Normal Developable Surfaces of Surfaces Along Curves

Title Normal developable surfaces of surfaces along curves

Author(s) Hananoi, Satoshi; Izumiya, Shyuichi

Proceedings of the royal society of edinburgh section a-mathematics, 147(1), 177-203 Citation https://doi.org/10.1017/S030821051600007X

Issue Date 2017-02

Doc URL http://hdl.handle.net/2115/66937

This article has been published in a revised form in Proceedings of the royal society of edinburgh section a-mathematics Rights http://dx.doi.org/10.1017/S030821051600007X. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. © copyright holder.

Type article (author version)

File Information 2015075.pdf

Instructions for use

Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP Normal developable surfaces of surfaces along curves

Satoshi HANANOI and Shyuichi IZUMIYA

October 6, 2015

Abstract We consider a developable surface normal to a surface along a curve on the surface. We call it a normal developable surface along the curve on the surface. We investigate the uniqueness and the singularities of such developable surfaces. We discover two new invariants of curves on a surface which characterize these singularities.

1 Introduction

In this paper we consider a curve on a surface in the Euclidean 3-space and a developable surface normal to the surface along the curve. Such a developable surface is called a normal developable surface along the curve if it exists. The notion of Darboux frames along curves on surfaces is well known. We have a special direction in the Darboux frame at each point of the curve which is directed by a vector in the rectifying plane of the surface along the curve. We call this vector field a rectifying Darboux vector field along the curve. On the other hand, there are three invariants associated with the Darboux frame of a curve on a surface. With a certain condition of those invariants, we can show that there exists a normal developable surface along the curve which is given as the envelope of rectifying spaces of the surface along the curve. It follows that a normal developable surface is a ruled surface whose rulings are directed by the rectifying Darboux vector field along the curve. By using the above invariants, we introduce two new invariants which are related to the singularities of normal developable surfaces. Actually, one of these invariants is constantly equal to zero if and only if the rectifying Darboux vector field has a constant direction which means that the normal developable surface is a cylindrical surface. We give a classification of the singularities of the normal developable surface along a curve on a surface by using those two invariants (Theorem 3.3). In §6 we consider the uniqueness of the normal developable surface. If the uniqueness does not hold, then the curve is a straight line (Corollary 6.3). In §7 we consider special curves on surfaces. If we consider a geodesic on a surface, the normal developable surface is a tangent surface of the curve. We also consider lines of curvatures of a surface. In this case the director curve of the normal developable surface is given by the normal vector field of the surface along the curve. It is known that the ruled surface along a curve on a surface whose director curve is the normal vector field of the surface is a developable surface if and only if the curve is a line

2010 Mathematics Subject classiﬁcation. Primary 57R45; Secondary 58Kxx Key Words and Phrases. Normal developable surfaces, Curves on surfaces

1 of curvature. If we consider a parametrization of a surface such that each coordinate curve is a line of curvature, then the locus of the singular values of normal developable surfaces for all coordinate curves is the focal surface of the surface. We give an example of an ellipsoid in §7. In [4] the second author and Saki Otani introduced an osculating developable surface of a surface along a curve. Such a developable surface is tangent to the surface along the curve and it gives a ﬂat approximation of the surface along the curve. The method we use in this paper is analogous to the method in the above paper. However, we consider the normal developable surface of the surface along the curve. Therefore, the situations are diﬀerent. Throughout this paper all curves, surfaces and maps will be of class C∞.

2 Basic concepts

We consider a surface M = X(U) given locally by an embedding X : U −→ R3, where R3 is the Euclidean space and U ⊂ R2 is an open set. Let γ : I → U be an embedding, where γ(t) = (u(t), v(t)) and I is an open interval. Then we have a regular curve γ = X ◦ γ : I −→ M ⊂ R3 on the surface M. On the surface, we have the unit normal vector field n defined by X × X n(p) = u v (u, v), ∥Xu × Xv∥ where p = X(u, v). Here, a × b is the exterior product of a, b in R3. Since γ is a space curve in R3, we adopt the arc-length parameter as usual and write γ(s) = X(u(s), v(s)). Then we have the unit tangent vector field t(s) = γ′(s) of γ(s), where γ′(s) = dγ/ds(s). We have nγ (s) = n ◦ γ(s) which is the unit normal vector field of M along γ. Moreover, we define b(s) = nγ (s) × t(s). Then we have an orthonormal frame {t(s), nγ (s), b(s)} along γ, which is called the Darboux frame along γ. Then we have the following Frenet-Serret type formulae:  ′  t (s) = κg(s)b(s) + κn(s)nγ (s), ′ −  b (s) = κg(s)t(s) + τg(s)nγ (s), ′ − − nγ (s) = κn(s)t(s) τg(s)b(s). By using the matrix representation, we have       ′ t 0 κg κn t  ′      b = −κg 0 τg b . ′ − − nγ κn τg 0 nγ Here,

′ ′ ′′ κg(s) = ⟨t (s), b(s)⟩ = det (γ (s), γ (s), nγ (s)) , ′ ′′ κn(s) = ⟨t (s), nγ (s)⟩ = ⟨γ (s), nγ (s)⟩, ⟨ ′ ⟩ ′ ′ τg(s) = b (s), nγ (s) = det(γ (s), nγ (s), nγ (s))

3 and ⟨a, b⟩ is the canonical inner product of R . We call κg(s) a geodesic curvature, κn(s) a normal curvature and τg(s) a geodesic torsion of γ respectively. It is known that 1) γ is an asymptotic curve of M if and only if κn = 0, 2) γ is a geodesic of M if and only if κg = 0, 3) γ is a principal curve of M if and only if τg = 0.

2 We deﬁne a vector ﬁeld Dr(s) along γ by

Dr(s) = τg(s)t(s) + κg(s)nγ (s),

2 2 ̸ which is called a rectifying Darboux vector along γ. If κg + τg = 0, we can deﬁne the spherical rectifying Darboux image by

τg(s)t(s) + κg(s)nγ (s) Dr(s) = √ . 2 2 κg(s) + τg (s)

On the other hand, we brieﬂy review the notions and basic properties of ruled surfaces and developable surfaces. Let γ : I −→ R3 and ξ : I −→ R3 \{0} be C∞-mappings. Then we 3 deﬁne a mapping F(γ,ξ) : I × R −→ R by

F(γ,ξ)(u, v) = γ(u) + vξ(u).

We call the image of F(γ,ξ) a ruled surface, the mapping γ a base curve and the mapping ξ a director curve. The line defined by γ(u) + vξ(u) for a fixed u ∈ I is called a ruling. We call the ruled surface with vanishing Gaussian curvature on the regular part a developable surface. It is known that a ruled surface F(γ,ξ) is a developable surface if and only if ( ) det γ˙ (u), ξ(u), ξ˙(u) = 0, where γ˙ (u) = (dγ/du)(u)(cf., [3]). If the direction of the director curve ξ is constant, we call e F(γ,ξ) a (generalized) cylinder. If we write ξ(u) = ξ(u)/∥ξ(u)∥, then we have F(γ,ξ)(I × R) = × R e˙ ≡ F(γ,ξe)(I ). In this case F(γ,ξ) is a cylinder if and only if ξ(u) 0. We say that F(γ,ξ) is e˙ non-cylindrical if ξ(u) ≠ 0. Suppose that F(γ,ξ) is non-cylindrical. Then a striction curve is defined to be ˙ ⟨γ˙ (u), eξ(u)⟩ σ(u) = γ(u) − eξ(u). ˙ ˙ ⟨eξ(u), eξ(u)⟩ It is known that a singular point of the non-cylindrical ruled surface is located on the striction curve [3]. A non-cylindrical ruled surface F(γ,ξ) is a cone if the striction curve σ is constant. In general, a wave front in R3 is a (singular) surface which is a projection image of a Legendrian submanifold in the projective cotangent bundle π : PT ∗(R3) −→ R3. It is known (cf., [3]) that a non-cylindrical developable surface F(γ,ξ) is a wave front if and only if ( ) det ξ(u), ξ˙ (u), ξ¨(u) ≠ 0.

