Spherical Images and Characterizations of Time-like Curve According to New Version of the Bishop Frame in Minkowski 3-Space

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1 Prespacetime Journal January 016 Volume 7 Issue 1 pp Article Spherical Images and Characterizations of Time-like Curve According to New Version of the Umit Z. Savcı 1 Celal Bayar University, Department of Mathematics Education, 45900, Manisa-Turkey Abstract In this work, we introduce a new version of Bishop frame using a common vector field as binormal vector field of a regular timelike curve and call this fame as Type- Bishop frame in E 3 1. Thereafter, by translating type- Bishop frame vectors to O the center of Lorentzian sphere of three-dimensional Minkowski space, we introduce new spherical images and call them as type- Bishop spherical images in E 3 1. Serret-Frenet apparatus of these new spherical images are obtained in terms of base curves s type- Bishop invariants. Additionally, we express some interesting theorems and illustrate an example of our main results. Keywords: Timelike curve, spherical image, Minkowski space, Bishop frame, general helix, slant helix, Bertrand mate. 1 Introduction The local theory of space curves are mainly developed by Serret-Frene theorem which expresses the derivative of a geometrically chosen basis of R 3 by the aid of itself is proved. In resent years, in the same and different space, researchers treat same clasical topics in analogy with the curves in the Euclidean, Minkowski, Galilean and complex space, see [1,13,14,15,17,18]. Then it is observed that by the solution of some special ordinary differential equations, further classical topics, for instance spherical curves, Bertrand curves, Mannheim curves, constant breadth curves, slant helix, general helix, involutes and evolutes are investigated. One of the mentioned works is spherical images of a regular curve in the Euclidian space. It is a well-known concept in the local differential geometry of the curves. Such curves are obtained in terms of the Serret-Frenet vector fields [6]. Bishop frame, which is also called alternative or parallel frame of the curves, was introduced by L.R. Bishop in 1975 by means of parallel vector fields. Recently, many research papers related to this concept have been treated in the Euclidian space [4,11], in Minkowski space [5,7]. Bishop and Serret-Frenet frames have a common vector field, namely the tangent vector field of the Serret-Frenet frame. In this work, we made spherical image for timelike curve. In smilar to research has been studied by Yılmaz [16]. Spherical indicatrices have been examined by authers, see [13,14,15,16]. We introduce a new version of the Bishop frame for timelike curves in E1 3. We call it is Type- Bishop frame of regular curves. Thereafter, translating new frames vector fields to the center 1 Correspondence: ziyasavci@hotmail.com ISSN: Prespacetime Journal

2 Prespacetime Journal January 016 Volume 7 Issue 1 pp of unit sphere, we obtain new spherical images. We call them as Type- Bishop Spherical Image of regular curves. We also distinguish them by the names, Ω 1, Ω and binormal Bishop spherical images. Besides, we investigate their Serret-Frenet apparatus according to type- Bishop invariants. We establish some results of spherical images and illustrate an example of main results. Preliminaries The Minkowski three dimensional space E 3 1 is a real vector space R3 endowed with the standard flat Lorentzian metric given by g = dx 1 + dx + dx 3 where x 1, x, x 3 ) is rectangular coordinate system of E1 3. Since g is an indefinite metric. Let u = u 1, u, u 3 ) and v = v 1, v, v 3 ) be arbitrary an vectors in E1 3, the Lorentzian cross product of u and v defined by i j k u v = det u 1 u u 3 v 1 v v 3 Recall that a vector v E1 3 can have one of three Lorentzian characters: it can be spacelike if gv, v) > 0 or v = 0; timelike if gv, v) < 0 and nulllightlike) if gv, v) = 0 for v 0. Similarly, an arbitrary curve δ = δs) in E1 3 can locally be spacelike, timelike or null lightlike) if all of its velocity vector δ are respectively spacelike, timelike, or null lightlike), for every s I R. The pseudo-norm of an arbitrary vector a E1 3 is given by a = ga, a). The curve δ = δs) is called a unit speed curve if velocity vector δ is unit i.e, δ = 1. For vectors v, w E1 3 it is said to be orthogonal if and only if gv, w) = 0. Denote by {T, N, B} the moving Serret-Frenet frame along the curve δ = δs) in the space E1 3. For an arbitrary timelike curve δ = δs) in E1 3, the following Serre-Frenet formulea are given in as follows T 0 κ 0 T N = κ 0 τ. N.1.1) B 0 τ 0 B where < T, T >= 1, < B, B >=< N, N >= 1, T s) = δ s), Ns) = T s), Bs) = T s) Ns) and first curvature and second curvature κs), τs) respectively. κs) = δ, τs) = detδ, δ, δ ) κs) κ [8]. Denoted by {T, N 1, N } moving Type-1 Bishop frame along to timelike curve α : I R E 3 1 in the Minkowski 3-space E 3 1. For an arbitrary timelike curve αs) in 3-space E3 1, the following Type-1 Bishop formulea are given by T 0 k 1 k T N1 = k N 1 N k 0 0 N ISSN: Prespacetime Journal

