arXiv:1706.05321v1 [math.DG] 16 Jun 2017
Legender Curves and Singularities of a Ruled Surfaces According to the RM Frame M.Bekar1 , F.Hathout2 , Y.Yayli3 1
Department of mathematics and computer sciences, Necmettin Erbakan University, 42090 Konya, Turkey. Email:
[email protected] 2
Department of mathematics, Saida University, 2000 Saida, Algeria. Email:
[email protected] 3
Department of mathematics, Ankara University, 06100 Ankara, Turkey. Email:
[email protected]
Abstract In this paper, Legender curves on the unit tangent bundle are obtained by using the rotation minimizing (RM) vector fields. Moreover, the ruled surfaces corresponding to these Legender curves are given. Finally, the singularities of these ruled surfaces are investigated. Key words: RM vectors, tangent bundle of sphere, Legender curves, ruled surface, singularity. 2010 AMS Mathematics Subject Classification: 53A04, 32S25
1
Introduction
One of the most orthonormal frame on a space curve is the Frenet frame, it comprising the unit tangent T , the unit normal N , and binormal B = T × N . With Frenet frame, the orient a body along a path have an angular velocity ω satisfies < ω, B >≡ 0, it has no component in the principal normal direction. This means that the body exhibits no instantaneous rotation about the principal normal vector p from point to point along the path. Bishop in [3] introduce an alternative to the Frenet moving frame along a curve in Rn called rotation minimizing frames (RM frame) that have no 1
instantaneous rotation about T . Currently, the RM frame have a large using in mathematical research (see [1], [11]...) and computer aided geometric design (see [6]). More precisely, in n-dimensional Riemannian manifold (M, g = ), an orthonormal frame defined by the tangent vector of the curve γ in M and n−1 normal vectors Ni , which do not rotate with respect to the tangent (i.e., ∇T Ni is proportional to T = γ 0 (t) and ∇ is the Levi Civita connection of g), is called an RM frame along a curve γ in M. Such a normal vector field along a curve is said to be a RM (i.e. a rotation minimizing) vector field. Any orthonormal basis {T (t0 ), N1 (t0 ), .., Nn−1 (t0 )} at a point γ(t0 ) defines a unique RM frame along the curve. The RM frame can be defined at any situation of the derivatives of the curve γ. The notion of RM frame particularizes to that of Bishop in Euclidian case (see [5]). The Frenet equations of RM frame is given by ∇T T (t) =
n−1 X
κi (s)Ni (s) and ∇T Ni (t) = κi (s)T (s)
i=1
where κi (t) are called the natural curvatures along the curve γ. In other hand, the Legender curves, especially in the tangent bundle of 2-sphere T S 2 , are studied by many authors (see [8], [9],...), we call the pair Γ = (γ, v) ⊂ T S 2 that satisfy < γ 0 , v >= 0 a Legender curve. We proof that any two RM vector fields correspond to a Legender curve in U T S 2 of some curves, in the theorems 3 and 4. We know from [9], that any Legender curve in T S 2 correspond a developable ruled surface. According to the RM Frame a long curve in 3dimensional manifold, one can define six ruled