In this case we call F(γ,ξ) a (non-cylindrical) developable front. Let M ⊂ R3 be a surface. We say that a developable surface N is a normal developable surface of M if N ∩ M ≠ ∅ and TpN and TpM are orthogonal at any point p ∈ N ∩ M. If N is a cylinder, we say that N is a normal cylinder of M. We also say that N is a normal cone of M if N is a cone. For a normal developable surface N of M, the intersection N ∩ M is a regular curve. In particular, we say that the intersection N ∩ M is a normal cylindrical slice if N is a normal cylinder of M. Moreover, N ∩ M is said to be a normal conical slice if N is a normal cone of M.

3 3 Normal developable surfaces

In this section we investigate normal developable surfaces of a given surface. For a regular ◦ −→ ⊂ R3 2 2 ̸ curve γ = X γ : I M on a surface M with κg(s) + τg (s) = 0, we deﬁne a map 3 NDγ : I × R −→ R by

τg(s)t(s) + κg(s)nγ (s) NDγ (s, u) = γ(s) + uDr(s) = γ(s) + u √ . 2 2 κg(s) + τg(s) This is a ruled surface and ( ′ ′ ) ′ κgτ − κ τg −κ t + τ n D = κ − g g √g g γ , r n 2 2 2 2 κg + τg κg + τg so that we have ( ) ( ′ ′ ) ′ τ t + κ n κgτ − κ τg −κ t + τ n det(γ′, D , D ) = det t, √g g γ , κ − g g √g g γ r r 2 2 n 2 2 2 2 κg + τg κg + τg κg + τg = 0.

This means that NDγ (I × R) is a developable surface. We call NDγ a normal developable surface of M along γ. Moreover, we introduce two invariants δr(s), σr(s) of (M, γ) as follows: κ (s)τ ′ (s) − κ′ (s)τ (s) − g g g g δr(s) = κn(s) 2 2 , κg(s) + τg (s)  ′ τg(s)  κg(s)  σr(s) = √ + √ , (when δr(s) ≠ 0). 2 2 2 2 κg(s) + τg (s) δr(s) κg(s) + τg (s)

′ By the above calculation, δr(s) = 0 if and only if Dr (s) = 0. We can also calculate that ( ) ∂ND ∂ND κ γ × γ = uδ − √ g b. r 2 2 ∂s ∂u κg + τg

Therefore, (s0, u0) ∈ I × R is a singular point of NDγ if and only if δr(s0) ≠ 0 and

κg(s0) u0 = √ . 2 2 δr(s0) κg(s0) + τg (s0)

If (s0, 0) is a regular point (i.e., κg(s0) ≠ 0), the normal vector of NDγ at NDγ (s0, 0) = γ(s0) is orthogonal to the normal vector of M at γ(s0). This is the reason why we call NDγ the normal developable surface of M along γ. On the other hand, these two invariants characterize normal cylindrical slices and normal conical slices of M, respectively.

−→ ⊂ R3 2 2 ̸ Theorem 3.1 Let γ : I M be a unit speed curves on M with κg(s) + τg (s) = 0. Then we have the following: (A) The following are equivalent:

(1) NDγ is a cylinder,

4 (2) δr(s) ≡ 0, (3) γ is a normal cylindrical slice of M.

(B) If δr(s) ≠ 0, then the following are equivalent: (1) NDγ a cone, (2) σr(s) ≡ 0, (3) γ is a normal conical slice of M.

Proof. (A) By deﬁnition, NDγ is a cylinder if and only if Dr(s) is constant. Since

′ −κn(s)t(s) + τg(s)nγ (s) Dr (s) = δr(s) √ , 2 2 κg(s) + τg (s)

Dr(s) is constant if and only if δr(s) ≡ 0. Therefore, (1) is equivalent to (2). Suppose that (3) holds. Then there exists a vector k ∈ S2 such that ⟨b(s), k⟩ ≡ 0, where k is the director of the ′ normal cylinder. Then there exist λ, µ ∈ R such that k = λt(s) + µnγ (s). Since ⟨b (s), k⟩ ≡ 0, we have −κg(s)λ + τg(s)µ = 0, so that we have k = Dr(s). Condition (1) holds. It is clear that (1) implies (3).

(B) Condition (1) means that the singular value set of NDγ is a constant vector. We consider a vector valued function f(s) deﬁned by

κg(s) f(s) = γ(s) − √ Dr(s). 2 2 δr(s) κg(s) + τg (s) Then the condition (1) is equivalent to the condition that f ′(s) ≡ 0. We can calculate that ( )′ κ κ ′ f ′ = t + √ g D + √ g D 2 2 r 2 2 r δ κg + τg δr κg + τg ( )′ κ κ κ t + τ b = t + √ g D + √ g √g g 2 2 r 2 2 2 2 δ κg + τg κg + τg κg + τg ( ( )′) τ κ = √ g + √ g D 2 2 2 2 r κg + τg δr κg + τg

= σrDo. It follows that (1) and (2) are equivalent. By the deﬁnition of the conical slice, (3) means that there exists c ∈ R3 such that ⟨γ(s) − c, b(s)⟩ ≡ 0. If (1) holds, then f(s) is constant. For the constant point c = f(s) ∈ R3, we have ⟨ ⟩ κg(s) ⟨γ(s) − c, b(s)⟩ = ⟨γ(s) − f(s), b(s)⟩ = √ Dr(s), b(s) = 0. 2 2 δr(s) κg(s) + τg (s)

This means that (3) holds. For the converse, by (3), there exists a point c ∈ R3 such that ⟨γ(s) − c, b(s)⟩ = 0. Taking the derivative of the both side, we have 0 = ⟨γ(s) − c, b(s)⟩′ = ⟨γ(s) − c, −κgt(s) + τgnγ (s)⟩. Then there exists λ ∈ R such that γ(s) − c = λDr(s). Taking the derivative again, we have √ ⟨ − ⟩′′ ⟨ − ⟩ ⟨ − − ′⟩ 2 2 0 = γ c, b = t, κgt + τgnγ + γ c, ( κgt + τgnγ ) = κg + λδr κg + τg .

