On the Differential Geometric Elements of Mannheim Darboux Ruled Surface in E 3

Transcription

1 Applied Mathematical Sciences, Vol. 10, 016, no. 6, HIKARI Ltd, On the Differential Geometric Elements of Mannheim Darboux Ruled Surface in E 3 Şeyda Kılıçoğlu 1 Faculty of Education, Department of Mathematics Basent University, Anara, Turey Süleyman Şenyurt Faculty of Arts and Sciences, Department of Mathematics Ordu University, Ordu, Turey Copyright c 016 Şeyda Kılıçoğlu and Süleyman Şenyurt. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor is properly cited. Abstract In this paper we consider two special ruled surfaces associated to Mannheim pair {α, α }. First, Mannheim Darboux Ruled surface (MDRS) of the curve α be defined and examined in terms of the Frenet-Serret apparatus of the Mannheim curve α, in E 3. Further we have examined the differential geometric elements such as, Weingarten map S, Gaus curvature K and mean curvature H of Darboux ruled surface (DRS) and Mannheim Darboux ruled surface (MDRS) relative to each other. Also the first, second and third fundamental forms of Mannheim Darboux ruled surface (MDRS) have been examined in terms of the Mannheim curve α too. Mathematics Subject Classification: 53A04, 53A05 Keywords: Ruled surface, Darboux vector, Mannheim curves 1 Introduction and Preliminaries Involute-evolute curves, Bertrand curves, and Mannheim partner curves are the curves derivyed based on the other curves in geometry. Mannheim curve 1 Corresponding author

2 3088 Şeyda Kılıçoğlu and Süleyman Şenyurt was firstly defined by A. Mannheim in A curve is called a Mannheim curve if and only if 1 is a nonzero constant, with curvatures 1 1 and. + Recently, a new definition of the associated curves was given by Liu and Wang [4]. According to this new definition, if the principal normal vector of first curve and binormal vector of second curve are linearly dependent, then first curve is called Mannheim curve, and the second curve is called Mannheim partner curve. As a result they called these new curves as Mannheim pair curves. For more detail see in [3] and [4]. The quantities {V 1, V, V 3, D,, } are collectively Frenet-Serret apparatus of the curve α : I E 3. Also V 1 V V 3 = V 1 V V 3. (1.1) are well nown the Frenet formulae. Darboux vector D is the areal velocity vector of the Frenet frame of a space curve. For any unit speed curve α,in terms of the Frenet-Serret apparatus, the Darboux vector can be expressed as D(s) = (s)v 1 (s) + (s)v 3 (s) (1.) where curvature functions are and [1]. Along curve α under the condition that 0, vector field D(s) = (s)v 1 (s) + V 3 (s) (1.3) is called the modified Darboux vector field of curve α in []. Let α : I E 3 and α : I E 3 be the C class differentiable unit speed and α : I E 3 be two curves and let V 1 (s), V (s), V 3 (s) and V1 (s ), V (s ), V3 (s ) be the Frenet frames of the curves α and α, respectively. If the principal normal vector V of the curve α is linearly dependent on the binormal vector V3 of the curve α, then the pair {α, α } is said to be Mannheim pair, then α is called a Mannheim curve and α is called Mannheim partner curve of α where < (V 1, V1 ) = cos θ and besides the equality = nonzero constant is nown 1 + the offset property. In [5] and [6] Mannheim partner curves and Mannheim offsets of ruled surfaces are defined and characterized. Mannheim pair {α, α } can be represented by α(s ) = α (s ) + λ(s )V 3 (s ) (1.4) for some function λ, since V and V 3 are linearly dependent. This equation can be rewritten as α (s) = α (s) λv (s) (1.5)

