CLAD HELICES AND DEVELOPABLE SURFACES( Title fulltext )

Author(s) TAKAHASHI,Takeshi; TAKEUCHI,Nobuko

Citation 東京学芸大学紀要. 自然科学系, 66: 1-9

Issue Date 2014-09-30

URL http://hdl.handle.net/2309/136938

Publisher 東京学芸大学学術情報委員会

Rights Bulletin of Tokyo Gakugei University, Division of Natural Sciences, 66: pp．1～ 9 ，2014

CLAD HELICES AND DEVELOPABLE SURFACES

Takeshi TAKAHASHI* and Nobuko TAKEUCHI**

Department of Mathematics

(Received for Publication; May 23, 2014)

TAKAHASHI, T and TAKEUCHI, N.: Clad Helices and Developable Surfaces. Bull. Tokyo Gakugei Univ. Div. Nat. Sci., 66: 1-9 (2014) ISSN 1880-4330

Abstract

We define new special curves in Euclidean 3-space which are generalizations of the notion of helices. Then we find a geometric invariant of a space curve which is related to the singularities of the special developable surface of the original curve.

Keywords: cylindrical helices, slant helices, clad helices, g-clad helices, developable surfaces, singularities

Department of Mathematics, Tokyo Gakugei University, 4-1-1 Nukuikita-machi, Koganei-shi, Tokyo 184-8501, Japan

1. Introduction

In this paper we define the notion of clad helices and g-clad helices which are generalizations of the notion of helices. Then we can find them as geodesics on the tangent developable(cf., §3). In §2 we describe basic notions and properties of space curves. We review the classification of singularities of the Darboux developable of a space curve in §4. We introduce the notion of the principal normal Darboux developable of a space curve. Then we find a geometric invariant of a clad helix which is related to the singularities of the principal normal Darboux developable of the original curve. In §5 we give examples of the principal normal Darboux developable of a space curve. All manifolds and maps considered here are of class C∞ unless otherwise stated.

2. Basic notions and properties

We now review some basic concepts on classical differential geometry of space curves in Euclidean space. For any two vectors x = (x , x , x ) and y = (y , y , y ), we denote x,y as the standard inner product. Let γ: I R3 be a curve with 1 2 3 1 2 3   → dγ/dt(t) ≠ 0. We also denote the norm of x by ||x||. The arc-length parameter s of a curve γ is determined such that ||γ´(s)|| = 1, where γ´(s) = dγ/ds(s). Let us denote e1(s) = γ´(s) and we call e1(s) a unit tangent vector of γ at s. We define the curvature of γ

* Takada High School (Joetsu-sh, Niigata, 943-8515, Japan) ** Tokyo Gakugei University (4-1-1 Nukuikita-machi, Koganei-shi, Tokyo, 184-8501, Japan)

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by κ(s) = ||γ´´ ( s)||. If κ(s) ≠ 0, then the unit principal normal vector e2(s) of the curve γ at s is given by γ´´ ( s) = κ(s)e2(s). The unit

vector e3(s) = e1(s) × e2(s) is called the unit binormal vector of γ at s. Then we have the Frenet-Serret formulae:   e'1(s) = κ(s)e2(s)  e'2(s) = κ(s)e1(s) + τ(s)e3(s) (2.1)  −  e' (s) = τ(s)e (s), 3 − 2 1 3 2 2 2 where τ (s) is the torsion of the curve γ at s. For any unit speed curve γ : I R , we define vector fields D˜ (s) = (1/(κ + τ ) )( s)(τ(s)e1(s) + κ(s)e3(s)) 1 1 → D˜ (s) = (1/(κ2 + τ2)2 )(s)(τ(s)e (s) + κ(s)e (s)) and Du(s) = (1/(κ2 + τ2)2 )(s)( κ(s)e (s) + τ(s)e (s)) along γ under the condition that κ(s) ≠ 0. Then we 1 3 − 1 3 have the following formulae :