5 It follows that

κg(s) c = γ(s) − λDo(s) = γ(s) + √ Dr(s) = f(s). 2 2 δr(s) κg(s) + τg (s)

Therefore, f(s) is constant, so that (1) holds. This completes the proof. 2

Corollary 3.2 The normal developable surface NDγ is non-cylindrical if and only if δr(s) ≠ 0. We remark that developable surfaces are classiﬁed into cylinders, cones or tangent surfaces of space curves (cf., [6]). Hartman and Nirenberg [2] showed that a cylinder is only one non- singular (complete) developable surface. Hence, (complete) tangent surfaces have always singularities. By the results of Theorem 3.1, two invariants δr(s) and σr(s) might be related to the singularities of normal developable surfaces. Actually, we can classify the singularities of normal developable surfaces of M along curves by using theses two invariants δr(s) and σr(s). −→ ⊂ R3 2 2 ̸ Theorem 3.3 Let γ : I M be a unit speed curve with κg(s) + τg (s) = 0. Then we have the following:

(1) The image of the normal developable surface NDγ of M along γ is non-singular at (s0, u0) if and only if κg(s0) √ − u0δr(s0) ≠ 0. 2 2 κg(s0) + τg (s0)

(2) The image of the normal developable surface NDγ of M along γ is locally diﬀeomorphic to the cuspidal edge C × R at (s0, u0) if (i) δr(s0) ≠ 0, σr(s0) ≠ 0 and

κg(s0) u0 = √ , 2 2 δr(s0) κg(s0) + τg (s0) or ′ ̸ (ii) δr(s0) = κg(s0) = 0, δr(s0) = 0 and κ′ (s ) ̸ − g 0 u0 = ′ ′ ′′ , 2κn(s0)τg(s0) + κn(s0)τg(s0) + κg (s0) or ′ ̸ (iii) δr(s0) = κg(s0) = 0 and κg(s0) = 0. ′ ̸ We remark that if δr(s0) = 0, then ′ ′ ′′ ̸ 2κn(s0)τg(s0) + κn(s0)τg(s0) + κg (s0) = 0.

(3) The image of the normal developable surface NDγ of M along γ is locally diﬀeomorphic to ̸ ′ ̸ the swallowtail SW at (s0, u0) if δr(s0) = 0, σr(s0) = 0, σr(s0) = 0 and

κg(s0) u0 = − √ . 2 2 δr(s0) κg(s0) + τg (s0)

2 3 Here, C×R = {(x1, x2, x3)|x1 = x2 } is the cuspidal edge ( c.f., Fig.1) and SW = {(x1, x2, x3)|x1 = 4 2 3 3u + u v, x2 = 4u + 2uv, x3 = v} is the swallowtail ( c.f., Fig.2).

6 Figure 1: The cuspidal edge Figure 2: The swallowtail

4 Support functions

In this section we introduce a family of functions on a curve which is useful for the study of invariants of curves on surfaces. For a unit speed curve γ : I −→ M ⊂ R3, we deﬁne a function G : I × R3 −→ R by G(s, x) = ⟨x − γ(s), b(s)⟩. We call G a support function on γ with respect ∈ R3 to b. We write gx0 (s) = G(s, x0) for any x0 . Then we have the following proposition.

−→ ⊂ R3 2 2 ̸ Proposition 4.1 Let γ : I M be a unit speed curve with κg + τg = 0. Then we have the following: ∈ R − (1) gx0 (s0) = 0 if and only if there exist u, v such that x0 γ(s0) = ut(s0) + vnγ (s0) ′ ∈ R (2) gx0 (s0) = gx0 (s0) = 0 if and only if there exists u such that

τg(s0)t(s0) + κg(s0)nγ (s0) x0 − γ(s0) = u √ . 2 2 κg(s0) + τg (s0)

(A) Suppose that δr(s0) ≠ 0. Then we have the following: ′ ′′ (3) gx0 (s0) = gx0 (s0) = gx0 (s0) = 0 if and only if

κg(s0) τg(s0)t(s0) + κg(s0)nγ (s0) x0 − γ(s0) = − √ √ . (*) 2 2 2 2 δr(s0) κg(s0) + τg (s0) κg(s0) + τg (s0)

′ ′′ (3) (4) gx0 (s0) = gx0 (s0) = gx0 (s0) = gx (s0) = 0 if and only if σr(s0) = 0 and (*). ′ ′′ (3) (4) (5) gx0 (s0) = gx0 (s0) = gx0 (s0) = gx (s0) = dx0 (s0) = 0 if and only if

′ σr(s0) = σr(s0) = 0 and (*).

(B) Suppose that δr(s0) = 0. Then we have the following: ′ ′′ (6) gx0 (s0) = gx0 (s0) = gx0 (s0) = 0 if and only if ′ − ∈ R κg(s0) = 0 (i.e., κg(s0) = 0, κg(s0) = κn(s0)τg(s0)) and there exists u such that

x0 − γ(s0) = ut(s0).

7 ′ ′′ (3) (7) gx0 (s0) = gx0 (s0) = gx0 (s0) = gx (s0) = 0 if and only if one of the following conditions holds: ′ ̸ (a) δr(s0) = 0, κg(s0) = 0 ′ − ′ ′ ′′ ̸ (i.e., κg(s0) = 0, κg(s0) = κn(s0)τg(s0), 2κn(s0)τg(s0) + κn(s0)τg(s0) + κg (s0) = 0) and ′ − − κn(s0) x0 γ(s0) = ′ ′ ′′ t(s0). 2κn(s0)τg(s0) + κn(s0)τg(s0) + κg (s0)

′ ′ (b) δr(s0) = 0, κg(s0) = κg(s0) = 0 ′ ′′ − ′ ∈ R (i.e., κn(s0) = κg(s0) = κg(s0) = 0, κg (s0) = κn(s0)τg(s0)) and there exists u such that x0 − γ(s0) = ut(s0). ⟨ − ⟩ Proof. Since gx0 (s) = x0 γ(s), b(s) , we have the following calculations: ⟨ − ⟩ (α) gx0 = x0 γ, b , ′ ⟨ − − ⟩ (β) gx0 = x0 γ, κgt + τgnγ , ′′ ⟨ − − ′ − 2 2 − ′ ⟩ (γ) gx0 = κg + x0 γ, (κg + τgκn)t (κg + τg )b (κgκn + τg)nγ , (δ) g(3) = 2κ′ + κ τ x0 g n g( ) +⟨x − γ, κ (κ2 + κ2 + τ 2) + (κ′ τ + 2κ τ ′ ) − κ′′ t 0 g g (n g n g n g g ) − ′ ′ 2 2 2 − ′ − ′′ ⟩ 3(κgκg + τgτg)b + τg(κg + κn + τg ) (κnκg + 2κnκng) τg nγ ,

(ϵ) g(4) = 3κ′′ + 2κ′ τ + 3κ τ ′ − κ (κ2 + κ2 + τ 2) x0 g ( n g n g g g n g +⟨x − γ, κ′ (3κ2 + κ2 + τ 2) + κ (3κ κ′ + 5κ κ′ + 5τ τ ′ ) 0 g n g g g n n g g g g ) +κ τ (κ2 + κ2 + τ 2) + (κ′′ τ + 3κ′ τ ′ + 3κ τ ′′) − κ′′′ t ( n g g n g n g n g n g g ) + (κ2 + τ 2)(κ2 + κ2 + τ 2) + 2κ (κ′ τ − κ τ ′ ) − 3(κ′2 + τ ′2) − 4(κ κ′′ + τ τ ′′) b ( g g g n g n g g g g g g g g g g + −τ ′ (3κ2 + κ2 + τ 2) − τ (3κ κ′ + 5κ κ′ + 5τ τ ′ ) g n g g g n n g g g g ) 2 2 2 − ′′ ′ ′ ′′ ′′′ ⟩ +τgκn(κg + κn + τg ) (κnκg + 3κgκn + 3κnκg ) + τg nγ By deﬁnition and (α), assertion (1) follows. ′ − − By (β), gx0 (s0) = gx0 (s0) = 0 if and only if x0 γ(s0) = ut(s0) + vnγ (s0) and κg(s0)u + τg(s0)v = 0. If κn(s0) ≠ 0, τg(s0) ≠ 0, then we have τ (s ) κ (s ) u = v g 0 , v = u g 0 , κg(s0) τg(s0) so that there exists λ ∈ R such that

