Darboux Iso-Geodesic Special Curve in Euclidean Space

Modern Applied Science; Vol. 13, No. 9; 2019 ISSN 1913-1844 E-ISSN 1913-1852 Published by Canadian Center of Science and Education Darboux Iso-Geodesic Special Curve in Euclidean Space M. M. Wageeda1, E. M. Solouma2 & M. Bary1 1 Department of Mathematics, Faculty of Science, Aswan University, Egypt 2 Department of Mathematics, Faculty of Science, Beni-Suef University, Egypt Correspondence: M. Bary, Department of Mathematics, Faculty of Science, Aswan University, Egypt. E-mail: [email protected] Received: August 5, 2019 doi:10.5539/mas.v13n9p98 Accepted: August 27, 2019 Online Published: August 28, 2019 URL: https://doi.org/10.5539/mas.v13n9p98 Abstract In this paper, by using Darboux frame we scrutinize the issues of reconstructing surfaces with given some unusual Smarandache curves in Euclidean 3-space, we make manifest the family of surfaces as a linear combination of the components of this frame and derive the necessary and sufficient conditions for coefficients to satisfy both the iso-geodesic and iso-parametric requirements. Keywords: Smarandache curve, Darboux frame, Euclidean space-time AMS Subject Classification (2010): Primary: 53A04; 53A05; 53C22; Secondary: 58E10. 1. Introduction Geodesic on the surface may be an essential geometric part that holds a paramount part in the different applications. Geometrically, the geodesic on any surface may be an indispensable part simple curve on the surface such that for any two points on the curve a geodesic will be shortest path between them on the surface. Many geometric operations would initially associate to geodesics, they are also important in many applications, the method finds its applications in computer vision and image processing, such as in object segmentation, and multi-scale more analysis (P. N. Azariadis & N. A. Aspragathos, 