3 Prespacetime Journal January 016 Volume 7 Issue 1 pp The relations between κ, τ, θ and k 1, k are given as follow κs) = k 1 + k, θs) = arctank k 1 ), k 1 0. So that k 1 and k effectively corrospond to cartasian coordinate system for the polar coordinates κ, θ with θ = τs)ds. The orientation of the parallel transport frame includes the arbitrary choice of integration constant θ 0, which disappears from τ due to the differentiation [7]. The Lorentzian sphere S 1 of radius r > 0 and with the center in the origin of the space E3 1 is defined by S 1 = { p = p 1, p, p 3 ) E 3 1 : gp, p) = r } Theorem.1.1: Let ϕs) be a unit speed timelike curve in E1 3. Then ϕ is a slant helix if and only if either one the next two functions κ τ κ ) 3/ τ κ ) or κ κ τ ) 3/ τ κ) is constant everywhere τ κ does not vanish. Theorem.1.: Let ϕs) be a timelike curve with curvatures κ and τ. The curve ϕ is a κ general helix if and only if = cons tan t. τ 3 Type- Bishop Frame of a Regular Curve in E 3 1 Theorem 3.1.1: Let α = αs) be timelike curve with a spacelike principal normal unit speed. If {Ω 1, Ω, B} is adapted frame, then we have defined by Type- Bishop Frame in E 3 1 Ω 1 Ω B = ξ ξ 0. Ω 1 Ω B 3.1.1) where gω 1, Ω 1 ) = 1, gω, Ω ) = gb, B) = 1, and gω 1, Ω ) = gω 1, B) = gω, B) = 0. If Ω 1 timelike Ω and B spacelike vectors. Thus we have equation 3.1.1) or shortly X = AX. Morever A is semi skew matrix where first curvature and ξ called second curvature of the curve, there the curvatures are defined by =< Ω 1, B >, ξ =< Ω, B >. 3.1.) Theorem 3.1.: Let {T, N, B} and {Ω 1, Ω, B} be Frenet ve Bishop frames, respectively. There exists a relation between them as T sinh θs) cosh θs) 0 Ω 1 N = cosh θs) sinh θs) 0. Ω B B where θ is the angle between the vectors N and Ω 1. Proof: We write the tangent vector according to frame {Ω 1, Ω, B} as T = sinh θs)ω 1 cosh θs)ω 3.1.3) ISSN: Prespacetime Journal

4 Prespacetime Journal January 016 Volume 7 Issue 1 pp and differentiate with respect to s T =κn= θ s) [cosh θs)ω 1 sinh θs)ω ] + sinh θs)ω 1 cosh θs)ω 3.1.4) Substituting Ω 1 = B and Ω = ξ B to equation 3.1.4), we get κn = θ s) [cosh θs)ω 1 sinh θs)ω ] +[sinh θs) cosh θs)ξ ]B 3.1.5) From equation 3.1.5) we get θs) = Arg tanh ξ, θ s) = κs), N = cosh θs)ω 1 sinh θs)ω, and T N B = sinh θs) cosh θs) 0 cosh θs) sinh θs) Ω 1 Ω B 3.1.6) Since there is a solition for θ satisyfing any initial condition, this show that localy relatively parallel normal fields exist. Besides equation 3.1.1) can also written as Taking the norm of both sides, we have B = τn = Ω 1 + ξ Ω 1 = τ = ξ ξ1 τ ) 3.1.7) ) ξ τ 3.1.8) and so by 3.1.5), we may express { = τs) cosh θs), ξ = τs) sinh θs) The frame {Ω 1, Ω, B} is properly oriented, and τ and θs) = s 0 κs)ds are polar coordinates for the curve α = αs). We shall call the set {Ω 1, Ω, B,, ξ } as type- Bishop invariants of the curve α = αs) in E New Spherical Images of a Regular Curve Let α = αs) be a regular timelike curve in E1 3. If we translate type- Bishop frame vectors to the center O of Lorentzian sphere of three-dimensional Minkowski space, we introduce new spherical images in E1 3. ISSN: Prespacetime Journal

