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46 Int'l Conf. Foundations of Computer Science | FCS'11 | 5 Conclusions In this paper two new methods for pseudo-triangulation of the set of points S were put forward. The trend was in a way that first a layer was generated for the set of points S on the plane of the convex hull and then performing one of the two methods or a combination of them on the layers the act of pseudo-triangulation was done. The surveys showed that the pseudo-triangulation performed were minimum i.e. the number of the produced pseudo-triangles is n-2 pseudotriangle and the number of edges in it is the least possible amount i.e. 2n-3. 6 References [1] G. Rote, F. Santos, and I. Streinu, “Pseudo- Triangulation – a Survey,” Discrete Comput. Geom. 2007. [2] O. Aichholzer, F. Aurenhammer, H. Krasser, and B. Speckmann, “Convexity minimizes pseudo-triangulations,” Computational Geometry 28, pp.3-10, 2004. [3] I. Streinu, “A combinatorial approach to planar noncolliding robot arm motion planning,” In: Proc. 41st Annu.IEEE Sympos. Foundat. Comput.Sci. (FOCS'00), pp.443-453, 2000. [4] S. Gerdjikov, and A. Wolff, “Decomposing a simple polygon into pseudo-triangles and convex polygons,” Computational Geometry 41, pp.21-30, 2008. [5] M. Pocchiola, and G. Vegter, “Topologically sweeping visibility complexes via pseudo-triangulations,” Discrete Compute.Geom. 16, pp.419-453, 1996. [6] M.T. Goodrich, and R. Tamassia, “Dynamic ray shooting and shortest paths in planar subdivisions via balanced geodesic triangulations,” J. Algorithms 23 (1), pp.51-73, 1997. [7] D.G. Kirkpatrick, J. Snoeyink, and B. Speckmann, “kinetic collision detection for simple polygons,” Internat. J. Comput. Geom. Appl. 12(1-2), pp. 3-27, 2002. [8] B. Speckmann, and C.D. T th, “Allocating vertex - guard in simple polygons via pseudo-triangulations,” Discrete Comput. Geom. 33 (2), pp.345-364, 2005.
Int'l Conf. Foundations of Computer Science | FCS'11 | 47 Generating Sunflower Random Polygons on a Set of Vertices L.Mohammadi 2 , A. Nourollah 1 1 Department of Electrical & Computer Engineering of Islamic Azad University, Qazvin, Iran 2 Faculty of Electrical & Computer Engineering of Islamic Azad University, Qazvin, Iran & Faculty of Electrical & Computer Engineering of Shahid Rajaee Teacher Training University, Tehran, Iran Abstract - Generating random polygons problem is important for verification of geometric algorithms. Moreover, this problem has applications in computing and verification of time complexity for computational geometry algorithms such as Art Gallery. Since it is often not possible to get real data, a set of random data is a good alternative. In this paper, a heuristic algorithm is proposed for generating sunflower random polygons using time. Keywords: convex hull, sunflower random polygon, visibility 1 Introduction Computational geometry is a very important research field in computer science in which most computations are performed on known geometrical objects as polygons. Polygons are a convenient representation for many real-world objects; convenient both in that an abstract polygon is often an accurate model of real objects and in that it is easily manipulated computationally. Examples of their applications include representing shapes of individual letters for automatic character recognition, of an obstacle to be avoided in a robot's environment, or a piece of a solid object to be displayed on a graphic screen [7]. The generation of random geometrical objects has received some attention by researchers [2][3][6]. A challenge of these problems is the generation of random simple polygons. Since no polynomial time algorithm is known to solve the problem, researchers either try to use heuristic algorithms which don't have uniformed distribution or restrict the problem to certain classes of polygon such as monotone and star-shaped polygons [1][3][4]. The importance of geometric objects application is the simplicity of testing geometric algorithms. Since a set of data may become both too large and too hard to define for practical purposes, what one might do is to use randomly generated data that has a high probability to cover all the different classes of inputs. Thus, since practical data may not be available for testing, it is natural to test the algorithm on randomly input data. Polygons are one of the fundamental building blocks in geometric modeling and they are used to present a wide variety of shapes and figures in computer graphics, vision, pattern recognition, robotics and other computational fields. Some recent applications address uniformed random generation of simple polygons with given vertices, in the sense that a polygon will be generated with probability if there exist a total of simple polygons with such vertices. One of the important geometry problems in which polygons play an important rule, is art gallery whose purpose is guarding a polygonal art gallery with the least number of guards (cameras). A well-known kind of art gallery problem is sunflower art gallery. The proposed question is this: What is the smallest number of guards required to protect the Sunflower Art Gallery? Figur 1 shows a sunflower art gallery which is protected by 4 stationary guards. Some of the guards can not see through walls around corners of art gallery. Every point is visible at least one guard and it would be more economical to protect the gallery with fewer guards, if possible [5]. In this paper a heuristic algorithm is proposed for the generation of random sunflower polygons to estimate such problems. Fig. 1. The sunflower art gallery [5] The following sections of this paper have been organized in this way: section 2 has been allocated to related works. In section 3 the needed preliminary concepts are stated. In section 4 the proposed algorithm is posed for generating random sunflower polygon and its performance and accuracy are investigated and finally in section 6 the conclusion will be discussed.
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