 � e' (s) = κ(s)2 + τ(s)2Du(s)  2  u � 2 2 (D )'(s) = κ(s) + τ(s) ( e2(s) + σ(s)D˜ (s)) (2.2)  � −  D'˜ (s) = κ(s)2 + τ(s)2σ(s)Du(s), − � �3 where σ(s) = (κ2 (τ/κ)'� κ2 + τ2 2 )(s). A curve γ : I R3 with κ(s) ≠ 0 is called cylindrical helix if the tangent lines of γ make a constant angle with a fixed −→ direction. It has been known that the curve γ is a cylindrical helix if and only if (τ/κ)(s)=constant. If both of κ(s) ≠ 0 and τ(s) ≠ 0 are constant, it is, of course, a cylindrical helix. In [5] S. Izumiya and N. Takeuchi defined new special curve as follow : A curve γ with κ(s) ≠ 0 is called a slant helix if the principal normal lines of γ make a constant angle with a fixed direction. We remark that the principal normal lines of a cylindrical helix is perpendicular to a fixed direction, so that a cylindrical helix is a slant helix. S. Izumiya and N. Takeuchi showed the following characterization of slant helices.

Proposition 2.1. Let γ be a unit speed space curve with κ(s) ≠ 0. Then γ is a slant helix if and only if

2 τ ' σ(s) =  κ (s)  3 κ   2 2 � �   κ + τ 2     � � 

is a constant function.

If σ(s) 0, then we have (τ/κ)'(s) 0. It follows that γ is a cylindrical helix. We now define a new special curve whose ≡ ≡ notion is a generalization of the notion of a slant helix as follows: A curve γ with κ(s) ≠ 0 is called a clad helix if the spherical image of the unit principal normal e (s) : I S2 is a part of a cylindrical helix in S2. Therefore we remark that a slant helix is a 2 −→ clad helix. We have the following characterization of clad helices.

Proposition 2.2. Let γ be a unit speed space curve with κ(s) ≠ 0. Then γ is a clad helix if and only if

σ' ϕ(s) = (s) � (κ2 + τ2)1/2(1 + σ2)3/2 �

is a constant function.

Proof. We respectively denote K(s) and T(s) as the curvature and the torsion of the spherical image of the unit principal normal

e2(s). By a straightforward computation, we have

σ' K(s) = ( √1 + σ2)(s), T(s) = (s). � (κ2 + τ2)1/2(1 + σ2) �

2 It follows that ϕ(s) = (T/K)(s). Therefore, the image of e2(s) is a part of a cylindrical helix in S if and only if ϕ(s) is a constant function. □

If ϕ(s) ≡ 0, then we have σ´(s) ≡ 0. It follows that γ is a slant helix. Moreover,we define a new special curve whose notion is

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generalization of the notion of a clad helix as follows: A curve γ with κ(s) ≠ 0 is called a g-clad helix if the spherical image of the unit principal normal e (s): I S2 is a part of a slant helix in S2. We have the following characterization of general clad 2 −→ helices.

Proposition 2.3. Let γ be a unit speed space curve with κ(s) ≠ 0. Then γ is a g-clad helix if and only if

ϕ' ψ(s) = (s) � (κ2 + τ2)1/2(1 + σ2)1/2(1 + ϕ2)3/2 �

is a constant function.

Proof. We respectively denote K(s) and T(s) as the curvature and the torsion of the spherical image of the unit principal normal

e2(s). Then we already have

σ' K(s) = ( √1 + σ2)(s), T(s) = (s) � (κ2 + τ2)1/2(1 + σ2) �

and K(s)2 + T(s)2 = ((1 + σ2)(1 + ϕ2))(s). Let s˜ be the arc-length parameter of the spherical image of the unit principal normal e (s). Then ds˜/ds(s)= e' (s) =(κ2 + τ2)1/2(s). Therefore we get 2 � 2 �

2 d T ϕ'  K ( )(s) = (s) = ψ(s).  3 ds˜ � K �  � (κ2 + τ2)1/2(1 + σ2)1/2(1 + ϕ2)3/2 �  K2 + T 2 2     � �    2 It implies that the spherical image of e2(s) is a part of a slant helix in S if and only if ψ(s) is a constant function. □

Now we consider the spherical image of Du(s) of γ.

Proposition 2.4. Let γ be a unit speed space curve with κ(s) ≠ 0. Then the spherical image of Du(s) of γ is a cylindrical helix if and only if ψ(s) is a constant function.