2001; R. J. Haw. 1985). Also, the geodesics are important in the relativistic description of gravity. (Where Einstein's guideline about proportionality advises us that geodesics represent the paths of freely falling particles in a given space, the geodesics principle states that the free trajectories additionally the geodesics of space, and therefore has not to be set as an independent equation). Also, the geodesics are important in architecture; a few exceptional curves bring decent properties As far as structural purpose Furthermore manufacturing expense. For instance, may be planar curves in perpendicular planes, which might be utilized similarly as help components (R. Brond et al., 1994; V. Caselles et al., 1997; F. Karakus & Y. Yayli. 2016; C. Y. Li et al., 2001; G. J. Wang et al., 2004). In this paper, by using Smarandache curve we will study what is the problem of the special curves which it is obtained with Darboux frame how to describe those surfaces that possess this curve as an iso-geodesic in three-dimension Euclidean space? and we will give some preliminary particulars and facts about this special curves in three- dimension Euclidean space, and define the iso-geodesic curve and we will study progressive surfaces as a linear combination of the Darboux frame of the presented curve, and derive necessary and sufficient conditions on marching scale functions to satisfy both iso-geodesic and Smarandache requirements. 2. Preliminaries Let we are given a spatial parametric curve 𝐶: Υ(𝜍)= (x(𝜍); y(𝜍); z(𝜍)); 0≤ 𝜍 ≤L and {T(𝜍), N(𝜍), B(𝜍)} are seret-ferrent, where Υ (𝜍) the arbitrarily parametrized geodesic, T(𝜍) is the unit tangent vector, N(𝜍) is the Frenet principal normal vector, and B(𝜍) is the Frenet binormal vector. Let {T(𝜍), 𝜇(𝜍), 𝜉(𝜍)} the Darboux frame, where {T(𝜍), 𝜇(𝜍), 𝜉(𝜍)} the {unit tangent, unit normal, unit