Smarandache Curves and Spherical Indicatrices in the Galilean. 3-Space

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1 arxiv: v [math.dg 2 Jan 205, 5 pages. DOI:0.528/zenodo Smarandache Curves and Spherical Indicatrices in the Galilean 3-Space H.S.Abdel-Aziz and M.Khalifa Saad Dept. of Math., Faculty of Science, Sohag Univ., Sohag, Egypt Abstract. In the present paper, Smarandache curves for some special curves in the threedimensional Galilean space G 3 are investigated. Moreover, spherical indicatrices for the helix as well as circular helix are introduced. Furthermore, some properties for these curves are given. Finally, in the light of this study, some related examples of these curves are provided. Keywords. Smarandache curves, Spherical indicatrices, Frenet frame, Galilean space. MSC A35, 53B30. Introduction Discovering Galilean space-time is probably one of the major achievements of non relativistic physics. Nowadays Galilean space is becoming increasingly popular as evidenced from the connection of the fundamental concepts such as velocity, momentum, kinetic energy, etc. and principles as indicated in [7. In recent years, researchers have begun to investigate curves and surfaces in the Galilean space and thereafter pseudo-galilean space. A regular curve in Euclidean space whose position vector is composed by Frenet frame vectors on another regular curve is called a Smarandache curve. Smarandache curves have been investigated by some differential geometers [, 9. M. Turgut and S. Yilmaz have defined a special case of such curves and call it Smarandache T B 2 curves in the space E 4 [9. They have dealed with a special Smarandache curves which is defined by the tangent and second binormal vector fields. Additionally, they have computed formulas of this kind curves. In [, the author has introduced some special Smarandache curves in the Euclidean space. He has studied Frenet-Serret invariants of a special case. In this paper, we investigate Smarandache curves for the position vector of an arbitrary curve in terms of the curvature and the torsion with respect to standard frame. Besides, special Smarandache curves such as T N, T B and T NB according to Frenet frame in Galilean 3-space are studied. Furthermore, we find differential geometric properties of these special curves and we calculate curvatures natural curvatures of these curves. Our main result in this work is to study Smarandache curves for some special curves in the Galilean 3-space G 3. address: mohamed khalifa77@science.sohag.edu.eg

2 2 Preliminaries Let us recall the basic facts about the three-dimensional Galilean geometry G 3. The geometry of the Galilean space has been firstly explained in [. The curves and some special surfaces in G 3 are considered in [3. The Galilean geometry is a real Cayley-Klein geometry with projective signature 0, 0, +, + according to [5. The absolute of the Galilean geometry is an ordered triple w, f, I where w is the ideal absolute plane x 0 = 0, f is a line in w x 0 = x = 0 and I is elliptic 0 : 0 : x 2 : x 3 0 : 0 : x 3 : x 2 involution of the points of f. In the Galilean space there are just two types of vectors, non-isotropic xx, y, z for which holds x 0. Otherwise, it is called isotropic. We do not distinguish classes of vectors among isotropic vectors in G 3. A plane of the form x = const. in the Galilean space is called Euclidean, since its induced geometry is Euclidean. Otherwise it is called isotropic plane. In affine coordinates, the Galilean inner product between two vectors P = p, p 2, p 3 and Q = q, q 2, q 3 is defined by [4: { p q if p 0 q 0, P, Q G 3 = p 2 q 2 + p 3 q 3 if p = 0 q = 0. And the cross product in the sense of Galilean space is given by: 0 e 2 e 3 p p 2 p 3 ; if p 0 q 0, q q 2 q 3 P Q G 3 = e e 2 e 3 p p 2 p 3 ; if p = 0 q = 0. q q 2 q The Galilean sphere of radius d and center c of the space G 3 is defined by S 2 Gc, d = {X c G 3 : X c, X c G 3 = ±d 2 }. A curve αt = xt, yt, zt is admissible in G 3 if it has no inflection points αt αt 0. An admissible curve in G 3 is an analogue of a regular curve in Euclidean space. For an admissible curve α : I form dt = ds, given by G 3, I R parameterized by the arc length s with differential αs = s, ys, zs. 2.3 The curvature κs and torsion τs of α are defined by κs = α s = y s 2 + z s 2, τs = y Note that an admissible curve has non-zero curvature. sz s + y sz s κ s 2