Proof. We respectively denote K(s) and T(s) as the curvature and the torsion of the spherical image of Du(s) of γ. Then by formulae(2.2) we have

ϕ' K(s) = ( 1 + ϕ2)(s), T(s) = (s). � � (κ2 + τ2)1/2(1 + σ2)1/2(1 + ϕ2) �

It follows that

ϕ' (T/K)(s) = (s) = ψ(s). � (κ2 + τ2)1/2(1 + σ2)1/2(1 + ϕ2)3/2 �

It implies that the spherical image of Du(s) is a part of a cylindrical helix in S2 if and only if ψ(s) is a constant function.

Thus we have that a curve γ with κ(s) ≠ 0 is a g-clad helix if and only if the spherical image of Du(s): I S2 is a part of a −→ cylindrical helix in S2.

Corollary 2.5. Let γ be a unit speed space curve with κ(s) ≠ 0. Then γ is a clad helix if and only if the spherical image of Du(s): I S2 is a part of a circle. −→

Proof. From proof of Prop.2.4, we see the spherical image of Du(s) of γ is a circle if and only if ψ(s) is a constant function. □

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Remark 2.6. We already know there exist cylindrical helices on S2.

3. Geodesics on the tangent developables

In this section we consider geodesics on the tangent developable surface associated to a space curve. Developable surfaces are ruled surfaces and have the vanishing Gaussian curvature on the regular part. A ruled surface in R3 is (locally) the map 3 3 3 F γ,δ : I R R defined by F γ,δ (t,u) = γ(t) + uδ(t), where γ: I R , δ: I R {0} are smooth mappings and I is an open ( ) × → ( ) −→ −→ \ interval or a unit circle S1. Let γ be a unit speed space curve with κ(s) ≠ 0 and τ(s) ≠ 0. Then we consider the tangent developable M:

F(γ,e1)(s,u) = γ(s) + ue1(s) and the curve γ̃ (s) = γ(s) + u(s)e1(s) on M. Now suppose that γ̃ (s) is a geodesic on M. We respectively denote K(s) and T(s) as the curvature and the torsion of γ̃ (s). By a straightforward computation, we have

(u''uκ (1 + u')2 u2κ3 (uκ)'(1+ u'))(s) = 0 − − −

and get

�uκτ �τ(1 + u' ) K(s) = (s), T(s) = (s), (1 + u' )2 + u2κ2 (1 + u' )2 + u2κ2

where ε = ± 1 and εuκτ > 0. Let s̃ be the arc-length parameter of γ̃ . It follows that

1 + u' d T ((1 + u')2 + u2κ2)1/2 (T /K)(s) = (s), (s) = (s). uκ ds˜ � K � u2κ

Therefore we get

2 d T κ  K  σ˜ (s) = 3 ( ) (s) = (s),  2 2 ds˜ � K �  τ  K + T 2     � �   d  σ˜ ϕ˜(s) = ds˜ (s) = σ(s)  (K2 + T 2)1/2(1 + σ˜ 2)3/2        and   d ϕ˜ ψ˜(s) = ds˜ (s) = ϕ(s).  (K2 + T 2)1/2(1 + σ˜ 2)1/2(1 + ϕ˜ 2)3/2        Therefore we see the followings:   (1) γ̃ is a slant helix, if γ is a cylindrical helix. (2) γ̃ is a clad helix, if γ is a slant helix. (3) γ̃ is a g-clad helix, if γ is a clad helix. Thus we can get slant helices, clad helices and g-clad helices.

4. Developable surfaces associated to a space curve

In this section we consider special developable surfaces associated to a space curve. Let γ be a unit speed space curve with

˜ κ(s) ≠ 0. A ruled surface F(e3 ,e1 )(s,u) = e3(s) + ue1(s) is called the Darboux developable of γ in [1]. Moreover, F(D˜ ,e2 )(s,u) = D(s) + ue2(s) ˜ F(D˜ ,e2 )(s,u) = D(s) + ue2(s) as the Darboux developable of the unit tangent vector e1(s) of γ is called the tangential Darboux developable of γ.

We consider the Darboux developable of the unit principal normal vector e2(s) of γ and call it the principal normal Darboux

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developable of γ.