vector} respectively (O. Bektas. & S. Yuce. 2013), then the relations between this frame and the Frenet frame can be described as the subsequent: 98 mas.ccsenet.org Modern Applied Science 𝑇(𝜍) 1 0 [𝜇(𝜍)] = [0 cos⁡𝜃 𝜉(𝜍) 0 −𝑠𝑖𝑛𝜃 Vol. 13, No. 9; 2019 𝑇(𝜍) 0 𝑠𝑖𝑛𝜃 ] [𝑁(𝜍)] 𝑐𝑜𝑠𝜃 𝐵(𝜍) (1) From the differential and applying the Frenet Serret formulas we get the subsequent. T(ς) = ⁡ Κ ξ ⁡ξ(ς) − Κ μ ⁡μ(ς) { ξ(ς) = −Κ ξ ⁡T(ς) + τ⁡μ(ς) μ(ς) = Κ μ ⁡T(ς) − τ⁡ξ(ς) Where Κ a curvature, 𝜏 a geodesic torsion, Κ 𝜉 = Κ⁡cos⁡𝜃 normal curvature of this curve. (2) a geodesic curvature and Κ𝜇 = Κ⁡sin⁡𝜃 is the There is a condition if and only if this condition is met we know the curve C: Υ(ς) will be a geodesic on the surface ψ(ς,v), if the principal normal μ(ς) to the curve C: Υ(ς) and the normal vector n(ς,v) to the surface ψ(ς,v) are parallel to each other at any point on the curve C: Υ(ς), then the curve C: Υ(ς) will be a geodesic on the surface ψ(ς,v). And the curve 𝐶: Υ(𝜍) is an iso-parametric curve if the parameteres {𝜍⁡𝑜𝑟⁡𝑣} are constant parameter value, i.e 𝜓(𝜍0 ⁡, 𝑣) = Υ(𝜍) or 𝜓(𝜍⁡, 𝑣0 ⁡) = Υ(𝜍). And the curve 𝐶: Υ(𝜍) is an iso-geodesic of a surface 𝜓(𝜍⁡, 𝑣) if it is both a geodesic and an iso-parametric curve on the surface. In other words, let can be taking the form 𝑛(𝜍⁡, 𝑣0 ⁡) = Φ1 (𝜍⁡, 𝑉0 )⁡𝑇(𝜍) + Φ2 (𝜍⁡, 𝑉0 )𝑁(𝜍) + Φ3 (𝜍⁡, 𝑉0 )𝐵(𝜍), if 𝑛(𝜍⁡, 𝑣0 ⁡) ∕∕ 𝑁(𝜍) where 𝑛(𝜍⁡, 𝑣) is the normal vector of the curve and 𝑁(𝜍) is the Frenet principal normal vector on the surface, then we can consider that 𝐶: Υ(𝜍) is an iso-geodesic on the surface, i.e. Φ2 (𝜍⁡, 𝑉0 ) ≠ 0. Suppose that Υ(ς) is a denote the unit speed regular curve in three-Dimension Euclidean space and the {TO(ς), ξ(ς), μ(ς)} be it's moving Darboux frame. The Smarandache Tμ curves are known by Υ(𝜍) = Υ ∗ (𝜍) = 1 1 ⁡(T(𝜍) + 𝜇(𝜍)). And Smarandache 𝑇𝜉 curves are known by Υ(𝜍) = Υ ∗ (𝜍) = ⁡(T(𝜍) + 𝜉(𝜍)). And √2 Smarandache 𝜇𝜉 curves are known by Υ(𝜍) = Υ ∗ (𝜍) = are known by Υ(𝜍) = Υ ∗ (𝜍) = 1 √3 ⁡(T(𝜍) + 𝜉(𝜍) + μ(ς)). 