5 Prespacetime Journal January 016 Volume 7 Issue 1 pp Ω 1 Bishop Spherical Image Definition 4.1.1: Let α = αs) be a regular timelike curve in E1 3. If we translate of the first vector field of type- Bishop frame to the center O of the unit sphere S1, we obtain a spherical image ϕ = ϕs ϕ ). This curve is called Ω 1 Bishop spherical image or indicatrix of the curve α = αs). Let ϕ = ϕs ϕ ) be Ω 1 Bishop spherical image of a regular timelikr curve α = αs). We shall investigate relations among type- Bishop and Serret-Frenet invariants. First, we differentiate ϕ = dϕ ds ϕ ds ϕ ds = B. Here, we shall denote differentiation according to s by a dash, and differentiation according to s ϕ by a dot. Taking the norm both sides the equation above, we have we differentiate 4.1.1) 1 as So, we have T ϕ = B ds ϕ ds = T ϕ = T ϕ ds ϕ ds = Ω 1 + ξ Ω. T ϕ = Ω 1 + ξ Ω. Since, we have the first curvature and principal normal of ϕ κ ϕ = T ϕ = ξ N ϕ = 1 κ ϕ Ω 1 + ξ Ω ) ) ) 4.1.) Cross product of T ϕ N ϕ gives us the binormal vector field of Ω 1 Bishop spherical image of α = αs) B ϕ = 1 ξ Ω 1 + Ω ) 4.1.3) κ ϕ Using the formula of the torsion, we write a relation τ ϕ = ) 7 ξ ξ ξ 1 ) 4.1.4) Considering equations 4.1.) 1 and 4.1.3), we give: Corollary 4.1.: Let ϕ = ϕs ϕ ) be Ω 1 Bishop spherical image of the curve α = αs). If the ratio of type- Bishop curvatures of α = αs) is constant ξ =constant, i.e.), then, the Ω 1 Bishop spherical indicatrix ϕ = ϕs ϕ ) is a circle in the osculating plane. ISSN: Prespacetime Journal

6 Prespacetime Journal January 016 Volume 7 Issue 1 pp Proof: Letϕ = ϕs ϕ ) be Ω 1 Bishop spherical image of a regular timelike curve α = αs). If the ratio of type- Bishop curvatures of α = αs) is constant, in terms of equations 4.1.) 1 and 4.1.4), we have κ ϕ =constant and τ ϕ = 0, respectively. Therefore, ϕ is a circle in the osculating plane. Theorem 4.1.3: Let ϕ = ϕs ϕ ) be Ω 1 Bishop spherical image of a regular curve α = αs). There exists a relation among Serret-Frenet invariants ϕs ϕ ) and type- Bishop curvatures of α = αs) as follow ξ = sϕ 0 κ ϕ ) τ ϕ ) 4 ds ϕ 4.1.5) Proof: Let ϕ = ϕs ϕ ) be Ω 1 Bishop spherical image of a regular timelike curve α = αs). Then equations 4.1.1) and 4.1.4) hold. Using 4.1.1) in 4.1.4) and by the chain rule, we have ) 7 d ) ξ dsϕ ds ϕ ds τ ϕ = ξ ξ ) 1 Substituting 4.1.) 1 to 4.1.6) and integrating both sides, we have 4.1.5) as desired. In the light of theorem.1.1 and.1., we express the following theorems without proofs. Theorem 4.1.4: Let ϕ = ϕs ϕ ) be Ω 1 Bishop spherical image of a regular curve α = αs). If ϕ is a general helix, then, type- Bishop curvatures of α = αs) a satisfy ) 5 ξ ) = constant. Theorem 4.1.5: Let ϕ = ϕs ϕ ) be Ω 1 Bishop spherical image of a regular curve α = αs). If ϕ is a slant helix, then, type- Bishop curvatures of α = αs) a satisfy ) ) 8 ξ ξ ξ1) 3 ξ 1 ξ ) 4 ξ1 =cons. [ [ ξ ) ) ] ] ξ1 ) 16 ξ Remark 4.1.6: Considering θ ϕ = sϕ 0 κ ϕ ds ϕ and using the transformation matrix, one can obtain the type- Bishop trihedral {Ω 1ϕ, Ω ϕ, B ϕ } of the curve ϕ = ϕs ϕ ). 4. Ω Bishop Spherical Image Definition 4..1: Let α = αs) be a regular curve in E1 3.If we translate of the second vector field of type- Bishop frame to the center of the unit sphere S1, we obtain a spherical image β = βs β ). This curve is called Ω Bishop spherical image or indicatrix of the curve α = αs). Let β = βs β ) be Ω Bishop spherical image of the timelike regular curve α = αs).we can write that β = dβ ds β ds β ds = ξ B. ISSN: Prespacetime Journal