Lemma 4.1. Let γ be a unit speed space curve with κ(s) ≠ 0. Then the principal normal Darboux developable of γ is given by

˜ σe2 + D u F σe2 +D˜ = + ( � , Du) (s,u) (s) uD (s). 1+σ2 √1 + σ2

σe2 +D˜ Moreover, (s0, u0) is a singular point of F( � , Du) (s,u) if and only if u0 = φ(s0). 1+σ2

Proof. We respectively denote E1(s), E2(s) and E3(s) as the unit tangent vector, the unit principal normal vector and the unit

binormal vector of the space curve e2(s). By (2.2), we have

˜ ˜ u e2 + σD σe2 + D E1(s) = D (s), E2(s) = − (s) and E3(s) = (s). √1 + σ2 √1 + σ2

Next we can calculate

˜ σe2 + D u 2 2 1/2 ( (s) + uD (s))' = ((κ + τ ) (e2 σD˜ ))(s)(ϕ(s) u) √1 + σ2 − −

F σe2 +D˜ (s u) so that (s0, u0) is a singular point of ( � , Du) , if and only if u0 = φ(s0). 1+σ2 Therefore we complete the proof. □

Remark that the Darboux developable of the unit binormal vector e3(s) of γ is the tangential Darboux developable of γ. Now we know a local classification of singularities of the Darboux developable of γ is given in the following theorem.

Theorem 4.2. [1, 2] Let γ: I R3 be a unit speed curve with κ(s) ≠ 0. Then we have the following: −→

(1) The Darboux developable of γ is locally diffeomorphic to C × R at F(e3 , e1)(s0,u0) if and only if τ(s0) ≠ 0, (τ/κ)′(s0) ≠ 0 and

u0=(τ/κ)(s0).

(2) The Darboux developable of γ is locally diffeomorphic to SW at F(e3 , e1)(s0,u0) if and only if τ(s0) ≠ 0, (τ/κ)′(s0) = 0, (τ/

κ)″(s0) ≠ 0 and u0 = (τ/κ)(s0).

(3) The Darboux developable of γ is locally diffeomorphic to CCR at F(e3 , e1)(s0,u0) if and only if u0 = τ(s0) = 0, (τ/κ)′(s0) ≠ 0.

2 3 4 2 3 Here, C × R = {(x1, x2)| x1 = x2 } × R is the cuspidal edge, SW = {(x1, x2, x3) | x1} = 3u + u v, x2 = 4u + 2uv, x3 = v} is the 3 2 swallowtail and CCR = {(x1, x2, x3)| x1 = u, x2 = uv , x3 = v } is the cuspidal crosscap.

As a corollary of Theorem 4.2, S. Izumiya and N. Takeuchi have given the following local classification theorem of the tangential Darboux developable of a generic space curve.

Theorem 4.3. [5] Let γ: I R3 be a unit speed curve with κ(s) ≠ 0. Then we have the following: −→ , (1) The tangential Darboux developable of γ is locally diffeomorphic to C × R at F(D˜ , e2)(s0 u0) if and only if u0 = σ(s0) ≠ 0,

σ′(s0) ≠ 0.

(2) The tangential Darboux developable of γ is locally diffeomorphic to SW at F(D˜ , e2)(s0,u0) if and only if u0 = σ(s0) ≠ 0,

σ′(s0) = 0, σ″(s0) ≠ 0.

(3) The tangential Darboux developable of γ is locally diffeomorphic to CCR at F(D˜ , e2)(s0,u0) if and only if u0 = σ(s0) = 0,

σ′(s0) ≠ 0. Therefore, as a corollary of Theorem 4.2, we have the following local classification theorem of the principal normal Darboux developable of a generic space curve.

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Theorem 4.4. Let γ: I R3 be a unit speed curve with κ(s) ≠ 0. Then we have the following: −→

γ F σe2 +D˜ (s ,u ) (1) The principal normal Darboux developable of is locally diffeomorphic to C × R at ( � , Du) 0 0 if and only if 1+σ2 u0 = ϕ(s0) ≠ 0, ϕ′(s0) ≠ 0.