surfaces. Finally, we want to describe what behavior of singularities on our ruled surfaces. The study of this singularities show that our six ruled surfaces can be Cuspidal edge C ×R, Swallowtail SW Cuspidal crosscap CCR or cone surface. This present paper is divided into two parts, we give section 2 some definitions and notions about the Legender curves in U T S 2 and the RM vectors and some relationships are given in the Theorems 3 and 4. In section 3 we show that the ruled surface obtained from the RM frame are developable and we study singularities of these ruled surfaces in the Theorems 6, 8 and 9. We close the section 3 with some corollaries and examples. All curves and manifolds considered here are of class C ∞ unless stated otherwise. 2
2
Legender curves and RM vectors fields
Let γ : I ⊂ R → M be a non-null curve with arc-length parameter s in three-dimensional Riemannian manifold (M, g = ). Then, there exists an accompanying three-frame {T, N, B} known as the Frenet-Serret frame of γ = γ(s). In this case, the moving Frenet-Serret formulas in M are given by; ∇T T (s) 0 κ(s) 0 T (s) ∇T N (s) = −κ(s) 0 τ (s) N (s) , (1) ∇T B(s) 0 −τ (s) 0 B(s) where κ(s) and τ (s) are called, respectively, the curvature and the torsion of the curve α at the point s. The set {T, N, B, κ, τ } is also called the Frenetframe apparatus. Definition 1. Let γ be a curve in (M, g). A normal vector field N over γ is said to be a rotation minimizing vector field (RM vector field, for short) if it is parallel with respect to the normal connection of γ. That means, ∇γ 0 N and γ 0 are proportional. A RM frame along a curve γ = γ(s) in (M 3 , g) is an orthonormal frame defined by the tangent vectors T and two normal vector fields N1 and N2 , where N1 and N2 are proportional to T . Any orthonormal basis {T, N1 , N2 } at a point γ(s0 ) defines a unique RM frame along the curve γ. Let ∇ be the Levi Civita connection of the metric g. Then, Frenet type equations read as ∇T T (s) 0 κ1 (s) κ2 (s) T (s) ∇T N1 (s) = −κ1 (s) 0 0 N1 (s) . (2) ∇T N2 (s) −κ2 (s) 0 0 N2 (s) Here, the functions κ1 (s) and κ2 (s) are called the natural curvatures of RM frame which are given by q κ1 (s)κ02 (s) − κ01 (s)κ2 (s) , κ(s) = κ21 (s) + κ22 (s) and τ (s) = θ0 (s) = κ21 (s) + κ22 (s) (s) where θ(s) = arg(κ1 (s), κ2 (s)) = arctan κκ12 (s) and θ0 (s) is the derivative of θ(s) with respect to the arc-length.
If (M, g) is the Euclidean 3-space (R3 , ), then the notion of RM frame particularizes to that of Bishop frame. 3
Let S2 be the unit 2-sphere in R3 . Then, the tangent bundle of S2 is given by T S2 = {(γ, v) ∈ R3 × R3 ; |γ| = 1 and < γ, v >= 0}. And the unit tangent bundle of S2 is given by U T S2 = {(γ, v) ∈ R3 × R3 ; |γ| = |v| = 1 and < γ, v >= 0} = {(γ, v) ∈ S2 × S2 ; < γ, v >= 0}
(3)
which is a 3-dimensional contact manifold and its canonical contact 1-form is θ, where ”” and ”|, |” denotes, respectively, the inner product and the norm in R3 . Definition 2. The smooth curve Γ(s) = (γ(s), v(s)) : I ⊂ R → U T S2 ⊂ S2 × S2 is called the Legender curve in U T S2 if < γ 0 (s), v(s) >= 0.
