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1. INTRODUCTION Ruled Surfaces Are One of the Most Important Topics

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1. INTRODUCTION Ruled Surfaces Are One of the Most Important Topics CONSTANT ANGLE RULED SURFACES IN EUCLIDEAN SPACES 1 2 Yusuf YAYLI , Evren ZIPLAR 1 Department of Mathematics, Faculty of Science, University of Ankara, Tando ğan, Ankara, Turkey [email protected] 2 Department of Mathematics, Faculty of Science, University of Ankara, Tando ğan, Ankara, Turkey [email protected] Abstract. In this paper, we study the special curves and ruled surfaces on helix hypersurface n whose tangent planes make a constant angle with a fixed direction in Euclidean n-space E . n Besides, we observe some special ruled surfaces in ΙR and give requirement of being developable of the ruled surface. Also, we investigate the helix surface generated by a plane curve in 3 Euclidean 3-space E . Keywords : Helix surfaces; Constant angle surfaces;Ruled surfaces Mathematics Subject Classification 2000: 53A04,53A05, 53A07, 53A10,53B25,53C40 1. INTRODUCTION Ruled surfaces are one of the most important topics of differential geometry. The surfaces were found by Gaspard Monge, who was a French mathematician and inventor of descriptive geometry. And, many geometers have investigated the many properties of these surfaces in [4,6,7]. Constant angle surfaces are considerable subject of geometry.There are so many types of these surfaces. Helix hypersurface is a kind of constant angle surfaces. An helix 1 hypersurface in Euclidean n-space is a surface whose tangent planes make a constant angle with a fixed direction.The helix surfaces have been studied by Di Scala and Ruiz- Hernández in [2]. And, A.I.Nistor investigated certain constant angle surfaces constructed on curves in Euclidean 3-space E 3 [1]. One of the main purpose of this study is to observe the special curves and ruled surfaces on a helix hypersurface in Euclidean n-space E n . Another purpose of this study is to obtain a helix surface by generated a plane curve in Euclidean 3-space E 3 . 2. PRELIMINARIES Definition 2.1 Let α : Ι ⊂ ΙR → E n be an arbitrary curve in E n . Recall that the curve α is said to be of unit speed ( or parametrized by the arc-length function s ) if α′(s), α′(s) = 1, where , is the standart scalar product in the Euclidean space E n given by n X ,Y = ∑ xi yi , i=1 n for each X = (x1 , x2 ,..., xn ), Y = ( y1 , y2 ,..., yn ) ∈ E . Let {V1 (s), V2 (s),..., Vn (s)} be the moving frame along α , where the vectors Vi are mutually orthogonal vectors satisfying Vi ,Vi = 1. The Frenet equations for α are given by V1′ 0 k1 0 0 ... 0 0 V1 ′ V2 − k1 0 k2 0 ... 0 0 V2 V3′ 0 − k2 0 k3 ... 0 0 V3 . = . . . . . . V ′ 0 0 0 0 ... 0 k V n−1 n−1 n−1 Vn′ 0 0 0 0 ... − kn−1 0 Vn Recall that the functions ki (s) are called the i -th curvatures of α [3]. 2 Definition 2.2 Given a hypersurface M ⊂ ΙR n and an unitary vector d ≠ 0 in ΙR n , we say that M is a helix with respect to the fixed direction d if for each q ∈ M the angle between d and Tq M is constant. Note that the above definition is equivalent to the fact that d,ξ is constant function along M , where ξ is a normal vector field on M [2]. Theorem 2.1 Let H ⊂ ΙR n−1 be an orientable hypersurface in ΙR n−1 and let N be an unitary normal vector field of H . Then, n fθ (x, s) = x + s(sin( θ )N(x) + cos( θ )d ) , fθ : H × ΙR → ΙR , θ = constant is a helix with respect to the fixed direction d in ΙR n , where x ∈ H and s ∈ΙR . Here n d is the vector 0,0( ,..., )1 ∈ ΙR such that d is orthogonal to N and H [2]. Definition 2.3 Let α : Ι ⊂ ΙR → E n be a unit speed curve with nonzero curvatures n ki ( i = 2,1 ,..., n) in E and let {V1 ,V2 ,...Vn } denote the Frenet frame of the curve of α . We call α a Vn -slant helix, if the n-th unit vector field Vn makes a constant angle ϕ with a fixed direction Χ , that is, π V , Χ = cos( ϕ), ϕ ≠ , ϕ = constant n 2 along the curve, where Χ is unit vector field in E n [3]. 