1 √2 √2 ⁡(𝜉(𝜍) + μ(ς)). And Smarandache 𝑇𝜉𝜇 curves 3. Surfaces with Common Smarandache Iso-Geodesic Curve Let the parametric surface 𝜓(𝜍⁡, 𝑣): [𝐿1 , 𝐿2 ] × [𝑉1 , 𝑉2 ] → ℜ3 is known stablished on the given curve Υ(𝜍) and the local coordinate frame known by 𝑇(𝜍), 𝜉(𝜍) and 𝜇(𝜍) as the subsequent: ψ(ς⁡, v) = Υ(ς) + {x(ς)T(ς) + Y(ς⁡, v)ξ(ς) + Z(ς⁡, v)μ(ς)}. (3) ; 𝐿1 ≤ ⁡𝜍 ≤ 𝐿2 ⁡⁡⁡,⁡⁡⁡𝑉1 ≤ 𝑣 ≤ ⁡ 𝑉2 Where 𝑋(𝜍⁡, 𝑣)⁡, 𝑌(𝜍⁡, 𝑣) and 𝑍(𝜍⁡, 𝑣) are 𝐶 1 functions, and 𝑋(𝜍⁡, 𝑣) the extension-like, 𝑌(𝜍⁡, 𝑣) exion-like and 𝑍(𝜍⁡, 𝑣) retortion-like influences by the point unit in the time 𝑣, starting from Υ(𝜍) and {𝑇(𝜍), 𝜉(𝜍), 𝜇(𝜍)} is the Darboux frame associated with the curve Υ(𝜍). The normal vector 𝑛(𝜍⁡, 𝑣) can be computed by taking the cross product of the partial differentials as the subsequent: n(ς⁡, v) = ∂ψ(ς,v) ∂ς × ∂ψ(ς,v) ∂v (4) Let the parameter ⁡⁡𝑉1 ≤ 𝑣 ≤ ⁡ 𝑉2 ⁡⁡, 𝐿1 ≤ ⁡𝜍 ≤ 𝐿2 ⁡⁡ such that 𝜓(𝜍⁡, 𝑣0 ⁡) = Υ(𝜍), and the principal normal to the curve and the normal vector to the surface are parallel, then the our goal and important purpose in this work is to discover carefully the most important and adquate conditions that make some special Smarandache curves of the unit space curve Υ(𝜍) on the surface 𝜓(𝜍⁡, 𝑣⁡) is an Iso-geodesic. 3.1 Surfaces with Common 𝑻𝝁 Curve Theorem 3.1. The Smarandache 𝑇𝜇 of the curve Υ1 (𝜍) is an iso-geodesic on a surface 𝜓(𝜍, 𝑣) if and only if this the subsequent condition is satisfed: 1. 2. 3. 4. 5. 𝑥(𝜍, 𝑣0 ) = 𝑦(𝜍, 𝑣0 ) = 𝑧(𝜍, 𝑣0 ) = 0 𝜕𝑧(𝜍,𝑣0 ) 𝜕𝑦(𝜍,𝑣0 ) = cot⁡(𝜃) 𝜕𝑣 𝜕𝑥(𝜍,𝑣0 ) 𝜕𝑦(𝜍,𝑣 ) 0 ≠− 𝜕𝑣 𝜕𝑣 Κ 𝜉 = cot⁡(𝜃)⁡Κ𝜇 ⁡Κ𝜇 ≠ 0. 