τg(s0)t(s0) + κg(s0)nγ (s0) x0 − γ(s0) = λ √ . 2 2 κg(s0) + τg (s0)

Suppose that κg(s0) = 0. Then we have τg(s0) ≠ 0, so that τg(s0)v = 0. Therefore we have

τg(s0)t(s0) + κg(s0)nγ (s0) x0 − γ(s0) = ut(s0) = ±u √ . 2 2 κg(s0) + τg (s0)

8 If τg(s0) = 0, then we have x0 − γ(s0) = vnγ (s0). Therefore the assertion (2) holds. ′ ′′ By (γ), gx0 (s0) = gx0 (s0) = gx0 (s0) = 0 if and only if

τg(s0)t(s0) + κg(s0)nγ (s0) x0 − γ(s0) = λ √ . 2 2 κg(s0) + τg (s0) and ′ ′ −τg(κnτg + κ ) − κg(κgκn + τ ) κ (s ) + λ √g g (s ) = 0. g 0 2 2 0 κg + τg

It follows that ( ′ ′ ) κ κgτ − κ τg √ g (s ) + λ κ − g g (s ) = 0. 2 2 0 n 2 2 0 κg + τg κg + τg Thus, ′ ′ κgτ − κ τg κ δ (s ) = κ (s ) − g g (s ) ≠ 0 and λ = − √ g (s ) r 0 n 0 2 2 0 2 2 0 κg + τg δr κg + τg or δr(s0) = 0, κg(s0) = 0. This completes the proof of the assertion (A), (3) and (B),(6). ̸ ′ ′′ (3) Suppose that δr(s0) = 0. By (δ), gx0 (s0) = gx0 (s0) = gx0 (s0) = gx0 (s0) = 0 if and only if ( κ τ ( ) 2κ′ + κ τ − √ g √ g κ (κ2 + κ2 + τ 2) + (κ′ τ + 2κ τ ′ ) − κ′′ g n g δ κ2 + τ 2 κ2 + τ 2 g g n g n g n g g r g g n g ) κ ( ) +√ g τ (κ2 + κ2 + τ 2) − (κ′ κ + 2κ′ κ ) − τ ′′ = 0 2 2 g g n g n g g n g κn + τg at s = s0. It follows that ( ) κ κ′ + τ τ ′ κ τ ′′ − κ′′τ ′ − κn(s0) ′ g g g g − g g g g 2κg(s0) + κg(s0)τg(s0) κn + 2κn 2 2 2 2 (s0) = 0. δr(s0) κg + τg κg + τg Since (κ κ′ + τ τ ′ )(κ τ ′ − κ′ τ ) κ τ ′′ − κ′′τ ′ ′ − g g g g g g g g − g g g g δr = κn 2 2 2 2 2 2 , (κg + τg ) κg + τg ′ κ (s )κ′ (s ) + τ (s )τ ′ (s ) ′ − δr(s0) − g 0 g 0 g 0 g 0 2κg(s0) + κn(s0)τg(s0) κg(s0) 2κg(s0) 2 2 = 0. δr(s0) κg(s0) + τg (s0) Moreover, we apply the relation

( )′ ′ ′ κ τ κgτ − κ τg τ √ g = √ g g g = √ g (δ − κ ) 2 2 2 2 2 2 2 2 r n κg + τg κg + τg κg + τg κg + τg to the above. Then we have ( ( ) ) √ ′ √ τ κ δ(s ) κ2 + τ 2(s ) √ g + √ g (s ) = δ (s )σ (s ) κ2 + τ 2(s ) = 0, 0 g g 0 2 2 2 2 0 r 0 r 0 g g 0 κg + τg δ κg + τg so that σr(s0) = 0. The converse assertion also holds.

9 ′ ′′ (3) Suppose that δr(s0) = 0. Then by (δ), gx0 (s0) = gx0 (s0) = gx0 (s0) = gx0 (s0) = 0 if and only ′ − ∈ R if κg(s0) = 0 (i.e., κg(s0) = 0, κg(s0) = κn(s0)τg(s0)), there exists u such that

x0 − γ(s0) = ut(s0) and ( ) ′ − ′ ′ ′′ 2κg(s0) + κn(s0)τg(s0) u 2κn(s0)τg(s0) + κn(s0)τg(s0) + κg (s0) = 0. − ′ Since δr(s0) = 0 and κg(s0) = 0, we have κn(s0)τg(s0) = κg(s0), so that ′ − ′ ′ ′′ κg(s0) u(2κn(s0)τg(s0) + κn(s0)τg(s0) + κg (s0)) = 0.

It follows that κ′ (s ) ′ ′ ′′ ̸ − g 0 2κn(s0)τg(s0) + κn(s0)τg(s0) + κg (s0) = 0 and u = ′ ′ ′′ 2κn(s0)τg(s0) + κn(s0)τg(s0) + κg (s0) or ′ ′ ′′ ′ 2κn(s0)τg(s0) + κn(s0)τg(s0) + κg (s0) = 0 and κg(s0) = 0. Therefore we have (B), (7), (a) or (b). By the similar arguments to the above, we have the assertion (A), (5). This completes the proof. 2

In order to prove Theorem 3.3, we use some general results on the singularity theory for families of function germs. Detailed descriptions are found in the book[1]. Let F :(R × r R , (s0, x0)) → R be a function germ. We call F an r-parameter unfolding of f, where f(s) = (p) ≤ ≤ Fx0 (s, x0). We say that f has an Ak-singularity at s0 if f (s0) = 0 for all 1 p k, (k+1) and f (s0) ≠ 0. Let F be an unfolding of f and f(s) has an Ak-singularity (k ≥ 1) ∂F (k−1) ∂F at s0. We write the (k − 1)-jet of the partial derivative at s0 by j ( (s, x0))(s0) = ∑ ∂xi ∂xi k−1 − j R × j=0 αji(s s0) for i = 1, . . . , r. Then F is called an -versal unfolding if the k r matrix of coeﬃcients (αji)j=0,...,k−1;i=1,...,r has rank k (k ≤ r). We introduce an important set concerning the unfoldings relative to the above notions. The discriminant set of F is the set ∂F D = {x ∈ Rr|there exists s with F = = 0 at (s, x)}. F ∂s Then we have the following classiﬁcation (cf., [1]).

r Theorem 4.2 Let F :(R × R , (s0, x0)) → R be an r-parameter unfolding of f(s) which has an Ak singularity at s0. Suppose that F is an R-versal unfolding. r−1 (1) If k = 2, then DF is locally diﬀeomorphic to C × R . r−2 (2) If k = 3, then DF is locally diﬀeomorphic to SW × R .

For the proof of Theorem 3.3, we have the following propositions.

−→ ⊂ R3 2 2 ̸ Proposition 4.3 Let γ : I M be a unit speed curve with κn + τg = 0 and G : ×R3 −→ R I the support function on γ with respect to b. If gx0 has an Ak-singularity (k = 2, 3) R ̸ at s0, then G is an -versal unfolding of gx0 . Here, we assume that δr(s0) = 0 for k = 3.