3. THE SPECIAL CURVES ON THE HELIX HYPERSURFACES IN EUCLIDEAN n-SPACE E n Theorem 3.1 Let M be a helix hypersurface with the direction d in E n and let α : Ι ⊂ ΙR → M be a unit speed geodesic curve on M . Then, the curve α is a V2 -slant helix with the direction d in E n . Proof: Let ξ be a normal vector field on M . Since M is a helix hypersurface with respect to d , d,ξ = constant . That is, the angle between d and ξ is constant on every point of the surface M . And, α ′′(s) = λξ | along the curve α since α is a α (s) geodesic curve on M . Moreover, by using the Frenet equation α′′(s) = V1′ = k1V2 , we 3 obtain λξ | α (s) = k1V2 , where k1 is first curvature of α . Thus, from the last equation, by taking norms on both sides, we obtain ξ = V2 or ξ = -V 2 . So, d,V2 is constant along the curve α since d,ξ = constant . In other words, the angle between d and V2 is constant along the curve α . Consequently, the curve α is a V2 -slant helix with the direction d in E n . Corollary 3.1 For n = 3, the following Theorem obtained. Theorem: Let α : Ι ⊂ ΙR → M be a curve on a constant angle surface M with unit normal N and the fixed direction k . If a curve α on M is a geodesic, then α is a slant helix with the axis k in E 3 (see [5]) . Theorem 3.2 Let M be a helix hypersurface in E n and let α : Ι ⊂ ΙR → M a unit speed curve on M . If the n-th unit vector field Vn of α equals to ξ or - ξ , where ξ is n a normal vector field on M , then α is a Vn -slant helix with the direction d in E . Proof: Let d ≠ 0 ∈ E n be a fixed direciton of the helix hypersurface M . Since M is a helix hypersurface with respect to d , d,ξ = constant . That is, the angle between d and ξ is constant on every point of the surface M . Let the n-th unit vector field Vn of α be equals to ξ or - ξ . Then d,Vn is constant along the curve α since d,ξ = constant . That is, the angle between d and Vn is constant along the curve α . n Finally, the curve α is a Vn -slant helix in E . Theorem 3.3 Let M be a helix hypersurface with the direction d in E n and let α : Ι ⊂ ΙR → M (α (t) ∈ M , t ∈ Ι ) be a curve on the surface M . If α is a line of curvature on M , then d ∈ Sp {T}⊥ along the curve α , where T is tangent vector field of α . Proof: Since M is a helix hypersurface with the direction d , 4 N oα,d = constant along the curve α , where N is the normal vector field of M . If we are taking the derivative in each part of the equality with respect to t , we obtain : (N oα ′,) d = 0. Since α is a line of curvature on M , (N oα )′ = S(T ) = λT , where S is the shape operator of the surface M . So, we have T,d = 0 . Finally, d ∈ Sp {T}⊥ along the curve α . 4. THE RULED SURFACES IN ΙR n Definition 4.1 Let H ⊂ ΙR n−1 be a orientable hypersurface in ΙR n−1 and let β be a curve on H , where β : Ι ⊂ ΙR → H ⊂ ΙR 1-n . t → β (t) Then, Φ t,( s) = β (t) + s(sin( θ )N(β (t))+ cos( θ )d ) n is a ruled surface with dimension 2 on fθ in ΙR ( fθ was defined in Theorem 2.1), where θ constant, N is a unitary normal vector field of H and d is constant vector as defined in Theorem 2.1. The surface Φ will be called the ruled surface generated by the curve β . Theorem 4.1 The ruled surface Φ t,( s) defined above is developable if and only if the curve β is a line of curvature on the surface H . Proof: We assume that β is a line of curvature on H . Let consider the surface Φ t,( s) = β (t) + s(sin( θ )N(β (t))+ cos( θ )d ) with rulings X (t) = sin( θ )N(β (t))+ cos( θ)d and directrix β . If we are taking the partial derivative in each part of the equality with respect to t , we obtain: 5 ∂Φ ′ Φ t = = β ′ + ()()s sin( θ ) N o β (1) ∂t And, S(T ) = λT since β is a line of curvature on H , where T is tangent vector field of dN β and S is the shape operator of the surface H . Besides, (N β )′ = = S(T). o dt Therefore, (N o β)′ = λT and by using (1), we obtain the equality Φ t = (1+ λ s.. sin( θ )) β ′ = (1+ λ s.. sin( θ )) T. Hence, the system {Φ t ,T} is linear dependent.
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