𝜕𝑣 99 mas.ccsenet.org Modern Applied Science Vol. 13, No. 9; 2019 proof: - Let Υ1 (𝜍) be a Smarandache 𝑇𝜇 curve on surface 𝜓(𝜍, 𝑣). From (3) we find the parametric surface is known by a Smarandache 𝑇𝜇 as the subsequent: 1 𝜓(𝜍, 𝑣) = (𝑇(𝜍) + 𝜇(𝜍)) + [𝑥(𝜍, 𝑣)𝑇(𝜍) + 𝑦(𝜍, 𝑣)𝜇(𝜍) + 𝑧(𝜍, 𝑣)𝜉(𝜍)] √2 Ever since a spatial Smarandache curve of 𝑇𝜇 is an iso-parametric curve on the surface⁡⁡𝜓(𝜍, 𝑣), there dwell a 1 parameter 𝑣 = 𝑣0 ∈ [𝑉1 ⁡, 𝑉2 ] such that 𝜓(𝜍, 𝑣) = (𝑇(𝜍) + 𝜇(𝜍))⁡ and 𝜍 ∈ [𝐿1 ⁡, 𝐿2 ] then √2 x(ς⁡, v0 ) = y(ς⁡, v0 ) = z(ς⁡, v0 ) = 0 (5) From (4) the normal vector can be expressed as 𝑛(𝜍⁡, 𝑣0 ) = 𝛼(𝜍⁡, 𝑣0 )𝑇(𝜍) + 𝛽(𝜍⁡, 𝑣0 )𝜇(𝜍) + 𝛾(𝜍⁡, 𝑣0 )𝜉(𝜍), Where ∂y(ς⁡,v0 ) ∂z(ς⁡,v0 ) √2 (Κ ξ − Κμ ), ∂v ∂v 2 ∂x(ς⁡,v0 ) ∂z(ς⁡,v0 ) √2 − (Κ ξ + Κμ ), 2 ∂v ∂v ∂x(ς⁡,v0 ) ∂y(ς⁡,v0 ) √2 + ). = Κμ ( ∂v ∂v 2 α(ς⁡, v0 ) = β(ς⁡, v0 ) = { From equation (1), (2) and (6) we find: γ(ς⁡, v0 ) (6) 𝑛(𝜍⁡, 𝑣0 ) = Φ1 (𝜍⁡, 𝑣0 )⁡𝑇(𝜍) + Φ2 (𝜍⁡, 𝑣0 )𝑁(𝜍) + Φ3 (𝜍⁡, 𝑣0 )𝐵(𝜍). Where Φ1 (ς⁡, v0 ) = α(ς⁡, v0 ) = 0, {Φ2 (ς⁡, v0 ) = β(ς⁡, v0 ) cos(θ) − γ(ς⁡, v0 )⁡sin⁡(θ) ≠ 0, Φ3 (ς⁡, v0 ) = β(ς⁡, v0 ) sin(θ) + γ(ς⁡, v0 ) cos(θ) = 0. (7) From (6) and (7) we find ⁡⁡⁡Φ1 (𝜍⁡, 𝑣0 ) = 0⁡⁡ and ⁡⁡⁡Φ3 (𝜍⁡, 𝑣0 ) = 0⁡⁡⁡ if and only if these the subsequent conditions are satisfied Then Φ2 ≠ 0 is given by Then we have, (1) { 𝜕𝑧(𝜍⁡,𝑣0 ) 𝜕𝑣 = cot⁡(𝜃) 𝜕𝑦(𝜍⁡,𝑣0 ) 𝜕𝑣 (2)Κ 𝜉 = cot(𝜃) Κ𝜇 . (8) 𝜕𝑥(𝜍⁡, 𝑣0 ) 𝜕𝑦(𝜍⁡, 𝑣0 ) 𝜇(𝜍) cos(𝜃) ( + )≠0 𝜕𝑣 𝜕𝑣 √2 1 (1) { 𝜕𝑥(𝜍⁡,𝑣0 ) 𝜕𝑣 ≠− 𝜕𝑦(𝜍⁡,𝑣0 ) (2)Κ𝜇 ≠ 0. 𝜕𝑣 (9) Then the Smarandache ⁡𝑇𝜇 of the curve is an iso-geodesic. From (5), (8) and (9) we have discovered the most important and adequate conditions for the ψ(ς ,v_0 ) have the spatial Smarandache Tμ curve of the curve is an iso-geodesic. ∎ 3.2 Surfaces with Common 𝑻𝝃 Curve Theorem 3.2. The Smarandache 𝑇𝜉 of the curve Υ2 (𝜍) is an iso-geodesic on a surface 𝜓(𝜍, 𝑣) if and only if this the subsequent condition is satisfed: 1. 2. 