(4)
Thus, the following theorem can be given; Theorem 3. Let γ : I ⊂ R → S2 be a regular unit speed curve with the frame apparatus {T, N, B, κ, τ }. Then, we have the following assertions: 1. If N1 (s) and N2 (s) are RM vector fields along the curve γ, then the curve (N1 (s), N2 (s)) is Legender in U T S 2 . 2. RIf N 1 (s) and N 2 (s) are RM vectors along a B-direction curve b(s) = B(s)ds, then the curve (N 1 (s), N 2 (s)) is Legender in U T S 2 . 3. If B(s)R and T (s) are RM vector fields along the N -direction curve β(s) = N (s)ds, then the curve (B(s), T (s)) is Legender in U T S 2 . Proof. Assume that γ : I ⊂ R → S2 is a regular unit speed curve with the frame apparatus {T, N, B, κ, τ }. Then, 1. Let us consider the curve Γ(s) = (N1 (s), N2 (s)) ∈ U T S2 . Since N1 (s) and N2 (s) are RM vector fields along the curve γ(s), from Eq. (2) we get that < N10 (s), N2 (s) >= −κ1 < T (s), N2 (s) >= 0. Thus, from Eq. (4) we can say that the curve Γ is Legender in U T S2 . 4
2. Let us consider the curve Γ(s) = (N 1 (s), N 2 (s)) ∈ U T S2 along the B-direction curve β(s). The Frenet type equations can be given as 0 B (s) B(s) 0 κ ¯ 1 (s) κ ¯ 2 (s) N 0 (s) = −¯ κ1 (s) 0 0 N 1 (s) (5) 1 0 −¯ κ2 (s) 0 0 N 2 (s) N2 (s) with the natural curvatures q κ ¯ 0 (s)¯ κ2 (s) − κ ¯ 01 (s)¯ κ2 (s) . ¯ 22 (s) and τ¯(s) = θ0 (s) = 1 κ ¯ (s) = κ ¯ 21 (s) + κ 2 2 ¯ 2 (s) κ ¯ 1 (s) + κ From Eq. (5), we get that 0
< N 1 (s), N 2 (s) >= −¯ κ1 < B(s), N 2 (s) >= 0. Thus, from Eq. (4), we can say that the curve Γ is Legender in U T S2 . The proof of Assertion 3 can be given by the similar way as Assertions 1 and 2. From the definition of the set U T S2 , we know that for a smooth curve Γ(s) = (γ(s), v(s)) in T S2 , the inner product of γ and v is zero, i.e. < γ(s), v(s) >= 0. We can define a new frame using the unit vector η(s) = γ(s) ∧ v(s). It is obvious that < γ(s), η(s) > = < v(s), η(s) >= 0. So, we have the following Frenet frame {γ(s), v(s), η(s)} along γ(s): 0 γ (s) 0 l(s) m(s) γ(s) v 0 (s) = −l(s) 0 n(s) v(s) . (6) 0 η (s) −m(s) −n(s) 0 η(s) Here l(s) =< γ 0 (s), v(s) >, m(s) =< γ 0 (s), µ(s) >, n(s) =< v 0 (s), µ(s) > . The triple (l, m, n) is called the curvature functions of Γ. We remark that, if l(s) = 0, then the curve Γ(s) = (γ(s), v(s)) is a Legender in U T S2 with the curvature functions (m, n). Theorem 4. Let Γ(s) = (γ(s), v(s)) be a smooth curve in U T S2 . If Γ(s) is Legender, then the vectors (γ(s), R v(s)) are RM vector fields along the η-direction curve β (i.e., β(s) = η(s)ds), and the triple vector field set {γ, v, η} is an RM frame. 5
Proof. Γ(s) = (γ(s), v(s)) be a smooth Legender curve in U T S2 . Then, the frenet frame Eq. (6) for Legender condition (that is, l(s) = 0) can be given by 0 η (s) 0 −m(s) −n(s) η(s) γ 0 (s) = m(s) 0 0 γ(s) . (7) 0 v (s) n(s) 0 0 v(s) Then, from Eq. (2), weRcan say that {η, γ, v} is an RM frame along the η-direction curve β(s) = η(s)ds.