γ F σe2 +D˜ (s ,u ) (2) The principal normal Darboux developable of is locally diffeomorphic to SW at ( � , Du) 0 0 if and only if u0 1+σ2 = ϕ(s0) ≠ 0, ϕ′(s0) = 0, ϕ″(s0) ≠ 0.

γ F σe2 +D˜ (s ,u ) (3) The principal normal Darboux developable of is locally diffeomorphic to CCR at ( � , Du) 0 0 if and only if 1+σ2 u0 = ϕ(s0) = 0, ϕ′(s0) ≠ 0.

Proof. We respectively denote K(s) and T(s) as the curvature and the torsion of the space curve e2(s). By a straightforward computation, we have

σ' K(s) = (√1 + σ2)(s), T(s) = (s). � (κ2 + τ2)1/2(1 + σ2) �

It follows that φ(s) = (T/K)(s). Applying the result in Theorem 4.2 to the space curve e2(s), we complete the proof. □

We also have the following proposition:

Proposition 4.5. For a unit speed curve γ: I R3 with κ(s) ≠ 0, the following are equivalent. −→ (1) The principal normal Darboux developable 3 F σe2 +D˜ : I × R of γ is a conical surface. ( � , Du) R 1+σ2 −→ (2) γ is a clad helix.

σe2 + D˜ u Proof. The singular locus of the principal normal Darboux developable F σe2 +D˜ (s,u) is given by σ(s)= (s) + ϕ(s)D (s) ( � , Du) √ 2 σ + ˜ 1+σ2 1 + σ e2 D (s) + ϕ(s)Du(s). √1 + σ2 u Therefore, F σe2 +D˜ (s,u) is a conical surface if and only if σ′(s) ≡ 0. By (2.2), we can show that σ′(s) = φ′(s)D (s). Hence, ( � , Du) 1+σ2 σ′(s) ≡ 0 if and only if φ′(s) ≡ 0. By Proposition 2.2, the assertion holds. □

We can see the Darboux developable of D˜ (s) of γ is the principal normal Darboux developable of γ. Also the Darboux developable of Du(s) of γ is given by

Y + ϕDu F Y+ϕDu (s,u) = (s) + uX(s), ( 2 , X) 2 �1+ϕ 1 + ϕ � e + σD˜ σe + D˜ where X(s) = − 2 (s) and Y(s) = 2 (s). We respectively denote K(s) and T(s) as the curvature and the torsion of the √1 + σ2 √1 + σ2 space curve Du(s). Then we already have (T/K)(s) = ψ(s) and get the following results:

Proposition 4.6. Let γ: I R3 be a unit speed curve with κ(s) ≠ 0. Then we have the following: −→ u γ F Y+ϕDu s ,u (1) The Darboux developable of D (s) of is locally diffeomorphic to C × R at ( , X)( 0 0) if and only if u0 = ψ(s0) 1+ϕ2 � ≠ 0, ψ′(s0) ≠ 0.

u γ F Y+ϕDu s ,u (2) The Darboux developable of D (s) of is locally diffeomorphic to SW at ( , X)( 0 0) if and only if u0 = ψ(s0) ≠ 0, 1+ϕ2 � ψ′(s0) = 0, ψ″(s0) ≠ 0.

u γ F Y+ϕDu s ,u (3) The Darboux developable of D (s) of is locally diffeomorphic to CCR at ( , X)( 0 0) if and only if u0 = ψ(s0) = 1+ϕ2 � 0, ψ′(s0) ≠ 0.

Proposition 4.7. For a unit speed curve γ: I R3 with κ(s) ≠ 0, the following are equivalent. −→ 3 u F Y+ϕDu R γ (1) The Darboux developable ( , X): I × R of D (s) of is a conical surface. +ϕ2 �1 −→ (2) γ is a g-clad helix.

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5. Examples

In this section we give examples of the principal normal Darboux developable and draw their pictures by using Mathematica.