3 Differential geometric elements of Mannheim Darboux ruled surface 3089 where λ =. Frenet-Serret apparatus of Mannheim partner curve α, 1 + based in Frenet-Serret vectors of Mannheim curve α are V1 = cosθ V 1 sinθ V 3 V = sinθ V 1 + cosθ V 3 V3 = V. The curvature and the torsion have the following equalyties, or (1.6) 1 = dθ ds = cos θ, (1.7) = sinθcosθ cosθcosθ =. (1.8) λ We use dot to denote the derivative with respect to the arc length parameter of the curve α. Also ds ds = 1 cos θ = λ sinθ, (1.9) where λ is the distance between the curves α and α, since d (α (s), α (s)) = λ. For more detail see in [5]. Also we can write V V 3 ds ds = 1. (1.10) 1 + λ The product of Frenet vector fields of the Mannheim pair {α, α } has the following matrix form; V1 [ ] V 1 V V 3 =. (1.11) cosθ 0 sinθ sinθ 0 cosθ Definition 1.1 Ruled surface is said to be Darboux Ruled surface if it is generated by moving Darboux vector fields, with the parametrization ϕ (s, u) = α (s) + u D(s)v = α (s) + u (s)v 1 (s) + uv 3 (s). (1.1) Also it has been called rectifying developable surface in []. Mannheim Darboux Ruled surface In this section we will define and wor on MDRS, which is nown as rectifying developable ruled surface, or D scroll as in [7] where the differential geometric elements of the involute D scroll.are examined too. Here first we will give Darboux vector field of the Mannheim partner α as in the following theorem.

4 3090 Şeyda Kılıçoğlu and Süleyman Şenyurt Theorem.1 The modified Darboux vector of Mannheim partner curve α of a Mannheim curve α, based on the Frenet apparatus of Mannheim curve α is D = + cos θv 1 + V + + cos θsinθ V 3. (.1) Proof. The desired result is obtained from equations (1.3), (1.6) and (1.8). Definition. The parametrization of MDRS, in terms of the Frenet-Serret apparatus of the Mannheim partner curve α is ϕ (s, v) = α v + cos θ V 1 + (v λ) V + v + cos θ sin θ V 3. (.) since where ϕ (s, v) = α (s) + v D (s), D = + cos θv 1 + V + + cos θ sin θ V 3. Theorem.3 DRS and MDRS are intersect each other along the curve ϕ (s) = α Proof. With the equations and sin θ cos θ V 1 + sin θ V 3. (.3) ϕ (s, v) = α + v cos θ V 1 + (v λ) V v cos θ sin θ λ λ under the conditions, we have This complete the proof. ϕ (s, u) = α + u V 1 + uv 3 v λ cos θ = u v λ = 0 v = u, cos θsinθ λ cos θ = u v = λ = u, cos θsinθ = tan θ. V 3

5 Differential geometric elements of Mannheim Darboux ruled surface 3091 Theorem.4 Normal vector field N of DRS is pependicular to the normal vector field N of MDRS. Proof. Since the normal vector field N of DR α is N = ϕ sλϕ u ϕ s Λϕ u = V [7], and the normal vector field N of MDRS of the curve α is it is trivial that V, V = 0. N = ϕ sλϕ u ϕ sλϕ u = V = sinθ V 1 + cosθ V 3. Theorem.5 The matrix corresponding to the Weingarten map (Shape Operator ) S of MDRS is S = [( cos θ 1+ v 1 + ) cos θ cos θ+ 1 ] 0 λ ( sin θ+ θ cos θ) 0 0. (.4) Proof. In the Euclidean 3 space, the matrix corresponding to the Weingarten map (Shape Operator ) S of DRS of curve α is [ ] ) 0 S = 1+u(. 0 0 [7]. Hence the matrix corresponding to the Weingarten map (Shape Operator ) S of MDRS is S = 1 ) 1+v( 1 s by substituing and 1 in matrix S we get S = cos θ 1 λ cos θ 1+v s Since the derivative with respect to parameter s is ( ) ( ) 1 cos θ d 1 cos θ λ ds = λ s ds ds [( ) 1 = 1 + cos θ ( cos θ sin θ + θ cos θ) ]. we have the proof. Where λ = +.