3 The associated trihedron is given by Ts = α s =, y s, z s, Ns = α s κs Bs = 0, z 0, y s, z s =, κs s, y s. 2.5 κs For derivatives of the tangent T, normal N and binormal B vector field, the following Frenet formulas in the Galilean space hold [ From 2.5 and 2.6, we derive an important relation T N Let us consider the following definitions: B 0 κ 0 T = 0 0 τ N τ 0 B α s = κ sns + κsτsbs. Definition 2. As in the three-dimensional Euclidean space, an admissible curve in G 3, whose position vector is composed by Frenet frame vectors on another admissible curve is called a Smarandache curve [9. In the light of the above definition, we adapt it to admissible curves in the Galilean space as follows: Definition 2.2 let Γ = Γs be an admissible curve in G 3 and {T, N, B} be its moving Frenet frame. Smarandache TN, TB and TNB curves are respectively, defined by Γ TN = T + N T + N, Γ TB = T + B T + B, Γ TNB = T + N + B T + N + B. 2.7 Definition 2.3 A helix is a geometric curve with non-vanishing constant curvature κ and nonvanishing constant torsion τ [0 Definition 2.4 A family of curves with constant curvature but non-constant torsion is called Salkowski curves and a family of curves with constant torsion but non-constant curvature is called Anti-Salkowski curves [6. Definition 2.5 As in Euclidean 3-space, let α be an admissible curve in Galilean 3-space with Frenet vectors T, N and B. The unit tangent vectors along the curve α generate a curve α T on a Galilean 3

4 sphere of radius about the origin. The curve α T is called the spherical indicatrix of T or more commonly, α T is called tangent indicatrix of the curve α. If α = αs is a natural representation of α, then α T s T = Ts will be a representation of α T. Similarly one considers the principal normal indicatrix α N s N = Ns and binormal indicatrix α B s B = Bs [8. 3 Smarandache curves of special curves In this section, we consider the position vector of an arbitrary curve with curvature κs and torsion τs in the Galilean space G 3 which computed from the natural representation form as follows [2 [ [ [ [ [ r s = s, κs cos τsds ds ds, κs sin τsds ds ds. 3. Its moving frame is T s = N s = Bs = [ [ [ [ [, κs cos τsds ds, κs sin τsds ds, [ [ [ 0, cos τsds, sin τsds, [ [ 0, sin τsds, cos τsds. 3.2 Let us investigate Frenet invariants of Smarandache curves according to definition TN-Smarandache curve Let r s be a TN-Smarandache curve of rswritten as [ [, κs cos τsds ds + cos τsds, r s = [ [ κs sin τsds ds + sin τsds. 3.3 By differentiating 3.3 with respect to s, we get where T = Also, differentiating 3.4, we obtain [ [ [ 0, κs cos τsds τs sin τsds, κ 2 + τ 2 κs sin [ τsds + τs cos [ τsds κ 2 + τ 2 2 κ N =, 3.4 ds ds = κ 2 + τ κ 2 + τ 2 2 0, λ, λ 2, 3.6 then, the curvature and principal normal vector field of r s are respectively, κ = λ 2 + λ2 2, N = where 0, λ λ 2 + λ 2, λ 2,

5 λ = κ 3 τ κ 2 τ τ 3 κ + κ κτ [ sin + κ 2 τ 2 + τ 2 κ τ 4 τκτ [ cos λ 2 = κ 3 τ + κ 2 τ + τ 3 κ κ κτ [ cos + κ 2 τ 2 τ 2 κ + τ 4 + τκτ [ sin On the other hand, the binormal vector of this curve is expressed as τsds τsds, 3.8 τsds τsds. 3.9 B = κ 2 + τ 2 λ 2 + λ2 2 [ [ [ λ2 κs cos τsds τs sin τsds [ [ λ κs sin τsds + τs cos τsds, 0, 0, 3.0 it follows that τ = TB-Smarandache curve Let r 2 s 2 be a TB-Smarandache curve given by [ [ [, κs cos τsds sin τsds, r 2 s 2 = [ [ κs sin τsds + cos τsds. 3.2 Differentiating this equation with respect to s yields [ [ T 2 = 0, cos τsds, sin τsds, 3.3 where Again, differentiating 3.3 gives κ 2 = N 2 = ds 2 ds = κ τ. 3.4 τ κ τ, 3.5 [ [ 0, sin τsds, cos τsds. 3.6 From the above mentioned, one can obtain B 2 =, 0, From which τ 2 =