7 Prespacetime Journal January 016 Volume 7 Issue 1 pp Similar to Ω Bishop spherical image, one can have T β = B ds β ds = ξ So, by differentiating of the formula 4..1) 1, we get or in other words Since, we express Cross product of T ϕ N ϕ gives us T β = T β ds β ds = Ω 1 + ξ Ω. T β = ξ Ω 1 + Ω. ) κ β = T β = 1 ξ1 ξ N β = 1 κ β ξ Ω 1 + Ω ) 4..1) 4..) By the formula of the torsion, we have B β = 1 κ β Ω 1 + ξ Ω ) 4..3) τ β = ξ ) 7 ξ1 ξ ξ 1 ξ ) 4..4) In terms of equations 4..) 1 and 4..4), we may give: Corollary 4..: Let β = βs β ) be Ω Bishop spherical image of a regular timelike curve α = αs). If the ratio of type- Bishop curvatures of α = αs) is constant ξ =constant, i.e.), then, the Ω Bishop spherical indicatrix β = βs β ) is a circle in the osculating plane. Theorem 4..3: Let β = βs β ) be Ω Bishop spherical image of a regular timelike curve α = αs).then, there exists a relation among Serret-Frenet invariants of βs β ) and type- Bishop curvatures of α = αs) as follow = s β κ β ) τ β 0 ξ ξ ) 4 ds β Proof: Similar to proof of theorem 4.1.3, above equation can be obtained by equations 4..1) and 4..) 1 and 4..4). In the light of theorem.1.1 and.1., we also give the following theorems for the curve β = βs β ) ISSN: Prespacetime Journal

8 Prespacetime Journal January 016 Volume 7 Issue 1 pp Theorem 4..4: Let β = βs β ) be Ω Bishop spherical image of a regular curve α = αs).if β is a general helix, then, type- Bishop curvatures of α = αs) a satisfy ) ξ ) 8 ξ1 ξ ξ ξ1) 3 = constant. Theorem 4..5: Let β = βs β ) be Ω Bishop spherical image of a regular curve α = αs).if β is a slant helix, then, type- Bishop curvatures of α = αs) a satisfy ) ξ ) 8 ξ ξ ξ ) 4 1 ξ ξ ξ1) 3 =const. [ [ ξ ) ) ] ] ξ ) 16 ξ1 ξ Remark 4..6: Considering θ ϕ = s β 0 κ β ds β and using the transformation matrix, one can obtain the type- Bishop trihedral {Ω 1β, Ω β, B β } of the curve β = βs β ). 4.3 Binormal Bishop Spherical Image Definition 4.3.1: Let α = αs) be a regular curve in E1 3. If we translate of the third vector field of type- Bishop frame to the center O of the unit sphere S1, we obtain a spherical image φ = φs φ ). This curve is called Binormal Bishop spherical image or indicatrix of the curve α = αs). Here, one question may come to mind about binormal spherical image, since, Serret-Frenet and type- Bishop frame in E1 3 have a common binormal vector field. Images of such binormal images are same as we shall demonstrate in section 6. But, here we are concerned with the Binormal Bishop spherical image s Serret-Frenet apparatus according to type- Bishop invariants. Let φ = φs φ ) be Binormal Bishop spherical image of a regular curve α = αs). One can differentiate of φ with respect to s φ = dφ ds φ ds φ ds = Ω 1 + ξ Ω. In terms of type- Bishop frame vector fields, we have tangent vector of the spherical image as follows T φ = Ω 1 + ξ Ω ξ 1 ξ ds φ ds = ξ ξ 1 In order to determine first curvature of φ, we write 4.3.1) T φ = P s)ω 1 + Q s)ω + P s) + Qs)ξ )B ISSN: Prespacetime Journal