6 309 Şeyda Kılıçoğlu and Süleyman Şenyurt Corollary.1 The Gaussian curvature of MDRS is Corollary. The mean curvature of MDRS is K = det S = 0. (.5) H = traces = ( u ( 1 ) ), (.6) H = cos θ [ ( ) 1 + v 1 + cos θ cos θ + λ ( sin θ + θ ) ]. cos θ where < (V 1, V 1 ) = cos θ and θ is not constant. Corollary.3 MDRS is not minimal surface since cos θ = 0 and v cos θ ( ) 1 + cos θ 1 + ( sin θ + θ ) (.7) cos θ with curvatures and of the Mannheim curve α. Proof. Minimal surfaces are classically defined as surfaces of zero mean curvature in the Euclidean 3 space. Since H 0 we have the proof. We now that a surface can be characterized by the basic intrinsic properties such as the fundamental forms of a surface; usually called the first, second and third fundamental forms. They are extremely important and useful in determining the metric properties of a surface, such as line element, area element, normal curvature, Gaussian curvature, and mean curvature. The third fundamental form is given in terms of the first and second forms by III HII + KI = 0 where H is the mean curvature and K is the Gaussian curvature. The fundamental forms of the involute D scroll are examined in [7] The first fundamental form characterizes the interior geometry of the surface in a neighbourhood of a given point M. Suppose that the surface is given by the equation ϕ(s, u); where s and u are parameters of the surface; and d ϕ = ϕ s ds + ϕ u du is the differential of the radius vector of ϕ along a chosen direction from a point M to an infinitesimally close point M, [8]. Theorem.6 The first fundamental form of MDRS is I = cos θdsds + dvdv (.8)

7 Differential geometric elements of Mannheim Darboux ruled surface 3093 Proof. The first fundamental form I of DRS can be calculated as I = d ϕ, d ϕ = dsds + dudu[8] Hence the first fundamental form I of MDRS is I = ds ds + dvdv Since ds = cos θds, it is trivial. Now we will examine the second fundemantal form of MDRS, already defined. Theorem.7 The second fundamental form of MDRS is II = cos θdsds + cos θdsdv. (.9) Proof. The second fundamental form II of DRS is given by II = d ϕ, dn = dsds + dsdu.[8], where N is the unit normal vector of the surface at the point M. Hence The second fundamental form II of MDRS is II = 1ds ds + ds dv = ( ) cos θ cos θdsds cos θdsdv. Since ds = cos θds, it is trivial. Theorem.8 The third fundamental form of MDRS is III = + (1 + ) cos θ dsds. (.10) Proof. The third fundamental form of of DRS is the square of the differential of the unit normal vector N of the surface at the point M which is denoted by III and given by III = dn, dn = ( 1 + ) dsds, [8] Hence third fundamental form of MDRS is III = ( ( ) ( ) ) 1 + ds ds = 1 + cos θ dsds cos θ λ ( ) ( ) = cos θ dsds cos θ ( = ) + (1 + ) cos θ cos θ dsds = + (1 + ) cos θ cos θ Since ds = cos θds, it is trivial. dsds.

8 3094 Şeyda Kılıçoğlu and Süleyman Şenyurt References [1] A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, nd ed., CRC Press, Boca Raton, FL, [] S. Izumiya, N. Taeuchi, Special curves and Ruled surfaces, Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry, 44 (003), no. 1, [3] M.M. Lipschutz, Differential Geometry. Schaum s Outlines, McGraw-Hill, New Yor, [4] H. Liu and F. Wang, Mannheim partner curves in 3-space, Journal of Geometry, 88 (008), no. 1-, [5] K. Orbay and E. Kasap, On mannheim partner curves, International Journal of Physical Sciences, 4 (009), no. 5, [6] K. Orbay, E. Kasap, İ. Aydemir, Mannheim Offsets of Ruled Surfaces, Mathematical Problems in Engineering, 009 (009), [7] S. Senyurt and S. Kılıcoglu On the differential geometric elements of the involute D-scroll in E 3, Adv. Appl. Clifford Algebras, 5 (015), no. 4, [8] Springerlin, Encyclopaedia of Mathematics, Springer-Verlag, Berlin, Heidelberg, New Yor 00. Received: August 8, 016; Published: November 1, 016