3
Singularities of ruled surfaces according to RM frame
A ruled surface in R3 is locally the map Φ(β,α) : I × R −→ R3 defined by Φ(β,α) (s, u) = β(s) + uα(s), where β : I−→ R3 , α : I−→ R3 \ {0} are smooth mappings and I is an open interval or a unit circle S1 . We call β the base curve (or directrix ) and α the director curve. The straight lines u−→β(s) + uα(s) are called rulings. The striction curve of the ruled surface Φ(β,α) (s, u) = β(s) + uα(s) is defined by hβ 0 (s), α0 (s)i ¯ α(s) (8) β(s) = β(s) − 0 hα (s), α0 (s)i ¯ coincides with the base and if hβ 0 (s), α0 (s)i = 0, then the striction curve β(s) curve β(s). A ruled surface Φ(β,α) (s, u) = β(s) + uα(s) is said to be developable if det (β 0 (s), α(s), α0 (s)) = 0. From Theorem 4 we can say that if Γ is a Legender curve, then the R vector set {η, γ, v} is an RM frame along the η-direction curve β(s) = η(s)ds. One can define by this frame the following six ruled surfaces: Φ(a
1i ,a2i )
(s, u) = a1i (s) + ui a2i (s); 6
for i = 1, ...,6
(9)
where a1i (s) and a2i (s) are different unit curves from the set {β(s), γ(s), v(s)} . Proposition 5. All ruled surfaces Φ(a ,a ) (s, u), for i = 1, ..., 6, defined by 1i 2i Eq. (9) are developable. Proof. Let Φ(a11 ,a21 ) (s, u) = β(s) + uγ(s) be a ruled surface defined by Eq. (9). Then using Eq. (7), we get det(β 0 (s), γ(s), γ 0 (s)) = det(η(s), γ(s), m(s)η(s)) = 0 which is the developability condition of Φ(a11 ,a21 ) . The proof of the other ruled surfaces Φ(a ,a ) for i = 2, ..., 6 can be given by the similar way. 1i
2i
Now, we recall the parametric equations of three types of surfaces in R3 , which are given in [10], see Figure 1: 1. Cuspidal edge; C × R = {(x1 , x2 ) ; x21 = x32 } × R. 2. Swallowtail ; SW= {(x1 , x2 , x3 ) ; x1 = 3u4 + u2 v, x2 = 4u3 + 2uv, x3 = v}. 3. Cuspidal crosscap; CCR= {(x1 , x2 , x3 ) ; x1 = u3 , x2 = u3 v 3 , x3 = v 2 }.
Figure 1: Left surface is the Cuspidal edge C × R, middle surface is the Swallowtail SW and right surface is the Cuspidal crosscap CCR. By the following theorem, we can give the local classification of singularities of the ruled surfaces defined by Eq. (7). 7
Theorem 6. Let Γ(s) = (γ(s), v(s)) be a smooth Legender curve in U T S2 . Then, according to RM frame {η, γ, v} along the η-direction curve β(s), we have the following: 1. Φ(β,γ) (s, u) = β(s) + uγ(s) which is locally diffeomorphic to; (a) Cuspidal edge C×R at Φ(β,γ) (s0 , u0 ) if and only if u0 = −m(s0 )−1 6= 0 and m0 (s0 ) 6= 0. (b) Swallowtail SW at Φ(β,γ) (s0 , u0 ) if and only if u0 = −m(s0 )−1 6= 0, m0 (s0 ) = 0 and (m(s0 )−1 )00 (s0 ) 6= 0. 2. Φ(β,v) (s, u) = β(s) + uv(s) which is locally diffeomorphic to; (a) Cuspidal edge C×R at Φ(β,v) (s0 , u0 ) if and only if u0 = −n(s0 )−1 6= 0 and u0 (s0 ) 6= 0. (b) Swallowtail SW at Φ(β,v) (s0 , u0 ) if and only if u0 = −n(s0 )−1 6= 0, n0 (s0 ) = 0 and (n(s0 )−1 )00 (s0 ) 6= 0. 