𝑥(𝜍, 𝑣0 ) = 𝑦(𝜍, 𝑣0 ) = 𝑧(𝜍, 𝑣0 ) = 0 𝜕𝑦(𝜍,𝑣0 ) 𝜕𝑧(𝜍,𝑣0 ) = tan⁡(𝜃) 𝜕𝑣 𝜕𝑥(𝜍,𝑣 ) 𝜕𝑧(𝜍,𝑣 ) 𝜕𝑣 0 0 3. ≠− 𝜕𝑣 𝜕𝑣 4. Κ𝜇 = tan(𝜃)⁡Κ 𝜉 5. ⁡Κ 𝜉 ≠ 0. Proof: Let Υ2 (𝜍) be a Smarandache 𝑇𝜉 curve on surface 𝜓(𝜍, 𝑣). From (3) we find the parametric surface is defined by a given Smarandache 𝑇𝜉⁡ as the subsequent: 1 (𝑇(𝜍) + 𝜉(𝜍)) + [𝑥(𝜍, 𝑣)𝑇(𝜍) + 𝑦(𝜍, 𝑣)𝜇(𝜍) + 𝑧(𝜍, 𝑣)𝜉(𝜍)] 𝜓(𝜍, 𝑣) = √2 100 mas.ccsenet.org Modern Applied Science Vol. 13, No. 9; 2019 Even since Smarandache curve of 𝑇𝜉 is an iso-parametric curve on the surface⁡⁡𝜓(𝜍, 𝑣), there dwell a parameter 1 v = v0 ∈ [V1 ⁡, V2 ] such that ψ(ς, v) = (T(ς) + ξ(ς))⁡ and ς ∈ [L1 ⁡, L2 ] then √2 𝑥(𝜍⁡, 𝑣0 ) = 𝑦(𝜍⁡, 𝑣0 ) = 𝑧(𝜍⁡, 𝑣0 ) = 0 (10) From (4) the normal vector can be expressed as 𝑛(𝜍⁡, 𝑣0 ) = 𝛼(𝜍⁡, 𝑣0 )𝑇(𝜍) + 𝛽(𝜍⁡, 𝑣0 )𝜇(𝜍) + 𝛾(𝜍⁡, 𝑣0 )𝜉(𝜍), Where ∂y(ς⁡,v0 ) ∂z(ς⁡,v0 ) √2 (Κ ξ − Κμ ), ∂v ∂v 2 ∂x(ς⁡,v0 ) ∂z(ς⁡,v0 ) √2 β(ς⁡, v0 ) = − Κ ξ ( + ), 2 ∂v ∂v ∂x(ς⁡,v0 ) ∂y(ς⁡,v0 ) √2 {γ(ς⁡, v0 ) = 2 (Κ μ ∂v + Κ ξ ∂v ) . α(ς⁡, v0 ) = (11) From equation (1), (2) and (11) we find: Where 𝑛(𝜍⁡, 𝑣0 ) = Φ1 (𝜍⁡, 𝑣0 )⁡𝑇(𝜍) + Φ2 (𝜍⁡, 𝑣0 )𝑁(𝜍) + Φ3 (𝜍⁡, 𝑣0 )𝐵(𝜍). Φ1 (ς⁡, v0 ) = α(ς⁡, v0 ) = 0, {Φ2 (ς⁡, v0 ) = β(ς⁡, v0 ) cos(θ) − γ(ς⁡, v0 )⁡sin⁡(θ) ≠ 0, Φ3 (ς⁡, v0 ) = β(ς⁡, v0 ) sin(θ) + γ(ς⁡, v0 ) cos(θ) = 0. (12) From (11) and (12) we find ⁡⁡⁡Φ1 (𝜍⁡, 𝑣0 ) = 0⁡⁡ and ⁡⁡⁡Φ3 (𝜍⁡, 𝑣0 ) = 0⁡⁡⁡ if the subsequent conditions are satisfies 𝜕𝑦(𝜍⁡,𝑣0 ) 𝜕𝑣 = tan(𝜃) 𝜕𝑧(𝜍⁡,𝑣0 ) 𝜕𝑣 ⁡,⁡⁡⁡⁡⁡⁡Κ𝜇 = tan(𝜃)⁡Κ 𝜉 . (13) From Φ2 ≠ 0 and by using (13) we have 𝜕𝑥(𝜍⁡, 𝑣0 ) 𝜕𝑧(𝜍⁡, 𝑣0 ) −1 Κ 𝜉 ⁡sec(𝜃) ( + )≠0 𝜕𝑣 𝜕𝑣 √2 Then we find, Κ 𝜉 ⁡ ≠ 0,⁡⁡⁡⁡⁡⁡⁡ 𝜕𝑥(𝜍⁡,𝑣0 ) 𝜕𝑣 + 𝜕𝑧(𝜍⁡,𝑣0 ) 𝜕𝑣 ≠0 (14) Then the ⁡⁡𝑇𝜉⁡ is an iso-geodesic. Using (10), (13) and (14) we have discovered the most important and adequate conditions for the ψ(ς ,v_0 ) have the spatial Smarandache 𝑇𝜉 curve of the curve is an iso-geodesic. ∎ 3.3 Surfaces with Common 𝝁𝝃 Curve Theorem 3.3. The Smarandache 𝜇𝜉 of the curve Υ3 (𝜍) is an iso-geodesic on a surface 𝜓(𝜍, 𝑣) if and only if these the subsequent conditions are satisfied: 1. 2. 