# A Method to Construct Approximate Fuzzy Voronoi Diagram for Fuzzy Numbers of Dimension Two

### Abstract

In this paper, we propose an approximate "fuzzy Voronoi" diagram(FVD)for fuzzy numbers of dimension two (FNDT) by designing an extension ofcrisp Voronoi diagram for fuzzy numbers. The fuzzy Voronoi sites are defined asfuzzy numbers of dimension two. In this approach, the fuzzy numbers have a convexcontinuous differentiable shape. The proposed algorithm has two stages: in the firststage we use the Fortune’s algorithm in order to construct a "fuzzy Voronoi" diagramfor membership values of FNDTs that are equal to 1. In the second stage, we proposea new algorithm based on the Euclidean distance between two fuzzy numbers in orderto construct the approximate "fuzzy Voronoi" diagram for values of the membershipof FNDTs that are smaller than 1. The experimental results are presented for aparticular shape, the fuzzy ellipse numbers.### References

http://dx.doi.org/10.1007/BF01840357

[2] Mcallisterm, M.; Kirkpatrick, D.; Snoeyink, J. (1996); A compact piecewise-linear Voronoi diagram for convex sites in the plane, Discrete Comput. Geom., 15-73.

[3] Karavelas, M.; Yvinec, M. (2003); Voronoi diagram of convex objects in the plane, In Proc Europ. Symp. Algorithms, LNCS Springer, 337-348.

[4] Yap, C. K. (1987); O(n log n) Algorithm for the Voronoi Diagram of a Set of Simple Curve Segments, Discrete Comput. Geom., 2:365-393.

http://dx.doi.org/10.1007/BF02187890

[5] Arya, S.; Malamatos, T. (2002); Linear-size approximate Voronoi diagrams, In: 13th Annual ACM-SIAM Symp. on Discrete algorithms, Society for Industrial and Applied Mathematics, 147-155.

[6] Emiris, I.; Hemmer, M.; Tsigaridas, E.; Tzoumas, G. (2008); Voronoi diagram of ellipses: CGAL-based implementation, ACS-TR-363603-01 Technical Report, University of Groningen.

[7] Aurenhammer, F.; Edelsbrunner, H.; An optimal algorithm for constructing the weighted Voronoi diagram in the plane, Pattern Recognition, 17(2): 251-257.

http://dx.doi.org/10.1016/0031-3203(84)90064-5

[8] Burnikel, C.; Mehlhorn, K.; Schirra, S. (1994); How to Compute Voronoi Diagram of Line Segments: Theoretical and Experimental Results, In Proc. 2nd Annual Symposium, Lecture Notes of Computer Science, 855:227-237.

[9] Buckley, J.J. ; Eslami, E. (1997); Fuzzy plane geometry II: Circles and polygons,Fuzzy Sets and Systems, 87:79-85.

http://dx.doi.org/10.1016/S0165-0114(96)00295-3

[10] Jooyandeh, M.; Mohades, A.; Mirzakah, M. (2009); Uncertain Voronoi Diagram, Information Processing Letters, 109:709-712.

http://dx.doi.org/10.1016/j.ipl.2009.03.007

[11] Jooyandeh, M.; Khorasani, A.M. (2009); Fuzzy Voronoi Diagram, Advances in Computer Science and Engineering, 13th International CSI Computer Conference, CSICC 2008, 6:82-89.

[12] Chaudhuri, B. B. (1991); Some shape definitions in fuzzy geometry of space, Pattern Recognition Letters, 12: 531-535.

http://dx.doi.org/10.1016/0167-8655(91)90113-Z

[13] Goetschel, R.; Voxman, W. (1986); Elementary fuzzy calculus, Fuzzy sets and systems, 18:31-43.

http://dx.doi.org/10.1016/0165-0114(86)90026-6

[14] Takahashi, O.; Schilling, R.J. (1989); Motion Planning in Plane Using Generalized Voronoi Diagrams, IEEE Transactions on Robotics and Automation, 5(2):143-150.

http://dx.doi.org/10.1109/70.88035

[15] Razavi, S.H.; H. Amoozad, H.; Zavadskas, E.K. ; Hashemi, S.S. (2013); A Fuzzy Data Envelopment Analysis Approach based on Parametric Programming, International Journal of Computers Communications & Control, ISSN 1841-9836, 8(4):594-607.

[16] Kaufmann, A.; Gupta, M.M. (1984); Introduction to Fuzzy Arithmetic: Theory and Applications, an Nostrand Reinhold Company, NY.

[17] Chakraborty, C.; Chakraborty, D. (2006); A theoretical development on a fuzzy distance measure for fuzzy numbers, Mathematical and Computer Modelling, 43:254-261.

http://dx.doi.org/10.1016/j.mcm.2005.09.025

[18] Grzegorzewski, P. (1998); Metrics and orders in space of fuzzy numbers, Fuzzy Sets and Systems, 97:83-94.

http://dx.doi.org/10.1016/S0165-0114(96)00322-3

[19] Woxman, W. (1998); Some remarks on distance between fuzzy numbers, Fuzzy Sets and Systems, 100:353-365.

http://dx.doi.org/10.1016/S0165-0114(97)00090-0

**INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL**, [S.l.], v. 9, n. 4, p. 389-396, june 2014. ISSN 1841-9844. Available at: <http://univagora.ro/jour/index.php/ijccc/article/view/31>. Date accessed: 07 july 2020. doi: https://doi.org/10.15837/ijccc.2014.4.31.

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