3. Φ(β,γ) (s, u) = β(s) + uγ(s) (resp. Φ(β,v) (s, u) = β(s) + uv(s)) which is a cone surface if and only if m(s) (resp., n(s)) is constant. Proof. Assume that Γ(s) = (γ(s), v(s)) is a smooth Legender curve in U T S2 according to the RM frame {η, γ, v} along the η-direction curve β(s). Using Eq. (7) and Φ(β,γ) (s, u) = β(s) + uγ(s), we get ∂Φ(β,γ) (s, u) = (1 + u m(s))η, ∂s ∂Φ(β,γ) (s, u) = γ, ∂u ∂Φ(β,γ) ∂Φ(β,γ) (s, u) ∧ (s, u) = (1 + u m(s))v, ∂s ∂u where ”∧” denotes the vector product in R3 . Singularities of the normal vector field of Φ(β,γ) = Φ(β,γ) (s, u) are u=
−1 . m(s)
From the Theorem 3.3 of the paper [10], we know that if there exists a paramm0 (s0 ) −1 0 eter s0 such that u0 = m(s = 6 0 and u = 6= 0 (i.e,. m0 (s0 ) 6= 0), then 0 m2 (s0 ) 0) 8
Φ(s, u) is locally diffeomorphic to the cuspidal edge C × R at Φ(β,γ) (s0 , u0 ). This completes the proof of Assertion 1.(a). Again from Theorem 3.3 of −1 6= 0, [10], we know that if there exists a parameter s0 such that u0 = m(s 0) 0
m (s0 ) −1 00 u00 = m ) (s0 ) 6= 0, then Φ(β,γ) is locally diffeomorphic 2 (s ) = 0 and (m(s0 ) 0 to swallowtail SW at Φ(β,γ) (s0 , u0 ). This completes the proof of Assertion 1.(b). Proof of Assertion 2 can be given similar to the proof of Assertion 1.Finally, for the proof of Assertion 3 can be given as: The singularity points is equal to the striction curve of Φ and can be given by
ϕ(β,γ) (s) = Φ(β,γ) (s,
1 −1 ) = β(s) − γ(s) m(s) m(s)
−1 1 resp., ϕ(β,v) (s) = Φ(β,v) (s, ) = β(s) − v(s) , m(s) m(s) Thus, we have ϕ0(β,γ) (s)
1 0 = −( ) γ(s) m(s)
resp,
ϕ0(β,v) (s)
1 0 = −( ) v(s) m(s)
This means that if m(s) is a constant function, then ϕ0(β,γ) (s) = ϕ0(β,v) (s) = 0. So, Φ(β,γ) (resp., Φ(β,v) ) has only one singularity point and thus it is a cone surface. Corollary 7. Let α : I ⊂ R → R3 be a smooth curve with frame apparatus {N1 , N2 , κ1 , κ2 } given by Eq. (5). If we choose Γ(s) = (γ(s), v(s)) = Γ(N1 (s), N2 (s)), then we obtain the Theorem 3.1 given in [7]. If we choose Γ(s) = (γ, v) = Γ(N2 (s), N1 (s)), then we obtain the Theorem 3.2 given in [7]. Proof. Let α : I ⊂ R → R3 be a smooth curve with frame apparatus {N1 , N2 , κ1 , κ2 } given by Eq. (2). Then, the vector fieldsR {T, N1 , N2 } is an RM frame along the T -direction curve β(s) = α(s) = T (s)ds. That means, Γ(N1 (s), N2 (s)) is a Legender curve in T S2 . Using Theorem 6, we complete the proof, where m(s) = κ1 (s) and n(s) = κ2 (s). Theorem 8. Let Γ(s) = (γ(s), v(s)) be a smooth Legender curve in U T S2 .Then, according to RM frame {η, γ, v} along the η-direction curve β(s), we have the foolowing: 9