10 Proof. We write that x = (x1, x2, x3), γ(s) = (x1(s), x2(s), x3(s)) and b(s) = (b1(s), b2(s), b3(s)). Then we have

G(s, x) = b1(s)(x1 − x1(s)) + b2(s)(x2 − x2(s)) + b3(s)(x3 − x3(s)), so that ∂G (s, x) = bi(s), (i = 1, 2, 3). ∂xi Therefore the 2-jet is ∂G 1 2 ′ − ′′ − 2 j (s0, x0) = bi(s0) + bi(s0)(s s0) + bi (s0)(s s0) . ∂xi 2 We consider the following matrix:     b1(s0) b2(s0) b3(s0) b(s0)  ′ ′ ′   ′  A = b1(s0) b2(s0) b3(s0) = b (s0) . ′′ ′′ ′′ ′′ b1(s0) b2(s0) b3(s0) b (s0) By the Frenet-Serret type formulae, we have

′ − ′′ − ′ − 2 2 ′ − b = κgt + τgnγ and b = (κnτg κg)t (κn + τg )b + (τg κgκn)nγ .

3 Since {t, b, nγ } is an orthonormal basis of R , the rank of   b(s0)   A = −κg(s0)t(s0) + τg(s0)nγ (s0) − − ′ − 2 2 ′ − ( κn(s0)τg(s0) κg(s0))t(s0) (κn(s0) + τg (s0))b(s0) + (τg(s0) κg(s0)κn(s0))nγ (s0) is equal to the rank of   0 0 1   −κg(s0) +τg(s0) 0 . − − ′ ′ − − 2 2 κn(s0)τg(s0) κg(s0) τg(s0) κn(s0)κg(s0) (κg(s0) + τg (s0)) Therefore rank A = 3 if and only if

̸ − ′ − ′ 2 2 − ′ − ′ 0 = κg(τg κnκg) + τg(κnτg + κg) = κn(κg + τg ) (κgτg κgτg) st s = s0. The last condition is equivalent to the condition δr(s0) ≠ 0. Moreover, the rank of ( ) ( ) b(s0) b(s0) ′ = b (s0) −κg(s0)t(s0) + τg(s0)nγ (s0) is always two. R If gx0 has an Ak-singularity (k=2,3) at s0, then G is -versal unfolding of gx0 . This completes the proof. 2

Proof of Theorem 3.3. By a straightforward calculation, we have ( ) ∂ND ∂ND κ γ × γ = − √ g + uδ n . 2 2 r γ ∂s ∂u κg + τg

11 Therefore, (s0, u0) is non-singular if and only if ∂ND ∂ND γ × γ ≠ 0. ∂s ∂u This condition is equivalent to

κg(s0) √ + u0δr(s0) ≠ 0. 2 2 κg(s0) + τg (s0)

This completes the proof of assertion (1).

By Proposition 4.1, (2), the discriminant set DG of the support function G of γ with respect to b is the image of the normal developable surface of M along γ. ̸ Suppose that δr(s0) = 0. It follows from Proposition 4.1, A, (3), (4) and (5) that gx0 has the A2-type singularity (respectively, the A3-type singularity) at s = s0 if and only if

κg(s0) u0 = − √ 2 2 δr(s0) κg(s0) + τg (s0)

̸ ′ ̸ and σr(s0) = 0 (respectively, σr(s0) = 0 and σr(s0) = 0). By Theorem 4.2 and Proposition 3.3, we have the assertions (2), (α) and (3).

Suppose that δr(s0) = 0. It follows from Proposition 4.1, B, (6) and (7) that gx0 has the A3-type singularity if and only if δr(s0) = 0, κg(s0) = 0 and ( ) ′ ̸ ′ ′ ′ ′′ ̸ κg(s0) = 0 or κg(s0) + u0 2κn(s0)τg(s0) + κn(s0)τg(s0) + κg (s0) = 0. By Theorem 4.3 and Proposition 4.3, we have the assertion (2), (β). This completes the proof. 2

5 Invariants of curves on surfaces

3 2 In this section we consider geometric meanings of the invariant σr. Let Γ : I −→ R × S be a regular curve and F : R3 ×S2 −→ R a submersion. We say that Γ and F −1(0) have contact of at ′ (k) least order k for t = t0 if the function g(t) = F ◦Γ(t) satisfies g(t0) = g (t0) = ··· = g (t0) = 0. −1 If γ and F (0) have contact of at least order k for t = t0 and satisfies the condition that (k+1) −1 g (t0) ≠ 0, then we say that Γ and F (0) have contact of order k for t = t0. For any 3 3 2 x ∈ R , we define a function gx : R × S −→ R by gx(u, v) = ⟨x − u, v⟩. Then we have −1 { ∈ R3 × 2 |⟨ ⟩ ⟨ ⟩} gx (0) = (u, v) S u, v = x, v . ∈ 2 −1 |R3 ×{ } ⟨ ⟩ ⟨ ⟩ If we fix v S , then gx (0) v is an affine plane defined by u, v = c, where c = x, v . 2 Since this plane is orthogonal to v, it is parallel to the tangent plane TvS at v. Here we have a representation of the tangent bundle of S2 as follows:

TS2 = {(u, v) ∈ R3 × S2 |⟨u, v⟩ = 1}.

| −1 −1 −→ 2 R3 × 2 −→ 2 We consider the canonical projection π2 gx (0) : gx (0) S , where π2 : S S . | −1 −1 −→ 2 2 Then π2 gx (0) : gx (0) S is a plane bundle over S . Moreover, we deﬁne a map Ψ :

12 −1 −→ 2 ⟨ ⟩ gx (0) TS by Φ(u, v) = (u/ x, v , v). Then Φ is a bundle isomorphism. Therefore, we 2 −1 2 −→ write TS (x) = gx (0) and call it an aﬃne tangent bundle over S through x. Let γ : I ⊂ R3 2 2 ̸ ̸ M be a unit speed curve on M with κg(s) + τg (s) = 0. Suppose that δr(s) = 0. By the proof of the assertion (B) of Theorem 3.1, the derivative of the vector valued function f of ′ NDγ is f (s) = σr(s)Dr(s). If we assume that σr(s) ≡ 0, then f is a constant vector x0. Then

κg(s) γ(s) − x0 = √ Dr(s). 2 2 δr(s) κg(s) + τg (s)

Therefore ⟨ − ⟩ gx0 (γ(s), b(s)) = gx0 (s) = γ(s) x0, b(s) = 0. ∈ R3 If there exists x0 such that gx0 (γ(s), b(s)) = 0, then we have

κg(s) γ(s) − x0 = √ Dr(s). 2 2 δr(s) κg(s) + τg (s)

3 2 and σr(s) ≡ 0. We consider a regular curve (γ, b): I −→ R × S . Then we have the following proposition.

−→ ⊂ R3 2 2 ̸ Proposition 5.1 Let γ : I M be a unit speed curve on M with κg(s)+τg (s) = 0 and 3 2 δr(s) ≠ 0. Then there exists x0 ∈ R such that (γ, b)(I) ⊂ TS (x0) if and only if σr(s) ≡ 0.

The result of the above proposition explains that the geometric meaning of the singularities of NDγ is related not only to the curve as a space curve but also to the behavior of the curve −→ ⊂ R3 2 2 ̸ in the surface. Let γ : I M be a unit speed curve with κg(s) + τg (s) = 0. Then we consider the support function gx0 (s) = gx0 (γ(s), b(s)). By assertion (2) of Proposition 4.1, 2 (γ, b) is tangent to TS (x0) at s = s0 if and only if x0 = NDγ (s0, u0) for some u0 ∈ R. Then we have the following proposition.

