stone [5, 6] offer a classic treatise of this area from an operational research perspective. Many variants and extensions of this problem, in a wide variety of directions, have been presented in both the statistical and operations research literature since Koopman [1] solved this problem for the unidimensional case. This problem has been discussed under some specific hypotheses by Richardson [4]. However, as pointed out by Koopman [2, 3], there is so much complexity in real search and rescue missions that any statistical model can only reflect part of the real-life situation and ours are no exception. On search theory in general, Stone [5] has given a good account of various results presently available, with some informative examples, and also has provided a rigorous mathematical treatment of the subject, for both discrete and continuous cases. On the other hand, Stone [6] provided an overview of different areas in the development of search theory, which could be designated as classical, mathematical, algorithmic, and dynamic. Also, exhaustive surveys of works realized on this topic have been given by Iida [7] and Benkoski et al. [8]. Studying of searching problem in the plane had found considerable interest among researchers due to its wide applications in our life. Feinerman et al. [9] introduced some search trajectories to find the desert Ants Nearby Treasure. They used multiple searchers without communication between them in the plane. Edelsbrunner and Maurer [10] obtained the optimal solutions for the postoffice problem. The certain point location problems in two dimensions are derived via geometric transforms from an optimal solution for the search problem in three dimensions: find the first point hit
%U http://www.hindawi.com/journals/isrn.mathematical.analysis/2013/151598/