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International Journal of Applications of Fuzzy Sets Vol.1 ( June 2011), 39-44 http://eclass.teipat.gr/RESE-STE138/document Bertrand Curves in the Sense of Fuzzy Logic Zühal Küçükarslan, Münevver Yıldırım Yılmaz, Mehmet Bektaş e-mail: [email protected], e-mail: [email protected], e-mail: [email protected]. Abstract In this paper, we give information about the fuzzy curves. We introduce the notion of fuzzy Bertrand curves by using ☺ level set and obtain characterizations. Keywords: Fuzzy sets; Fuzzy geometry; Curves; Differential Geometry. 1 Introduction Fuzzy sets theory was introduced by Zadeh in 1965. A fuzzy set is characterized by a membership function which assigns to each object a grade of membership ranging between zero and one. One of the main goals in the fuzzy sets theory is to give fuzzification of different classical mathematical structures and to study properties of these fuzzy objects. Many scientists have studied fuzzy sets theory by different aspects for many years. Also, the concepts of fuzzy set, fuzzy logic, fuzzy number, fuzzy topology and fuzzy geometry were studied. The notion of Bertrand curves was introduced by J. Bertrand in 1850. Bertrand curve is a curve which shares the normal line with another curve in Euclidean 3-space. These curves have an important role in differential geometry of curves. There are many works related with Bertrand curves in different areas, [1,2,16]. The study of curves and their characterizations have not been considered in fuzzy set theory yet. In this regard, the aim of this work is to give a contribution to a kind of fuzzy space curves in terms of the fuzzy Frenet frame. In this paper, firstly we give the basic notions and properties of a fuzzy Frenet curve by using the de Blasi differential. Subsequently we study fuzzy Bertrand curves. 39 2 Preliminaries Let denote the family of all nonempty compact convex subsets of . Addition and scalar multiplication are defined as usual in Moreover, let be a compact interval defined by , where (i) u is normal i.e. there exist an given by , (ii) u is fuzzy convex, (iii) u is upper semicontinuous, (iv) is compact. The level set of a fuzzy set u is given by for each Hence from (i)-(iv) it follows that level set Furthermore its support is defined by , for each . Hausdorff metric is defined as follows , where , [7]. and Definition 2.1 For be the , is called a fuzzy vector such that level set, respectively. . Let . The length of is defined by , [15]. Definition 2.2 For defined as follows level set , semi- inner product of u,v is , [3]. Definition 2.3 A mapping is the de Blasi differentiable at , if there exists an upper semi-continuous, homogeneous mapping such that 1 d ∞ [ F (t 0 + ∆t ), F (t 0 + Dto (∆t )] = 0 || ∆t || The mapping is called the de Blasi differential of Let be a fuzzy function. Then for all lim|||∆t|| → 0 . the level set mapping 40 defined by the de Blasi differentiable . The level set mapping is . We shall also along in this work, [4]. Remark 2.4 From the above definition follows that if is Hukuhara differentiable for all multivalued mapping , where denotes the Hukuhara derivative of is differentiable then the and , [7]. Proposition 2.5 (i) If is Hukuhara differentiable at , then is nondecreasing in for all . (ii) If the level set mapping exist Hukuhara differences, there exist the de Blasi differences. The converse is not always true, [4]. Definition 2.6 Let curve if level set be a differentiable functions. is called a fuzzy is defined by , for all . be a fuzzy curve with Definition 2.7 Let , the velocity vector of at . Then, for all is defined by ). Definition 2.8 be Let a fuzzy curve. Assume is a fuzzy Frenet frame in the - fuzzy curvature function is called Theorem 2.9 Let the fuzzy curve , for - for . Then, for is defined as follows , where that , fuzzy curvature. denote the moving 3-fuzzy Frenet frame along . The Frenet formulae are given by the following equations 41 (2.1) is called the where 3 - fuzzy Ferenet curvature. Fuzzy Bertrand Curves Definition 3.1 Let are be fuzzy curves. The fuzzy Frenet frames of denoted by and , respectively. If there exist a relationship between the fuzzy curves that the principal normal lines of coincide with the principal normal lines the corresponding points of the fuzzy curves, then curve and such at is called a fuzzy Bertrand is called as a fuzzy Bertrand partner curve of The pair of is said to be a fuzzy Bertrand pair. Theorem 3.2 Let be a fuzzy Bertrand pair. The distance between corresponding points of the fuzzy Bertrand partner curves is constant. Proof. From Definition 3.1 we can write (3.1) for some fuzzy functions By the de Blasi differentiating equation (3.1) , we obtain (3.2) Since This means that are linearly dependent, and from (3.2) we have is a nonzero constant, for . Then The following corollary can be easily obtained from the above theorem: Corollary 3.3 The distances between corresponding points of all α − level sets of the fuzzy Bertrand partner curves are equal . 42 Theorem 3.4 Let are fuzzy curves and curvature and the torsion of and only if Proof. Assume that the angle between are the . and be a fuzzy Bertrand pair if is then we get (3.3) By taking the de Blasi derivative of equation (3.3), we obtain Using the linear dependency of , for , we obtain (3.5) From equation (3.3) and (3.5), we get References [1] Balgetir H., Bektaş M., Ergüt M., (2004). Bertrand curves for nonnull curves in 3dimensional Lorentzian space: Hadronic Journal, 27, 229-236 . [2]Balgetir H., Bektaş M., Inoguchi,J., (2004). Null Bertand curves in Minkowski 3space and their characterizations: Note Di Matematica, 23, no.1, 7-13 . [3] Cho E. W., Kim Y. K., Shin C. S., (1995). Fuzzy semi-inner product space: J. Korea Soc. Math. Educ. Ser. B: Pure Appl. Math. 2, no. 2, 163-171. [4] Diamond P., Kloeden P., (1993) . Metric spaces of fuzzy sets: World Scientific . [5] El-Ghoul M., El-Deen M. R. Z., (1999) . On some local properties of fuzzy manifold and its folding: Le Matematiche Vol. LIV Fasc. II, pp. 201-209. 43 fuzzy manifolds: Fuzzy Sets and Systems [6] Feraro M., Foster D. H., (1993) . Vol. 54 , pp. 99-106 .301-317. [7] Kaleva O., (1987) . Fuzzy differential equations: Fuzzy Sets and Systems 24, 301-317 North-Holland. [8] Kaufmann A., Gupta M. M., (1991) . Introduction to fuzzy arithmetic: Van Nostrand Reinhold , New York. [9] Kühnel W., (2005) . Differential geometry curves- surfaces- manifolds: American Mathematical Society. [10] Lakshmikantham V., Mohopatra R. N., (2003) . 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