−→ ⊂ R3 2 2 ̸ Proposition 5.2 Let γ : I M be a unit speed curve with κg(s) + τg (s) = 0 and δr(s) ≠ 0. For x0 = NDγ (s0, u0), we have the following: 2 (1) The order of contact of (γ, b) with TS (x0) at s = s0 is two if and only if

κg(s0) u0 = − √ . (**) 2 2 δr(s0) κg(s0) + τg (s0) and σr(s0) ≠ 0. 2 (2) The order of contact of (γ, b) with TS (x0) at s = s0 is three if and only if (**) and ′ ̸ σr(s0) = 0 and σr(s0) = 0.

′ ′′ Proof. By assertions (3), (4) of Proposition 4.1, the conditions gx0 (s0) = gx0 (s0) = gx0 (s0) = 0 (3) ̸ ̸ ◦ and gx0 (s0) = 0 if and only if (**) and σr(s0) = 0. Since gx0 = gx0 (γ, b), the above condition 2 means that (γ, b) and TS (x0) have contact of order two at s = s0. For the proof assertion (2), we use assertions (4), (5) of Proposition 4.1 exactly the same way as the above case. 2

Therefore the geometric meaning of the classiﬁcation results of Theorem 3.3 are given as follows.

13 −→ ⊂ R3 2 2 ̸ ̸ Theorem 5.3 Let γ : I M be a unit speed curve with κg(s)+τg (s) = 0 and δr(s) = 0. (1) The image of the normal developable surface NDγ of M along γ is locally diﬀeomorphic to the cuspidal edge C × R at (s0, u0) if

κg(s0) u0 = − √ 2 2 δ(s0) κg(s0) + τg (s0)

2 and the order of contact of (γ, b) with TS (x0) at s = s0 is two. (2) The image of the normal developable surface NDγ of M along γ is locally diﬀeomorphic to the swallowtail SW at (s0, u0) if

κg(s0) u0 = − √ 2 2 δr(s0) κg(s0) + τg (s0)

2 and the order of contact of (γ, b) with TS (x0) at s = s0 is three.

6 Uniqueness of normal developable surfaces

In this section we consider existence and uniqueness of normal developable surfaces along curves on a surface.

Theorem 6.1 Let M ⊂ R3 be a regular surface and γ : I −→ M ⊂ R3 a unit speed curve with 2 2 ̸ κg(s) + τg (s) = 0. Then there exists a unique developable surface which is normal to M along γ.

Proof. For existence, we have the normal developable surface NDγ along γ. On the other hand, let Nγ be a developable surface which is normal to M along γ. Since Nγ is a ruled surface, we assume that Nγ(s, u) = γ(s) + uξ(s). Here we write ξ(s) = λ(s)t(s) + µ(s)b(s) + ν(s)nγ (s), so that ′ ′ ′ ′ ξ = (λ − µκg − νκn) t + (µ + λκg − ντg) b + (ν + λκn + µτg) nγ . ′ ′ Since Nγ is a developable surface, we have det (γ , ξ, ξ ) = 0. The last condition is equivalent to   t   det λt + µb + νnγ = 0, ′ ′ ′ (λ − µκg − νκn) t + (µ + λκg − ντg) b + (ν + λκn + µτg) nγ which is also equivalent to

′ ′ µ (ν + λκn + µτg) − ν (µ + λκg − ντg) = 0. (1)

On the other hand, since Nγ is a developable surface which is normal to M along γ, we have ∂N ∂N γ (s, u) × γ (s, u) = θ(s, u)b(s). (2) ∂s ∂u

14 If Nγ is nonsingular at (s, 0), then θ(s, 0) ≠ 0. By a straightforward calculation, we have ∂N γ = γ′ + uξ′ ∂s ′ ′ ′ = (1 + u (λ − µκg − νκn)) t + u (µ + λκg − ντg) b + u (ν + λκn + µτg) nγ

∂N γ = ξ = λt + µb + νn , ∂u γ so that ∂N ∂N γ × γ = u (µ (ν′ + λκ + µτ ) − ν (µ′ + λκ − ντ )) t ∂s ∂u n g g g ′ ′ + (uλ (ν + λκn + µτg) − ν {1 + u (µ + λκg − ντg)}) b ′ ′ + (µ {1 + u (λ − µκg − νκn)} − uλ (µ + λκg − ντg)) nγ . If we substitute u = 0, then ∂N ∂N γ × γ = −νb + µn . ∂s ∂u γ By (2), we have −ν(s) = θ(s, 0) , µ(s) = 0. It also follows from (1) that

ν(s)(λ(s)κg(s) − ν(s)τg(s)) = 0.

Suppose that Nγ is non-singular along γ. Then θ(s, 0) ≠ 0, so that ν(s) ≠ 0. Thus, we have λ(s)κ (s) − ν(s)τ (s) = 0. If κ (s) ≠ 0, then λ(s) = τg(s) ν(s). Therefore, we have g g g κg(s) τ (s) ξ(s) = g ν(s)t(s) + ν(s)n (s) κ (s) γ g √   2 2 κg(s) + τg (s) τ (s)t(s) + κ (s)n (s) = ν(s)  g √ g γ  κg(s) 2 2 κg(s) + τg (s) √ 2 2 κg(s) + τg (s) = ν(s) Dr(s). κg(s)

This means that the direction of ξ(s) is equal to the direction of Dr(s). If τg(s) ≠ 0, we have the same result as the above case. On the other hand, suppose that Nγ has a singular point at (s0, 0). Then θ(s0, 0) = 0. It follows that ν(s0) = 0, µ(s0) = 0. Thus we have ξ(s0) = λ(s0)t(s0). If the singular point γ(s0) is in the closure of the set of points where the normal deveoplable is regular on γ, then there exists a point s in any neighborhood of s0 such that the uniqueness of the normal developable surface holds at s. Passing to the limit s → s0, uniqueness of the normal developable surface holds at s0. Suppose that there exists an open interval J ⊂ I such that Nγ is singular at γ(s) for any s ∈ J. Then Nγ(s) = γ(s) + uλ(s)t(s) for any s ∈ J. This means that µ(s) = ν(s) = 0 for s ∈ J. It follows that ∂N ∂N γ × γ (s, u) = uλ2(s)(κ (s)b(s) − κ (s)n (s)). ∂s ∂u n g γ

The above vector is directed to b(s), so that κg(s) = 0 for any s ∈ J. In this case, Dr(s) = ±t(s). This means that uniqueness holds. 2

15 Proposition 6.2 Let γ : I −→ M be a regular curve with κg(s) ≡ τg(s) ≡ 0. Then Nγ is a normal developable along γ if and only if γ is a normal slice of M.

Proof. We remark that the torsion of the curve γ as a space curve is ′ − ′ κgκn κgκn τ = τg + 2 2 . κg + τg

′ If κg(s) ≡ τg(s) ≡ 0, then τ ≡ 0, so that γ is a plane curve. Moreover, we have b = −κgt + τgnγ ≡ 0. Thus, Nγ is a plane normal to M. Since γ is the intersection of M and Nγ, γ is a normal slice of M. For the converse, if γ is a normal slice of M, then there exists a plane Π such that γ(I) = M ∩ Π and nγ (s), t(s) ∈ Π for any s ∈ I. Therefore, Π is orthogonal to b(s) for any s ∈ I. Since Π is a plane (i.e. a developable surface), Π is a normal developable surface of M along γ. 2

Corollary 6.3 Let γ : I −→ M ⊂ R3 be a unit speed curve on M. If there are two normal developable surfaces along γ, then γ is a straight line.