9 Prespacetime Journal January 016 Volume 7 Issue 1 pp where P s) = ξ 1 ξ ξ1 and Qs) = ξ ξ ξ1. Since, we immediately arrive at κ φ = T φ = 4.3.) Qs)) + [P s) + Qs)ξ )] P s)) Therefore, we have the principal normal N φ = 1 κ φ {P s)ω 1 + Q s)ω + [P s) Qs)ξ )]}B 4.3.3) By the cross product of T φ N φ, we obtain the binormal vector field B φ = 1 κ φ {[P s)qs) Q s)ξ ]Ω 1 + [ P s) P s)qs)ξ ] Ω [ P s) ) ] Qs) B} P s) 4.3.4) where P s) = ξ 1 ξ 1 ξ and Qs) = ξ ξ 1 ξ. By means of obtained equations, we express the torsion of the Binormal Bishop spherical image τ φ = 1 κ { [3ξ ξ1 +ξ ξ )- φ ξ1 + ξ ) + ξ1 3 + ξ )] 4.3.5) + ξ [3ξ1 ξ1 +ξ ξ )) ξ 1 + ξ ) + ξ ξ )]} Consequently, we determined Serret-Frenet invariants of the Binormal Bishop spherical image according to type- Bishop invariants in E1 3. In terms of equations 4.3.) and 4.3.5), we have : Corollary 4.3.: Let φ = φs φ ) be binormal Bishop spherical image of a regular curve α = αs). If the ratio of type- Bishop curvatures of α = αs) is constant =constant, i.e), ξ then, Binormal Bishop spherical image φs φ ) is a circle in the osculating plane. Remark 4.3.3: Considering φ ϕ = s φ 0 κ φ ds φ and using the transformation matrix, one can obtain the type- Bishop trihedral {Ω 1φ, Ω φ, B φ } of the φ = φs φ ). ISSN: Prespacetime Journal

10 Prespacetime Journal January 016 Volume 7 Issue 1 pp Main Results Theorem 5.1.1: Let α = αs) be a regular timelike curve in 3-dimensional Minkowski space. Both of Ω 1, Ω and B spherical image of α. Both of Ω 1 and Ω spherical images of α are spherical involutes for binormal spherical image of α. Proof: Let us denote the tangent vectors of Ω 1 and Ω spherical images as T ϕ, T β and T φ respectively. These tangent vectors are given in 4.1.1) 1,4..1) 1 and 4.3.3) 1. T ϕ = B T β = B T φ = Ω 1 + ξ Ω ξ 1 ξ where {Ω 1, Ω, B} is type- Bishop frame and and ξ are Bishop curvatures of α. If the inner products are calculated, we get < T ϕ, T φ >= 0 < T β, T φ >= 0 The tangent vectors of Ω 1 and Ω spherical images is orthogonal to tangent vectors of binormal spherical images. So the proof is completed. Theorem 5.1.: Let α = αs) be a regular timelike curve in 3-dimensional Minkowski space. Both of Ω 1, Ω and B spherical images of α. Binormal vector of Ω 1 are orthogonal to normal vector Ω. Proof: Let us denote the binormal vectors of Ω 1 and principal normal vector of Ω, B ϕ and N β respectively. From 4.1.3), 4..) this vectors are given B ϕ = 1 κ ϕ ξ Ω 1 + Ω ) N β = 1 κ β ξ Ω 1 + Ω ) If the Lorentzian inner product of B ϕ and N β are calculated, we get < B ϕ, N β >= 0 It can be seen that, Binormal vector of Ω 1 and normal vector Ω are perpendicular. 6 Example In this section, first we show how to find a regular curve s type- Bishop trihedral and thereafter illustrate one example of new spherical images in E 3 1. Theorem 6.1.1: Next, let us consider the following unit speed curve ws) of E 3 1 by w = ws) = sinh s, cosh s, s) ISSN: Prespacetime Journal