6 3.3 TNB-Smarandache curve By definition 2.2, the TNB-Smarandache curve is given by [ [ [ [, κs cos τsds + cos τsds sin τsds, r 3 s 3 = [ [ [ κs sin τsds + sin τsds + cos τsds. 3.9 Similarly, its tangent, the principal normal and the binormal vectors are respectively, [ [ [ 0, κ τ cos τsds τs sin τsds, T 3 = κ τ 2 + τ 2 κ τ sin [ τsds + τs cos [ τsds, 3.20 in which N 3 = ds 3 ds = κ τ 2 + τ 2, 3.2 0, µ cos [ τsds µ 2 sin [ τsds, µ 2 cos [ τsds + µ sin [ τsds µ 2 + µ 2 2, 3.22 where µ s = µ 2 s = µ 2 κ τ τµ B 3 = κ τ 2 + τ 2, 0, 0, 3.23 µ 2 + µ2 2 κ τ 2 + τ 2 κ τ τ 2 κ τ κ τ κ τ + τ τ, κ τ 2 + τ 2 τκ τ 2 + τ τ κ τ κ τ + τ τ The curvature and the torsion of this curve are respectively, given by κ 3 = µ 2 + µ 2 2 κ τ 2 + τ 2 2, τ 3 = Theorem 3. Let rs given by 3. be a helix βs in G 3 τ/κ = m = const.which can be written as βs = s, m [ sin m κsds ds, m [ cos m κsds ds, 3.26 then TN, TB and TNB-Smarandache curves of β are plane curves with constant curvatures. Proof. We prove this theorem for TN-Smarandache curve only. The straightforward computations on β give respectively, the tangent, the principal normal and the binormal vectors of β as follows T β s =, m [ sin m κsds, [m m cos κsds, N β s = [ 0, cos m [ κsds, sin m κsds, 6

7 [ B β s = 0, sin m [ κsds, cos m κsds The TN-Smarandache curve of β can be expressed as, m β s = sin [ m κsds + cos [ m κsds, m cos [ m κsds + sin [ m κsds It has the moving Frenet frame T β = N β = + m 2 0, cos [ m κsds m sin [ m κsds, sin [ m κsds + m cos [ m κsds [ [ 0, sin m κsds m cos m κsds, + m 2 cos [ m κsds m sin [ m κsds, B β =, 0, m 2 From the above data, the curvature and torsion of β are κ β = m, τ β = The curvature is constant and torion is vanished. The same results for TB and TNB-Smarandache curves of β are valid. Hence the proof is completed. Remark 3. If rs is a circular helix κ = const., τ = const., then TN, TB and TNB-Smarandache curves are also plane curves with constant curvatures. Theorem 3.2 If rs is a family of Salkowski curves γs in G 3 τ = τs, κ = a = const.which can be written as γs = [ s, a cos [ τsds ds ds, a sin, τsds ds ds, 3.3 then TN, TB and TNB-Smarandache curves of γ are plane curves with non-constant curvatures. Proof. After some calculations on γ, the tangent, the principal normal and the binormal vectors of γ are, respectively T γ s = [, a cos N γ s = B γ s = [ 0, cos [ 0, sin [ τsds ds, a sin [ τsds, sin [ τsds, cos τsds, τsds ds, τsds Here, TN-Smarandache curve of γ is [ [, a cos τsds ds + cos τsds, γ s = a [ sin τsds ds + sin [ τsds,