1. Φ(γ,β) (s, u) = γ(s) + uβ(s) which is locally diffeomorphic to; (a) Cuspidal edge C ×R at Φ(γ,β) (s0 , u0 ) if and only if u0 = −m(s0 ) 6= 0 and m0 (s0 ) 6= 0. (b) Swallowtail SW at Φ(γ,β) (s0 , u0 ) if and only if u0 = −m(s0 ) 6= 0, m0 (s0 ) = 0 and m00 (s0 ) 6= 0. (c) Cuspidal crosscap CCR at Φ(γ,β) (s0 , u0 ) if and only if u0 = −m(s0 ) = 0 and m0 (s0 ) 6= 0. 2. Φ(v,β) (s, u) = v(s) + uβ(s) which is locally diffeomorphic to; (a) Cuspidal edge C ×R at Φ(v,β) (s0 , u0 ) if and only if u0 = −n(s0 ) 6= 0 and n0 (s0 ) 6= 0. (b) Swallowtail SW at Φ(v,β) (s0 , u0 ) if and only if u0 = −n(s0 ), n0 (s0 ) = 0 and n00 (s0 ) 6= 0. (c) Cuspidal crosscap CCR at Φ(v,β) (s0 , u0 ) if and only if u0 = −n(s0 ) = 0 and n0 (s0 ) 6= 0. 3. Φ(γ,β) (s, u) = γ(s) + uβ(s) (resp., Φ(v,β) (s, u) = v(s) + uβ(s)) which is a cone surface if and only if m(s) (resp., n(s)) is constant. Proof of the Theorem 8 can be given similar to the proof of Theorem 6. Theorem 9. Let Γ(s) = (γ(s), v(s)) be a smooth Legender curve in U T S2 with curvature functions (m, n). Then we have the following: 1. Ruled surface Φ(γ,v) (s, u) = γ(s) + uv(s) is locally diffeomorphic to; (a) Cuspidal edge C×R at Φ(γ,v) (s0 , u0 ) if and only if u0 = − m (s0 ) 6= n m 0 0 and ( n ) (s0 ) 6= 0. (b) Swallowtail SW at Φ(γ,v) (s0 , u0 ) if and only if u0 = − m (s0 ) 6= 0, n m 00 0 (m ) (s ) = 0 and ( ) (s ) = 6 0. 0 0 n n (c) Cuspidal crosscap CCR at Φ(γ,v) (s0 , u0 ) if and only if u0 = − m (s0 ) = n m 0 0 i.e. m(s0 ) = 0 and ( n ) (s0 ) 6= 0. 2. Ruled surface Φ(v,γ) (s, u) = v(s) + uγ(s) is locally diffeomorphic to; 10
n (s0 ) 6= (a) Cuspidal edge C×R at Φ(v,γ) (s0 , u0 ) if and only if u0 = − m n 0 0 and ( m ) (s0 ) 6= 0. n (b) Swallowtail SW at Φ(v,γ) (s0 , u0 ) if and only if u0 = − m (s0 ) 6= 0, n 0 n 00 ( m ) (s0 ) = 0 and ( m ) (s0 ) 6= 0 n (s0 ) = (c) Cuspidal crosscap CCR at Φ(v,γ) (s0 , u0 ) if and only if u0 = − m n 0 0 i.e. n(s0 ) = 0 and ( m ) (s0 )(s0 ) 6= 0.
3. Ruled surfaces Φ(γ,v) (s, u) = γ(s) + uv(s) (resp., Φ(v,γ) (s, u) = v(s) + n (s) (resp., m (s)) is constant. uγ(s)) is a cone surface if and only if m n Corollary 10. Let α : I ⊂ R → R3 be a smooth curve with frame apparatus {T, N, B, κ, τ }. If we choose Γ(s) = (γ(s), v(s)) = Γ(B(s), T (s)), then we obtain the Theorem 3.2 given in [10]. Proof. Since TR and B are RM vector fields along the T -direction curve β(s) = α(s) = T (s)ds, the curve Γ(B(s), T (s)) is a Legender in T S2 . Using Theorem 8 and taken m(s) = κ1 (s) and n(s) = κ2 (s), we get the proof. 3 Corollary 11. Let α : I ⊂ R → R be a smooth curve with frame apparatus N, C, W = D, f, g , (see [2]). If we choose Γ(s) = (γ(s), v(s)) = Γ(D(s), N (s)), then we obtain the Theorem 3.3 given in [10], where
W = D(s) =
τ (s)T (s) + κ(s)B(s) p κ2 (s) + τ 2 (s)
is the unit Darboux vector field. 3 Proof. Let α : I ⊂ R → R be a smooth curve with frame apparatus N, C, W = D, f, g . Then, the curve Γ(s) = (γ(s), v(s)) = Γ(D(s), N (s)) is a Legender in T S2 . Using Theorem 9 and taking account that we get the slant helix condition ! m κ2 τ 0 (s) = ) (s) = σ(s) 3 ( n (κ2 + τ 2 ) 2 κ
which completes the proof.