2 2 ̸ Proof. Under the assumption of κg + τg = 0, the normal developable surface along γ is unique by Theorem 6.1. If κg ≡ 0 and τg ≡ 0, γ is a normal slice. In this case, a normal plane Π of M at γ(s0) is an normal developable surface along γ. If there is another normal developable surface Nγ along γ, then Nγ is tangent to Π along γ. This means that Π is a tangent plane of the developable surface Nγ. By deﬁnition, Π is tangent to Nγ along a ruling of Nγ, which is γ. Thus γ is a line. If κg = τg = 0 at a point s0 in the closure of the set of points where 2 2 ̸ κg + τg = 0, then there exists a point s in any neighborhood of s0 such that the uniqueness of the normal developable surface holds at s. Passing to the limit s → s0, uniqueness of the normal developable surface holds at s0. This completes the proof. 2

7 Special curves on a surface

In this section we consider special curves on a surface.

7.1 Geodesics We consider geodesics on surfaces. Let γ : I −→ M be a unit speed curve on a suface M ⊂ R3. It is a geodesic of M if and only if κg ≡ 0. Therefore, if τg ≠ 0, then

NDγ (s, u) = γ(s) + ut(s), which is known as the tangent surface of γ. In this case

′ δr(s) = κn(s) , σr(s) = ±1 , σr (s) ≡ 0.

By Theorem 3.3, we have the following proposition.

Proposition 7.1 Let γ : I −→ M be a geodesic on a surface M with τg(s) ≠ 0. Then we have the following:

16 1. NDγ is a regular surface at (s0, u0) if u0 ≠ 0,

2. The image of (NDγ, (s0, u0)) is locally diﬀeomorphic to the cuspidal edge if (a) κ (s ) ≠ 0 and u = 0, or n 0 0 ( ) ′ ′ ̸ (b) κn(s0) = 0 and u0 2τgκn + τgκn (s0) = 0.

′ Since σr (s) ≡ 0, swallowtails never appear. We now consider the following example.

Example 7.2 We consider a surface parametrized by ( ) cos(t) − t sin(t) sin(t) + t cos(t) 5 X(t, u) = (t cos(t), t sin(t), 5t) + u √ , √ , √ . t2 + 26 t2 + 26 t2 + 26 This surface is a ruled surface such that the base curve is γ(t) = (t cos(t), t sin(t), 5t) . Thus, γ is a regular curve on the surface M = Im X. It follows that

γ˙ (t) = ( − 2 sin(t) − t cos(t) , 2 cos(t) − t sin(t) , 0 )

∂X ∂X α(t) + β(t, u) (( ) ( ) × = t2 + 52 sin(t) + t t2 + 27 cos(t) , ∂t ∂u 4 2 7/2 (t + 29t + 104)( ) ( ) ) t t2 + 27 sin(t) − t2 + 52 cos(t) , 5t , where α(t) = t12 + 87t10 + 2835t8 + 42485t6 + 294840t4 + 940992t2 + 1124864, ( √ √ 4 2 − 7 4 2 β(t, u) = u 31460t t + 29√t + 104 5t t + 29t +√ 104 ) −320t5 t4 + 29t2 + 104 − 5850t3 t4 + 29t2 + 104 . In this case the Darboux frame of γ is

Xt × Xu nγ (t) = ∥Xt × Xu∥ 1 (( ) ( ) = t2 + 52 sin(t) + t t2 + 27 cos(t) , t6 + 55t4 + 858t2 + 2704 ( ) ( ) ) t t2 + 27 sin(t) + −t2 − 52 cos(t) , 5t , ( ) ˙ − γ(t) cos(√t) t sin(t) sin(√t) + t cos(t) √ 5 t(t) = = , , , γ(˙t) t2 + 26 t2 + 26 t2 + 26

b(t) = nγ (t) × t(t).

Since γ¨(t) = (−2 sin(t) − t cos(t), 2 cos(t) − t sin(t), 0), we have

det (γ˙ (t), γ¨(t), n (t)) κ (t) = γ = 0, g ∥γ˙ (t)∥3 so that γ is a geodesic of M. Moreover, we have

⟨γ¨(t), nγ (t)⟩ 1 ′ 4t κn(t) = = − , κ (t) = , ∥γ˙ (t)∥2 (t2 + 26)2 n (t2 + 26)3

17 2 det (γ˙ (t), nγ (t), n˙ γ (t)) 5 (t + 6) τg(t) = = , ∥γ˙ (t)∥2 (t2 + 26) (t4 + 29t2 + 104)2 20t (2t6 + 83t4 + 950t2 + 3484) τ ′ (t) = − . g (t2 + 26)2 (t4 + 29t2 + 104)3 We can draw the pictures of γ and M in Fig. 3,4,5.

Figure 3: M and γ Figure 4: NDγ Figure 5: M and NDγ

7.2 Lines of curvature We consider lines of curvature. Let γ : I −→ M be a unit speed line of curvature of M with κg(s) ≠ 0. Since τg(s) ≡ 0, we have

NDγ (s, u) = γ(s) + unγ (s) ′ ′ 2 − ′′ ̸ −κn ′ 2κn κnκn and δr(s) = κn(s). Moreover, if κn(s) = 0, then σr(s) = 2 (s), σr (s) = 3 (s). κn κn Then we have the following proposition as a corollary of Theorem 3.3.

Proposition 7.3 Let γ : I −→ M be a line of curvature on a surface M with κg(s) ≠ 0. Then we have the following:

1) NDγ is a regular surface at (s0, u0) if u0κn(s0) ≠ ±1. 1 2) NDγ is locally diﬀeomorphic to the cuspidal edge at (s0, u0) if κn(s0) ≠ 0 and u0 = ± . κn(s0) ̸ ′ ′′ ̸ 3) NDγ is locally diﬀeomorphic to the swallowtail at (s0, u0) if κn(s0) = 0, κn(s0) = 0, κn(s0) = 1 0 and u0 = ± . κn(s0)

Example 7.4 We now consider an ellipsoid as an example. The following parametrization of an ellipsoid is known. Let X(u, v) = (x(u, v), y(u, v), z(u, v)) be a surface in R3 deﬁned by √ √ √ a(a − u)(a − v) b(b − u)(b − v) c(c − u)(c − v) x(u, v) = ± , y(u, v) = ± , z(u, v) = ± , (a − b)(a − c) (b − c)(b − a) (c − a)(c − b) where 0 < c ≤ v ≤ b ≤ u ≤ a. We can easily show that the image of X is an ellipsoid deﬁned x2 y2 z2 by + + = 1. Moreover, it is classically known that each u-curve and v-curve are a b c

18 lines of curvature, respectively. However, we do not know the references, so that we can show as follows: For a u-curve deﬁned by γ(u) = X(u, v0) = (x(u, v0), y(u, v0), z(u, v0)), the unit normal vector along γ is   yu(u, v0)zu(u, v0)(b − c)(u − v0) −   ωu(u, v0)(u − b)(u − c)(v0 − b)(v0 − c)    X × X  z (u, v )x (u, v )(c − a)(u − v )  u v − u 0 u 0 0  nγ (u) = ∥ × ∥(u, v0) =  − − − −  , Xu Xv  ωu(u, v0)(u a)(u c)(v0 a)(v0 c)  − −  − xu(u, v0)yu(u, v0)(a b)(u v0) ωu(u, v0)(u − a)(u − b)(v0 − a)(v0 − b) where √ uv(u − v)2 ω(u, v) = − . (a − u)(a − v)(u − b)(b − v)(u − c)(c − v) Since ( ) a(a − v ) b(b − v ) c(c − v ) γ˙ (u) = 0 , 0 , 0 , 2xu(u, v0)(a − b)(a − c) 2yu(u, v0)(b − a)(b − c) 2zu(u, v0)(c − a)(c − b) ( ) a2(a − v )2 b2(b − v )2 c2(c − v )2 γ¨(u) = 0 , 0 , 0 , 3/2 2 2 3/2 2 2 3/2 2 2 4xu(u, v0) (a − b) (a − c) 4yu(u, v0) (b − a) (b − c) 4zu(u, v0) (c − a) (c − b) det (γ˙ (u), n (u), n˙ (u)) we have τ (u) = γ γ = 0. This means that each u-curve is a line of g ∥γ˙ (u)∥2 curvature. By the same calculation as the above, we can show that each v-curve is a line of curvature. Suppose that a = 20, b = 10, c = 5. Then we can draw the pictures of the ellipsoid, the u-curve for v = 7 and the normal developable surface along the u-curve with v = 7.