𝑥(𝜍, 𝑣0 ) = 𝑦(𝜍, 𝑣0 ) = 𝑧(𝜍, 𝑣0 ) = 0, 𝜕𝑧(𝜍,𝑣0 ) 𝜕𝑦(𝜍,𝑣0 ) = cot⁡(𝜃) 𝜕𝑣 𝜕𝑦(𝜍,𝑣 ) 𝜕𝑣 0 3. ≠0 𝜕𝑣 4. Κ𝜇 + Κ 𝜉 ≠ 0. Proof: Let Υ3 (𝜍) be a Smarandache 𝜇𝜉 curve on surface 𝜓(𝜍, 𝑣). From (3) we find the parametric surface is defined by a given Smarandache 𝜇𝜉⁡ as the subsequent: 1 𝜓(𝜍, 𝑣) = (𝜇(𝜍) + 𝜉(𝜍)) + [𝑥(𝜍, 𝑣)𝑇(𝜍) + 𝑦(𝜍, 𝑣)𝜇(𝜍) + 𝑧(𝜍, 𝑣)𝜉(𝜍)] √2 Even since Smarandache curve of 𝜇𝜉 is an iso-parametric curve on the surface⁡⁡𝜓(𝜍, 𝑣), there dwell a parameter 1 𝑣 = 𝑣0 ∈ [𝑉1 ⁡, 𝑉2 ] such that 𝜓(𝜍, 𝑣) = (𝜇(𝜍) + 𝜉(𝜍))⁡ and 𝜍 ∈ [𝐿1 ⁡, 𝐿2 ] then √2 x(ς⁡, v0 ) = y(ς⁡, v0 ) = z(ς⁡, v0 ) = 0 From (4) the normal vector can be expressed as Where 𝑛(𝜍⁡, 𝑣0 ) = 𝛼(𝜍⁡, 𝑣0 )𝑇(𝜍) + 𝛽(𝜍⁡, 𝑣0 )𝜇(𝜍) + 𝛾(𝜍⁡, 𝑣0 )𝜉(𝜍). 101 (15) mas.ccsenet.org Modern Applied Science α(ς⁡, v0 ) = 0, ∂z(ς⁡,v0 ) √2 (Κ μ + Κ ξ ) , ∂v 2 ∂y(ς⁡,v0 ) √2 (Κ μ + Κ ξ ) . ∂v 2 β(ς⁡, v0 ) = − γ(ς⁡, v0 ) = { From equation (1), (2) and (16) we find: Vol. 13, No. 9; 2019 (16) 𝑛(𝜍⁡, 𝑣0 ) = Φ1 (𝜍⁡, 𝑣0 )⁡𝑇(𝜍) + Φ2 (𝜍⁡, 𝑣0 )𝑁(𝜍) + Φ3 (𝜍⁡, 𝑣0 )𝐵(𝜍). Where Φ1 (ς⁡, v0 ) = α(ς⁡, v0 ) = 0, {Φ2 (ς⁡, v0 ) = β(ς⁡, v0 ) cos(θ) − γ(ς⁡, v0 )⁡sin⁡(θ) ≠ 0, Φ3 (ς⁡, v0 ) = β(ς⁡, v0 ) sin(θ) + γ(ς⁡, v0 ) cos(θ) = 0. From (16) and (17) we find⁡⁡⁡Φ3 (𝜍⁡, 𝑣0 ) = 0⁡⁡⁡ we get, 𝜕𝑧(𝜍⁡,𝑣0 ) 𝜕𝑣 = cot(𝜃) 𝜕𝑦(𝜍⁡,𝑣0 ) 𝜕𝑣 ⁡. (17) (18) From Φ2 ≠ 0 and by using (18) we have −1 𝜕𝑦(𝜍⁡, 𝑣0 ) (Κ 𝜇 + Κ 𝜉 )⁡csc(𝜃) ≠0 𝜕𝑣 √2 Then we have, 𝜕𝑦(𝜍⁡,𝑣0 ) 𝜕𝑣 And ≠0 (Κ 𝜇 + Κ 𝜉 ) ≠ 0 (19) (20) Then from (15), (18), (19) and (20) we have discovered the most important and adequate conditions for the ψ(ς,v_0 ) have the spatial Smarandache 𝜇⁡𝜉 curve of the curve is an iso-geodesic. ∎ 3.4 Surfaces with Common 𝑻𝝁𝝃 Curve Theorem3.4. The Smarandache 𝑇𝜇𝜉 curve of the curve Υ4 (𝜍) is an iso-geodesic on a surface 𝜓(𝜍, 𝑣) if and only if these the subsequent conditions are satisfied: 1. 2. 3. 4. 5. 𝑥(𝜍, 𝑣0 ) = 𝑦(𝜍, 𝑣0 ) = 𝑧(𝜍, 𝑣0 ) = 0 𝜕𝑦(𝜍,𝑣0 ) 𝜕𝑣 = Κ𝜇 𝜕𝑧(𝜍,𝑣0 ) Κ𝜉 𝜕𝑣 Κ 𝜉 = tan(𝜃)⁡Κ𝜇 𝜕𝑥(𝜍,𝑣0 ) 𝜕𝑣 ≠ −(1 + tan⁡(𝜃)) Κ𝜇 ≠ 0. 