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

## Authors

• Dragos Arotaritei Grigore T. Popaâ€ University of Medicine and Pharmacy

## Keywords:

approximated fuzzy Voronoi diagram, fuzzy numbers of dimension two, computational geometry, fuzzy arithmetic, bisector median, path planning

## Abstract

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

## References

Fortune, S. (1887); A sweep algorithm for Voronoi Diagrams, Algoritmica, 2:153-174. http://dx.doi.org/10.1007/BF01840357

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

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

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

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.

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.

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

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.

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

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

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.

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

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

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

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.

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

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

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

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

2014-06-15

## Section

Articles

### Most read articles by the same author(s)

Obs.: This plugin requires at least one statistics/report plugin to be enabled. If your statistics plugins provide more than one metric then please also select a main metric on the admin's site settings page and/or on the journal manager's settings pages.