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

Home , Ruled surface

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

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].

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 d is the vector 0,0( ,..., )1 ∈ ΙR n 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, ′′ | along the curve since is a α (s) = λξ α (s) α α 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 ,

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:

∂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. And, we know that a tangent plane along a ruling is spanned by Φ = X (t ) and Φ . Finally, the tangent planes are parallel along s t the ruling sin( θ )N(β (t))+ cos( θ )d passing from the point β (t) . That is, the surface Φ t,( s) is developable.

We assume that the ruled surface Φ t,( s) is developable. Then, the system {Φ t ,T} is linear dependent. So, from the equality ∂Φ Φ = = β ′ + ()()s sin( θ ) ( N β )′ = T + s sin( θ) ( N β)′ , t ∂t o o we get (N o β)′ = λT . Therefore (N o β)′ = S(T) = λT , where S is the shape operator of the surface H . That is, β is a line of curvature on H .

Corollary 4.1 Let H be the hypersphere

n−1 → n−2  2  1-n S = x = (x1 , x2 ,..., xn−1 ) : f (x) = ∑ xi = 1 , ∇f ≠ 0 ⊂ ΙR .  i=1 

Let β be a curve on H = S n−2 where

β : Ι ⊂ ΙR → H = S n−2 ⊂ ΙR n−1 . t → β(t)

Then, the ruled surface Φ t,( s) = β(t) + s(sin( θ )β(t) + cos( θ) d ) ⊂ ΙR n is always developable from Theorem 4.1. Because, each curve on the hypersphere S n−2 is a line of curvature.

5. HELIX SURFACES GENERATED BY A PLANE CURVE IN EUCLIDEAN 3- SPACE E 3

Let α : Ι ⊂ ΙR → E 3

u → α(u) be a plane curve in Euclidean 3-space E 3 . And, we denote the tangent, principal normal, the binormal of α by V1 = T , V2 and V3 = B , respectively. Note that binormal of a plane curve in E 3 is constant.

Definition 5.1 We can obtain a ruled surface by using the plane curve α such that φ :U ⊂ E 2 → E 3 u v u v u v V u B ( , ) → φ( , ) = α( ) + ()sin( θ ) 2 ( ) + cos( θ ) . The ruled surface will be called as the surface generated by the curve α .

Theorem 5.1 The ruled surface φ :U ⊂ E 2 → E 3

(u,v) → φ(u,v) = α(u) + v()sin( θ)V2 (u) + cos( θ)B is a helix surface with the direction B in E 3 , where θ is constant , α is a plane curve and B is a constant vector which is perpendicular to the plane of the curve α .

Proof: We want to show that Z, B is a constant function along φ , where Z is a normal vector field of φ . First, we are going to find a normal vector field Z . To do this, we will compute the partial derivatives of φ with respect to u and v . Note that k2 = 0 for the curve α since α is a planer curve in E 3 .

φu (u,v) = (1− k1 v.. sin( θ))T and φv (u,v) = sin( θ )V2 + cos( θ )B . (2) Using the equalities in (2), a normal to the surface φ is given by

φu ×φv Z = = −cos( θ )V2 + sin( θ)B . φu ×φv

So, we have Z, B = sin( θ) = const . Finally, φ is a helix surface with the direction B in E 3 .

This completes the proof.

Corollary 5.1 The helix surface φ :U ⊂ E 2 → E 3

(u,v) → φ(u,v) = α(u) + v()sin( θ )V2 (u) + cos( θ )B . is always developable.

Proof: We know that If det( T , X , X ′) = ,0 where X = sin( θ )V2 + cos( θ )B and T tangent of α , then φ is developable. So, we will compute det( T , X , X ′ :) T = α′

X = sin( θ )V2 + cos( θ )B

X ′ = −k1.sin( θ ). T and so, we have det( T , X , X ′) = 0 . This completes the proof.

Theorem 5.2 Let α : Ι ⊂ ΙR → E 3

s →α(s) be a plane curve (not a straight line) with unit speed in Euclidean 3-space E 3 . We consider the helix surface (generated by the curve α(s))

φ :U ⊂ E 2 → E 3

(s, v) → φ(s,v) = α(s) + v()sin( θ)V2 (s) + cos( θ)B . Then, the Gauss curvature of φ is zero, and the mean curvature of φ : 1 k .cos( θ) H = . 1 , 2 1− k1 v.. sin( θ) where k1 is the first curvature of the curve α .

Proof: From corollary 5.1, the surface φ is developable. So, the Gauss curvature of φ is zero.

1 k .cos( θ) Now, we are going to prove that H = . 1 . 2 1− k1 v.. sin( θ) x x The system { 1 , 2 } is an orthonormal basis for the tangent space of φ at the point

φs φ(s,v) , where x1 = and x2 = φv (φ s is partial derivative of φ with respect to s φs and φv is partial derivative of φ with respect to v ). Recall that a normal vector field of

φ is Z (s,v) = − cos( θ )V2 (s) + sin( θ )B by Theorem 5.1. And, we know that the mean curvature of φ at a point φ(s,v) :

1 2 H ()φ(s,v) = ∑ S(xi ), xi , 2 i=1 where S is the shape operator of φ .

So firstly, we will compute S(x1 ) and S(x2 ) :

1 1 dZ  k .cos( θ )  S(x ) = D Z = D Z = D Z = =  1 T 1 x1 φS φ φs φ ds  1− k v.. sin( θ )  φS s s  1  and dZ S(x ) = D Z = D Z = = 0 , 2 x2 φv dv where D is standard covariant derivative in E 3 . Therefore, we have

k1.cos( θ) S(x1), x1 = and S(x2 ), x2 = 0 . 1− k1 v.. sin( θ) Finally,

2 1 1 k1.cos( θ ) H = ∑ S(xi ), xi = . , 2 i=1 2 1− k1 v.. sin( θ ) where 1− k1 v.. sin( θ) ≠ 0 . This completes the proof.

Corollary 5.2 The surface φ defined above is minimal if and only if θ = π 2 where

1− k1 v.. sin( θ) ≠ 0 . In that case (whenever θ = π 2 ), the surface φ is a plane.

Example 5.1 Let the curve α (u) be a plane curve parametrized by the vector function

 3 4  α(u) =  sin( u 1), + cos( u), sin( u), u ∈[ 5,0 π ].  5 5  Then,  3 4  V = − sin( u), −cos( u), − sin( u) 2  5 5 

 4 3  B =  ,0, −   5 5  where V2 is the principal normal and B is the binormal of α , respectively. So, If we choose θ = π 6 and v ∈ [ ,0 π ], the helix surface generated by the curve α (u) has the parametric representation:

3 3v 2 3 x = ( − )sin( u) + v 5 10 5 v y = 1( − ) cos( u) +1 2

4 2v 3 3v z = ( − )sin( u) − . 5 5 10 And, the surface generated by the curve α is shown the following Figure.

2 1.5 0 1 1 0.5 2 0

-1

-2

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[3] Gök, I., Camcı, Ç., Hacısaliho ğlu, H.H., 2009, Vn -slant helices in Euclidean n- space E n , Math. Commun., Vol. 14, No. 2, pp. 317-329.

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International electronic journal of geometry., Volume 4 No. 1 pp. 70-78.

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Contemp. Math. Sci., Vol. 2, no. 1, pp. 1-11.

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