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Title CLAD HELICES and DEVELOPABLE SURFACES( Fulltext ) 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) － 1 － Bulletin of Tokyo Gakugei University, Division of Natural Sciences, Vol. 66 (2014) TAKAHASHI and TAKEUCHI: CLAD HELICES AND DEVELOPABLE SURFACES 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 － 2 － Bulletin of Tokyo Gakugei University, Division of Natural Sciences, Vol. 66 (2014) TAKAHASHI and TAKEUCHI: CLAD HELICES AND DEVELOPABLE SURFACES 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.
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