Example 5.1. We consider a space curve defined by γ(t) = (t, t2, t3). (5.1)

We respectively denote E1(s), E2(s) and E3(s) as the unit tangent vector, the unit principal normal vector and the unit binormal vector of the space curve γ. Then, we have

1 2t 3t2 E1(s) = ( , , ), √9t4 + 4t2 + 1 √9t4 + 4t2 + 1 √9t4 + 4t2 + 1

t()9t2 + 2 1 9t4 32()t3 + t E2(s) = ( , − , ) − √9t4 + 4t2 + 1 √9t4 + 9t2 + 1 √9t4 + 4t2 + 1 √9t4 + 9t2 + 1 √9t4 + 4t2 + 1 √9t4 + 9t2 + 1

and

3t2 3t 1 E3(s) = ( , , ). √9t4 + 9t2 + 1 − √9t4 + 9t2 + 1 √9t4 + 9t2 + 1

We can calculate the curvature and the torsion as follows:

2 √9t4 + 9t2 + 1 3 κ(t) = ,τ(t) = . 9t4 + 4t2 + 1 3/2 9t4 + 9t2 + 1 �  By Theorem 4.4 (2), we can see the principal normal Darboux developable F σe2 +D˜ (s,u) of γ is locally diffeomorphic to ( � , Du) +σ2 √ √ 1 SW at (s,u) = (0, 90 ),( 1 , 6 6 ),( 1 , 6 6 ). − 169 √3 25 − √3 25 We now draw the picture of the principal normal Darboux developable of γ in Fig.1.

Fig.1

Example 5.2. We consider a space curve defined by γ(t) = (t2, sin t, cos t). (5.2) Then, we have

2t cost sint E1(s) = ( , , ), √4t2 + 1 √4t2 + 1 − √4t2 + 1 2 (4t2 + 1)sint + 4t cost (4t2 + 1)cost 4t sint E2(s) = ( , , − ), √4t2 + 1 √4t2 + 5 − √4t2 + 1 √4t2 + 5 − √4t2 + 1 √4t2 + 5

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1 2(t cost sint) 2(t sint + cost) E3(s) = ( , − , ). − √4t2 + 5 √4t2 + 5 − √4t2 + 5

We can calculate the curvature and the torsion as follows:

√4t2 + 5 2t κ(t) = ,τ(t) = . 4t2 + 1 3/2 − 4t2 + 5 �  By Theorem 4.4 (3), we can see the principal normal Darboux developable

F σe2 +D˜ (s,u) of γ is locally diffeomorphic to CCR at (s, u) = (0, 0). ( � , Du) 1+σ2 We now draw the picture of the principal normal Darboux developable of γ around (s, u) = (0, 0) in Fig.2.

Fig.2

REFERENCES

[1] S. Izumiya, H. Katsumi and T. Yamasaki, The rectifying developable and the spherical Darboux image of a space curve, Geometry and topology of caustics-Caustics ‘98- Banach Center Publications 50 (1999), 137-149. [2] S. Izumiya and N. Takeuchi, Geometry of Ruled Surfaces, Applicable Mathematics in the Golden Age (2003), Narosa Publishing House, New Delhi, 305-338. [3] S. Izumiya and N. Takeuchi, Generic properties of helices and Bertrand curves, Journal of Geometry 74 (2002), 97-109. [4] S. Izumiya and N. Takeuchi, Special Curves and Ruled Surfaces, Beitrage zur Algebra und Geometrie Contributions to Algebra and Geometry 44 (no.1) (2003), 203-212. [5] S. Izumiya and N. Takeuchi, New Special Curves and Developable Surfaces, Turkish Journal of mathematics 28 (no.2) (2004), 153-163. [6] S. Izumiya, T.Sano, O.Saeki and K.Sakuma, “Geometry and Singularities” (in Japanese), Kyoritsu Shuppan (2001) [7] M. do Carmo, “Differential Geometry of Curves and Surfaces”, Prentice-Hall, New Jersey (1976). [8] M. Spivak, “A Comprehensive Introduction to Differential Geometry”, vol.3 Second Edition, Publish Or Perish (1979).

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ク ラ ッ ド 螺 旋 と 可 展 面

高 橋 岳 志・竹 内 伸 子

数学分野

要 旨

3 次元ユークリッド空間内で常螺旋の一般化として新しい曲線を定義した。その曲線を特徴づける不変量はある種 の可展面の特異点に関係していることがわかった。

キーワード: 定傾曲線，スラント螺旋，クラッド螺旋，ジークラッド螺旋，可展面，特異点

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