𝜕𝑧(𝜍,𝑣0 ) 𝜕𝑣 Proof: Let Υ4 (𝜍) be a Smarandache 𝑇𝜇𝜉 curve on the surface𝜓(𝜍, 𝑣). From (3) we find the parametric surface is defined by a given Smarandache 𝑇𝜇𝜉⁡ as the subsequent: 1 (𝑇(𝜍) + 𝜇(𝜍) + 𝜉(𝜍)) + [𝑥(𝜍, 𝑣)𝑇(𝜍) + 𝑦(𝜍, 𝑣)𝜇(𝜍) + 𝑧(𝜍, 𝑣)𝜉(𝜍)] 𝜓(𝜍, 𝑣) = √3 Even since Smarandache curve of 𝑇𝜇𝜉 is an iso-parametric curve on the surface⁡⁡𝜓(𝜍, 𝑣), there dwell a 1 parameter 𝑣 = 𝑣0 ∈ [𝑉1 ⁡, 𝑉2 ] such that 𝜓(𝜍, 𝑣) = (𝑇(𝜍) + 𝜇(𝜍) + 𝜉(𝜍))⁡ and 𝜍 ∈ [𝐿1 ⁡, 𝐿2 ] then √3 x(ς⁡, v0 ) = y(ς⁡, v0 ) = z(ς⁡, v0 ) = 0 From (4) the normal vector can be expressed as Where 𝑛(𝜍⁡, 𝑣0 ) = 𝛼(𝜍⁡, 𝑣0 )𝑇(𝜍) + 𝛽(𝜍⁡, 𝑣0 )𝜇(𝜍) + 𝛾(𝜍⁡, 𝑣0 )𝜉(𝜍), 102 (21) mas.ccsenet.org Modern Applied Science Vol. 13, No. 9; 2019 ∂y(ς⁡,v0 ) ∂z(ς⁡,v0 ) √3 (Κ ξ − Κμ ), ∂v ∂v 3 ) ) ∂x(ς⁡,v ∂z(ς⁡,v ∂z(ς⁡,v0 ) √3 0 0 + ) + Κμ ⁡], − [⁡Κ ξ ( ∂v ∂v ∂v 3 α(ς⁡, v0 ) = β(ς⁡, v0 ) = { γ(ς⁡, v0 ) = From equation (1), (2) and (22) we find: Where ∂x(ς⁡,v0 ) √3 [⁡Κ μ ( 3 ∂v + ∂y(ς⁡,v0 ) ∂v ) + Κξ ∂y(ς⁡,v0 ) ∂v 𝑛(𝜍⁡, 𝑣0 ) = 𝛷1 (𝜍⁡, 𝑣0 )⁡𝑇(𝜍) + 𝛷2 (𝜍⁡, 𝑣0 )𝑁(𝜍) + 𝛷3 (𝜍⁡, 𝑣0 )𝐵(𝜍). Φ1 (ς⁡, v0 ) = α(ς⁡, v0 ) = 0, {Φ2 (ς⁡, v0 ) = β(ς⁡, v0 ) cos(θ) − γ(ς⁡, v0 )⁡sin⁡(θ) ≠ 0, Φ3 (ς⁡, v0 ) = β(ς⁡, v0 ) sin(θ) + γ(ς⁡, v0 ) cos(θ) = 0. (22) ⁡]. (23) From (22) and (23) we find ⁡⁡⁡Φ1 (𝜍⁡, 𝑣0 ) = 0⁡⁡ and ⁡⁡⁡Φ3 (𝜍⁡, 𝑣0 ) = 0⁡⁡⁡ if and only if these the subsequent conditions are satisfied 𝜕𝑦(𝜍⁡,𝑣 ) Κ 𝜕𝑧(𝜍⁡,𝑣 ) 0 0 (1)⁡ = 𝜇 ⁡⁡ 𝜕𝑣 Κ𝜉 𝜕𝑣 { (2)⁡⁡⁡⁡⁡⁡Κ 𝜉 = 𝑐𝑜𝑡(𝜃)⁡Κ𝜇 (24) From Φ2 ≠ 0 and by using (24) we have Then we find, −1 √3 Κ𝜇 {csc(𝜃) 𝜕𝑥(𝜍⁡, 𝑣0 ) 𝜕𝑧(𝜍⁡, 𝑣0 ) + (csc(𝜃) + sec(𝜃)) }≠0 𝜕𝑣 𝜕𝑣 𝜕𝑥(𝜍⁡,𝑣0 ) 𝜕𝑣 ≠ −(1 + tan⁡(𝜃)) Κ𝜇 ⁡ ≠ 0,⁡⁡ 𝜕𝑧(𝜍⁡,𝑣0 ) 𝜕𝑣 (25) (26) Then from (21), (24), (25) and (26) we have discovered the most important and adequate conditions for the ψ(ς,v_0 ) have the spatial Smarandache 𝑇𝜇𝜉 curve of the curve is an iso-geodesic. ∎ 4. 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