11
We close this section with some examples. The following example is an application of the Theorem 9. Example 12. Let γ : I ⊂ R → R3 be a smooth curve given by 1 γ(s) = √ (− cos s, − sin s, 1) 2 and the unit vector
1 v(s) = √ (cos s, sin s, 0). 2
Then, we have < γ 0 (s), v >= 0. Hence, Γ(s) = (γ, v) is a LegenderRcurve in U T S2 . The RM frame {η, γ, v} along the η-direction curve β(s) = η(s)ds is given by 0 √1 √1 0 − η (s) η(s) 2 2 1 γ 0 (s) = 0 − √2 0 γ(s) . √1 v 0 (s) v(s) 0 0 2 The ruled surface 1 Φ(v,γ) (s, u) = v(s) + uγ(s) = √ (cos s − u cos s, sin s − u sin s, u) 2 is a cone surface, see Figure 2. The following example is an application of the Theorem 8. Example 13. Let α : I ⊂ R → R3 be a smooth curve defined by s s s γ(s) = (cos √ , sin √ , √ ). 2 2 2 Then, the speed and binormal vectors are, respectively, 1 s s T (s) = √ (− sin √ , cos √ , 1), 2 2 2 1 s s B(s) = √ (sin √ , cos √ , 1) 2 2 2 12
Figure 2: The ruled surface Φ(v,γ) (s, u) is the cone surface with one singularity point. with the curvature κ = 12 and the torsion τ = 12 . So, γ is a helix. The curve Γ(s) = (B, T ) is a Legender curve in U T S2 and the ruled surface Φ(B,T ) (s, u) = B(s) + uT (s) s s 1 = √ ((1 − u) sin √ , (u + 1) cos √ , 1 + u) 2 2 2 is a cone surface. √ In fact, the singularity point is for u = 1 on the point Φ(B,T ) (s, 1) = (0, 0, 2), see Figure 3. Finally, the following example is an application of the Theorem 6. Example 14. Let α : I = [0, A] → R3 be a smooth curve (for 0 < A 6 2π) defined by √ 1 (3 cos s − cos 3s, 3 sin s − sin 3s, 2 3 cos s), 4 √ 1 v(s) = (3 sin s − sin 3s, −3 cos s − cos 3s, −2 3 sin s), 4 √ 1 √ η(s) = ( 3 cos 2s, 3 sin 2s, −1). 2
γ(s) =
13
Figure 3: The ruled surface Φ(B,T ) (s, u) is the cone surface with helix singularity curve. Γ(s) = (γ(s), v(s)) is a Legender curve with Legender curvature function √ m(s) = 3 sin s and we have the following: 1. If A = π, then m( π2 ) = The ruled surface
√
√ 3 6= 0, m0 ( π2 ) = 0 and m00 ( π2 ) = − 3 6= 0.
Φ(β,γ) (s, u) = β(s) + uγ(s) √ 3 3 1 = ( sin 2s + u cos s − u cos 3s, (10) 2 4 4 √ √ 3 3 1 s 3 − cos 2s − u sin s − u sin 3s, − + u cos s) 2 4 4 2 2 −1 is locally diffeomorphic to the cuspidal edge C × R at Φ(β,γ) ( π2 , √ ), see 3 Figure 4.
2. If A = π2 , then u0 = m−1 (s0 ) 6= 0, (m−1 )0 (s0 ) 6= 0. The ruled surface given in Eq. (10), is locally diffeomorphic to the swallowtail SW at Φ(β,γ) ( π2 , u0 ), see Figure 5. 14
−1 Figure 4: The cuspidal edge C × R at Φ(β,γ) ( π2 , √ ) 3
Figure 5: The swallowtail SW at Φ(β,γ) ( π2 , u0 )
15
4
Conclusions
In this paper, we have given the Legender curves on the unit tangent bundle by using the rotation minimizing (RM) vector fields. We presented the ruled surfaces corresponding to these Legender curves and discussed their singularities. For some special cases we obtained the studies in [7] and [10], which we have given by the Corollaries 7, 10 and 11.
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