11 Prespacetime Journal January 016 Volume 7 Issue 1 pp It is rendered in figure 1. And this curves s curvature function is expressed as in E 3 1 {κs) = The Serret-Frenet frame of the w = ws) may be written by the aid Mathematical program as follows T = cosh s, sinh s, 1) N = sinh s, cosh s, 0) B = cosh s, sinh s, ) θs) = s 0 ds = s Using transformation matrix equation 3.1.6) we get w = ws) and tangent, normal, binormal spherical images of unit speed curve with respect to Serret-Frenet frame. respectively Fig 1, a, b, c.we have type- Bishop spherical images of the unit speed curve w = ws),see figures 3a, 3b, 3c Ω 1 = sinh θ cosh s + cosh θ sinh s, sinh θ sinh s + cosh θ cosh s, sinh θ) Ω = cosh θ cosh s + sinh θ sinh s,, cosh θ sinh s + sinh θ cosh s, cosh θ) B = cosh s, sinh s, ) where θ = s. F ig.1 ISSN: Prespacetime Journal

12 Prespacetime Journal January 016 Volume 7 Issue 1 pp F ig.a F ig.b F ig.c ISSN: Prespacetime Journal

13 Prespacetime Journal January 016 Volume 7 Issue 1 pp F ig.3a F ig.3b F ig.3c ISSN: Prespacetime Journal

14 Prespacetime Journal January 016 Volume 7 Issue 1 pp REFERENCES [1] Ali, A.T. and Lopez, R. Slant helices in Minkowski space E 3 1, J. Korean Math. Soc. 48 1), , 011. [] Bishop, L. R. There is more than one way to frame a curve, Amer. Math. Monthly 8 3), 46-51, [3] Barros, M., Ferrandez, A., Lucas, P. and Merono, M. A. General helices in three dimensional Lorentzian space forms, Rocky Mountain J. Math. 31 ), , 001. [4] Bükcü, B. and Karacan, M.K. On the slant helices according to Bishop frame of the timelike curve in Lorentzian space, Tamkang J. Math. 39 3), 55-6, 008. [4] Bükcü, B. and Karacan, M.K. The slant helices according to Bishop frame, Int. J. Math.Comput. Sci. 3 ), 67-70, 009. [5] Bükcü, B. and Karacan, M.K. Special Bishop motion and Bishop Darboux rotation axis of the timelike curve in Minkowski 3-space, Kochi J.Math. 4, , 009. [6] do Carmo, M. P. Differential Geometry of Curves And Surfaces Prentice Hall, Englewod Cliffs, NJ, 1976). [7] Karacan, M.K. and Bükcü, B. Bishop frame of the timelike curve in Minkowski 3-space, S.D.Ü. Fen-Edebiyat Fak. Fen Dergisi e-dergi) 3 1), 80-90, 008. [8] O Neil, B. Elemantery Differential Geometry Academic Press Inc., New York, 1966). [9] Pekmen, U. and Pasalı, S. Some characterizations of Lorentzian Spherical spacelike curves, Math. Morav. 3, 33-37, [10] Petroviç- Torgasev, M. and Sucuroviç, E. Some characterizations of the spacelike, the timelike and Null Curves on the Pseudohyperbolic space H0 in E3 1, Kraguevac J. Math., 71-8, 000. [11] Yılmaz, S. and Turgut, M. A new version of Bishop Frame and An Application to Spherical Images, J. Math. Anal. Appl. 371 ), , 010. [1] Yılmaz S. Contributions to differential geometry of isotropic curves in the complex space, J. Math. Anal. Appl. 374, , 011. [13] Yılmaz S., Özyılmaz E., Yaylı Y., Turgut M. Tangent and trinormal spherical images of a time-like curve on the pseudohyperbolic space H 3 0. Proceedings of the Estonian Academy of Sciences, 59 3), 16 4, 010. [14] Yılmaz S., Özyılmaz E., Turgut M. New spherical indicatrices and their characterizations. Analele Stiintifice ale Universitatii Ovidius Constanta. 18 ), , 010. [15] Yılmaz S. Bishop spherical images of a spacelike curve in Minkowski 3-space, International Journal of Physical Sciences, 5 6), , 010. [16] Yılmaz S. and Turgut M. A new version of Bishop frame and an application to spherical images, Journal of Mathematical Analysis and Applications, 371 ), , 010. [17] Yılmaz S., Turgut M. Some characterizations of isotropic curves in the Euclidean space, Int. J. Comput. Math. Sci. ), , 008. [18] Yılmaz S., Savcı Ü. Z., Turgut M. Characterizations of curves of constant breadth in Galilean 3-space G3, Journal of Advanced Research in Pure Mathematics, Vol 6 1), 19-4, 014. ISSN: Prespacetime Journal