Figure 6: The ellipsoid and Figure 7: The ellipsoid and Figure 8: The normal devel- the u-curve the normal developable along opable along the u-curve the u-curve

We can observe that there appears the swallowtail on the normal developable surface. We also have ⟨ ⟩ − γ¨(u), nγ (u) − bcxu(u, v0)(u v0) κn(u) = 2 = ∥γ˙ (u)∥ 4uyu(u, v0)zu(u, v0)(a − u)(a − v0)(b − c) and show that the singular point is the swallowtail. We can also show that the normal developable surface along a v-curve has the swallowtail. Since the director of the normal developable

19 surface along a line of curvature γ is nγ , if we consider all u-curves and v-curves, then we have the evolute as the trajectory of the singular value sets of corresponding normal developable surfaces. We can draw the picture of the trajectory of the singular value sets of normal developable surfaces along u-curves in Fig. 9, which gives one of the branch of the evolute of the ellipsoid. The other branch of the evolute is depicted in Fig. 10, which is given by the normal developable suﬃces along v-curves.

Figure 9: u-curves and the trajectory of the sin- Figure 10: v-curves and trajectory of the singu- gular values lar values

The picture of the union of two branches of the evolute is drawn in Fig. 11. It is known that the purse PS appears as a singular point of the evolute of the ellipsoid [5]. Here PS = {(3u2 + uv, 3v2 + wu, w) ∈ R3 | w2 = 36uv }. Moreover, the purse corresponds to an umbilical point of the ellipsoid. We can observe the purse in Fig.12.

Figure 11: The union of the Fig.9 and Fig. 10 Figure 12: The purse of the evolute

8 Curves on a surface of revolution

In this section we consider curves on surfaces of revolution. A surface of revolution is deﬁned to be X(u, v) = (f(u) cos v, f(u) sin v, g(u)) for (u, v) ∈ U ⊂ R2. We assume that f(u) ≠ 0. It is easy to show that the unit normal vector ﬁeld alone M = X(U) is ( ) f(u)g′(u) cos v f(u)g′(u) sin v f(u)f ′(u) n(u, v) = −√ , −√ , √ . f(u)2 (f ′(u)2 + g′(u)2) f(u)2 (f ′(u)2 + g′(u)2) f(u)2 (f ′(u)2 + g′(u)2)

20 For a curve γ(t) = ( f(u(t)) cos v(t) , f(u(t)) sin v(t) , g(u(t)) ) on M, the Darboux frame is given by

1 ′ ′ ′ nγ (t) = √ (−fg cos tv , − fg sin tv , ff ) f 2h

1 ′ ′ ′ t(t) = √ (f u˙ cos tv − fv˙ sin tv , f u˙ sin tv + fv˙ cos tv , g u˙) f 2v˙ 2 + hu˙ 2

1 ′ ′ ′ b(t) = √ (−hu˙ sin tv − ff v˙ cos tv , hu˙ cos tv − ff v˙ sin tv , − fg v˙) , h (f 2v˙ 2 + hu˙ 2) where d d d d f ′ = f , g′ = g , h(t) = f ′(u(t))2 + g′(u(t))2, u˙(t) = u(t), v˙(t) = v(t) , t = v(t). du du dt dt v Moreover, we can calculate that det (γ˙ (t), γ¨(t), n(t)) κ (t) = g ∥γ˙ (t)∥3 ( ( ) 1 ( ) = √ 2u ˙ 2vf˙ ′3 + ff ′2 (u ˙v¨ − u¨v˙) +vf ˙ ′ f fv˙ 2 − u˙ 2f ′′ + 2u ˙ 2g′2 , h (f 2v˙ 2 + hu˙ 2)3 ( )) +fg′ −u˙ 2vg˙ ′′ − u¨vg˙ ′ +u ˙vg¨ ′ ⟨γ¨(t), n(t)⟩ u˙ 2 (f ′g′′ − f ′′g′) + fv˙ 2g′ κn(t) = = √ ( ), ∥γ˙ (t)∥2 h u˙ 2f ′2 + f 2v˙ 2 +u ˙ 2g′2 ( ) det (γ˙ (t), n(t), n˙ (t)) u˙v˙ ff ′′g′ − ff ′g′′ + f ′2g′ + g′3 τ (t) = γ = ( ) . g ∥γ˙ (t)∥2 h u˙ 2f ′2 + f 2v˙ 2 +u ˙ 2g′2

For a meridian curve γ(u) = X(u, v0) = (f(u) cos v0, f(u) sin v0, g(u)) , we have f ′g′′ − f ′′g′ κg(u) ≡ 0 , τg(u) ≡ 0 , κn(u) = ( ) . f ′2 + g′2 3/2 In this case the normal developable surface along γ is a normal slice of M (cf. Fig.13 and 14).

Figure 13: A meridian of M Figure 14: The normal slice of M

21 For a parallel circle γ(v) = X(u0, v) = (f(u0) cos v, f(u0) sin v, g(u0)) , we have 10v κg(v) = , τg(v) = 0 , κn(v) = 1. (v2 + 2)2

Since nγ (v) = −γ(v), the normal developable surface along γ is NDγ (v, w) = wγ(v). It is the circular cone through the origin (cf. Fig. 15, 16 and 17).

Figure 15: A parallel circle Figure 16: The circular cone Figure 17: The union of Fig.15 and 16

References

[1] J. W. Bruce and P. J. Giblin, Curves and singularities (second edition). Cambridge University press (1992)

[2] P. Hartman and L. Nirenberg, On spherical image maps whose Jacobians do not change sign. Amer. J. Math. 81 (1959), 901–920

[3] S. Izumiya and N. Takeuchi, Geometry of ruled surfaces, Applicable math. in the golden age (2003) 305-338.

[4] S. Izumiya and S. Otani, Flat approximations of surfaces along curves, DEMONSTRATIO MATHEMATICA XLVIII (2015), DOI: 10.1515/dema-2015-0016

[5] I. R. Porteous, Geometric diﬀerentiation for the intelligence of curves and surfaces (second edition). Cambridge University press (2001)

[6] I. Vaisman, A ﬁrst course in diﬀerential geometry. Pure and applied Mathematics, A series of Monograph and Textbooks, MARCEL DEKKER, (1984)

Shyuichi Izumiya, Department of Mathematics, Hokkaido University, Sapporo 060-0810,Japan e-mail:[email protected]

Satoshi Hananoi, Fudooka high school, 1-7-45, Fudooka, Kazo, Saitama 347-8513, Japan e-mail:[email protected]