8 with the Frenet vectors N γ = Ω T γ = 0, a cos τsds τs sin τsds, a 2 τ 2 s a sin τsds + τs cos τsds 0, θ s sin τsds + θ 2 s cos τsds, θ 2 s sin τsds + θ s cos τsds B γ = Ω From the aforementioned calculations, the curvatures of γ s are κ γ = ; Ω =, θ 2 + θ2 2, aθ τθ 2 a 2 τ 2 s, 0, Ω a 2 τ 2 s 2, τ γ = 0, 3.35 where θ s = aτ a 2 τ 2 + ττ 2, θ 2 s = a ττ τ 2 a 2 τ As above, we can do similar calculations for TB and TNB-Smarandache curves of γ. In all cases, the curvatures are non-constants they depend on the torsion of the Salkowski curve and the torions are vanished. Thus, this completes the proof. Remark 3.2 If rs is a family of Anti-Salkowski curves δs in G 3 τ = b = const., κ = κs, which can be written as δs = [ s, [ κs cosbsds ds, κs sinbsds ds, 3.37 then TN, TB and TNB-Smarandache curves of δ are also plane curves with non-constant curvatures. 4 Spherical indicatrices of a general helix in G 3 In what follows, we investigate the spherical indicatrix of the tangent of the helix βs. By differentiating 3.26 with respect to s, we have the tangent spherical curve as follows: Γ T = β T s T =, m sinm κsds, m cosm κsds. 4. The Frenet vectors of β T s T are T T s T = N T s T = 0, cosm 0, sinm κsds, sinm κsds, cosm κsds, κsds, B T s T =, 0, 0, 4.2 and its curvatures are given by s T = m, τ T s T =

9 Theorem 4. Let β = βs be a helix in the Galilean space G 3 with κs 0. The curvatures of the tangent spherical curve β T s T of β for each s T I R satisfy the following equalities T T G 3 = 0, N T G 3 =, B T G 3 = κ T κ κ 2 T τ T. Proof. By assumption we have β T G 3 = d 2, 4.4 for every s T I R and d is the radius of the Galilean sphere S 2 G respect to s, we have where s T dβt ds T ds T ds, β = 0, T G 3 is the arc length of the tangent spherical curve in G 3. So, we get. By differentiating 4.4 with then ds T ds = κs, κs T T G 3 = 0, By a new differentiation of 4.5, we find that T T G 3 = κ N T G 3 + κ T T, T T G 3 = 0. From which More differentiation of 4.6 gives N T G 3 =. 4.6 τ T κ B T G 3 + κ N T, T T G 3 = κ 2, T since then Thus, the proof is completed. N T, T T G 3 = 0, dn T ds T = τ T B T, B T G 3 = κ τ T κ 2 T From aforementioned information, we have the following proposition. Proposition 4. The Galilean spherical images of a general helix or circular helix in the threedimensional Galilean space are plane curves with constant curvatures. Proposition 4.2 We can also do in similar way the calculations for the other spherical images of TN, TB and TNB-Smarandache curves, we find that they are plane curves with non-constant curvatures. 9.

10 5 Examples Example 5. Let βs be a general helix in three-dimensional Galiliean space with κ = τ = 2 s, parameterized by βs = s, 2 [ sin s s cos s, 2 [ cos s + s sin s The TN- Smarandache curve of βs is given by Then, the Frenet vectors are β s =, sin s + cos s, cos s + sin s. T s = N s = The derivatives of these vectors give 0, cos s sin s, cos s + sin s, 2 2 0, cos s + sin s, cos s sin s, 2 2 B s = cos 2 s + sin 2 s, 0, 0. κ s = 2, τ s = 0. The TB and TNB-Smarandache curves of β s can be calculated as above. The tangent spherical image of βs is defined by From which, one can obtain β T =, sin s, cos s. T T = 0, cos s, sin s, N T = 0, sin s, cos s, sin 2 2 B T = s + cos s, 0, 0. It follows that =, τ T = 0. In an analogous way, one can obtain the normal and binormal spherical images of β and their curvatures. The general helix βs and its T N-Smarandache curve and tangent spherical image are shown in Figures,2,3. 0

11 Figure : The general helix βs. Figure 2: The T N-Smarandache curve of βs. Figure 3: The tangent spherical image of βs. Example 5.2 Consider a circular helix νs in G 3 with non-zero constant curvature and torsion, which is given by νs = s, cos s, sin s, the representation of its tangent spherical curve is ν T =, sin s, cos s.

12 Then, simple calculations lead to T T = 0, cos s, sin s, N T = 0, sins, coss, B T = sins 2 + coss 2, 0, 0. By the aid of the derivatives of the above formulas, we obtain the curvatures of ν T =, τ T = 0. The curve νand its tangent spherical image are shown in Figures 4,5. Figure 4: A circular helix νs. Figure 5: The tangent spherical image of νs. 6 Conclusion In the three-dimensional Galilean space, Smarandache curves of some special curves such as helix, circular helix, Salkowski and Ant-Salkowski curves are studied. Furthermore, the spherical images of the helix are given. Some interesting results are presented. Finally as an application for this work, some examples are given and plotted. 2

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