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50 Int'l Conf. Foundations of Computer Science | FCS'11 | part, all its point set are visible from 2 3 on , because of polar sorting of points around 2 3, it is verified simply that a random sunflower polygon is inevitably generative by this proposed algorithm on any set of given points with number of assumption convex hull layers . 5 Conclusions In this paper, a heuristic algorithm with time complexity of was proposed for the generation of random sunflower polygon. This algorithm can be used to estimate many algorithms and geometric problems such as art gallery. Also it was proved that this algorithm has properly on any set of given points with intricate assumption convex hulls (with the number of layers ) and it generates a random sunflower polygon. 6 References [1] McDonald, B., Smith, I. “Partial symmetry breaking”; In: P. Van Hentenryck(Ed.), Proc. Of CP'02 LNCS2470, Springer-verlag, 431—445, 2002. [2] Auer, T., Held, M. “Heuristics for the Generation of Random Polygons”; Proc. 8 th Canadian on Computational Geometry(CCCG'96), 38—41, 1996. [3] Zhu, C., Sundaram, G., Soneyink, J., Mitchell, J.S.B. “Generating Random Polygon with Given Vertices”; Comput. Geom. Theory and Appl., Vol 6, Issues 5, 277— 290, 1996. [4] Sohler, C. “Generating random star-shaped polygons”; In: Proc. 11 th Canadian Conference on Computational Geometry(CCCG'99), 174—177, 1999. [5] Epstein, P. “Generating Geometric Objects at Random”; Master's thesis, CS Dept., Carleton University Ottawa K1S5B6, Canada, 1992. [6] O'rourke, J., Virmani, M. “Generating Random Polygons”; Technical Report 011, CS Dept. Smith Colege, Northhampton, MA 01063 1991. [7] Berg, M.D. “Computational Geometry Algorithms and Applications”; 3 rd , Published by springer-verlag, 2008. [8] Michael, T.S. “How to Guard an Art Gallery and other Discerete Mathematical Adventures”; the Book of the Johns Hobkins University Press, printed in the United Stetes of American, 2009.
Int'l Conf. Foundations of Computer Science | FCS'11 | 51 The Generation of Pseudo-Triangulated Spiral Polygon Using Convex Hull Layers F. Taherkhani 1 , A. Nourollah 1,2 1 Department of Computer Engineering & IT, Islamic Azad University, Qazvin, Iran 2 Department of Electrical & Computer Engineering, Shahid Rajaee Teacher Training University, Tehran, Iran Abstract - The generation of random simple polygon and the pseudo-triangulation of a polygon are regarded as the proposed problems in computational geometry. The production of a random polygon is used in the context of the consideration of the accuracy of algorithms. In this paper, a new algorithm is presented to generate a simple spiral polygon on a set of random points S in the plane using convex hull layers in a way that pseudo-triangulation is also performed on it simultaneously. The new algorithm can be done in O(nlogn) time, so it is considered as one of the optimal algorithms. Keywords: simple spiral polygon, pseudo-triangulation, convex hull layers, convex and concave chain 1 Introduction Polygons are suitable shapes to demonstrate the objects of real world and every object in nature is demonstrable as a set of polygons. The generation of random simple polygons has two main areas of application: a) testing the correctness and b) evaluating the CPU-time consumption of algorithms that operate on polygons. The generation of random geometric objects has received some attention by researchers. Epstein studied the uniformly random generation of polygon triangulation [2]. Polygon pseudo-triangulation is a generalized form of polygon triangulation. The names pseudo-triangle and pseudotriangulation were coined by Pocchiola and Vegter in 1993[3]. A pseudo-triangle is a simple polygon with exactly three convex vertices, called corners and three concave chains of edges joining the corners [4]. Let S be a set of random n points p0,…, pn-1 in the plane. The goal is generation of a random simple polygon with a uniform distribution. A uniformly random polygon on S is a polygon generated with probability of 1/k if there exist k simple polygons on S in total [2, 5]. Since the generation of a polygon from a set of random points is frequently used to consider the performance of proposed algorithms in the context of polygons such as the Art gallery problem; therefore, the generation of similar polygons cannot show well the quality of the performance of the algorithms. Thus, we are looking for algorithms having the production ability of kinds of polygons with different structures, and up to now no solutions with the polynomial time to generate uniform random polygons have been known. The following subjects of this paper have been organized in this way: In section 2 the initial definitions are presented. In section 3 the generation manner of convex hull layers are expressed. In section 4 the suggested algorithm to generate pseudo-triangulated spiral simple polygon and the analysis of its time complexity is proposed. Finally in section 5, conclusion will be presented. 2 Preliminaries A sequence of line segments, such that the end of each one is the beginning of the following one, is referred as polygon and a polygon whose edges don’t intersect one another is called simple polygon. A simple polygon is called a convex polygon when all the internal angles are less than . According to this definition, the set of points S on a plane is called convex if and only if in exchange for both the points p,q∈ S, the line segment pq completely lies inside S (pq ⊆ S). The most applicable structure in robatic geometry is convex hull. Convex hull of the given points p0,…, pn-1 is the smallest convex set on the plane which contains the points. Let three points p1(x1, y1), p2(x2, y2) and p3(x3, y3) are given in the plane. Hence matrix A is defined as follows: Let det (A) refers to determinant of matrix A. Three cases can be occurred. (1) • Case a: det (A) > 0, Sequence p1, p2, p3 are counterclockwise (left turn).
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