GEOMETRY AND TOPOLOGY OF CAUSTICS — CAUSTICS ’98 BANACH CENTER PUBLICATIONS, VOLUME 50 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 1999 THE RECTIFYING DEVELOPABLE AND THE SPHERICAL DARBOUX IMAGE OF A SPACE CURVE SHYUICHIIZUMIYA Department of Mathematics, Hokkaido University Sapporo 060-0810, Japan E-mail: [email protected] HARUYO KATSUMI and TAKAKO YAMASAKI Department of Mathematics, Ochanomizu University Bunkyou-ku Otsuka Tokyo 112-8610, Japan Dedicated to the memory of Professor Yosuke Ogawa Abstract. In this paper we study singularities of certain surfaces and curves associated with the family of rectifying planes along space curves. We establish the relationships between singularities of these subjects and geometric invariants of curves which are deeply related to the order of contact with helices. 1. Introduction. There are several articles concerning singularities of the tangent developable (i.e., the envelope of osculating planes) and the focal developable (i.e., the envelope of normal planes) of a space curve ([3{12]). In these papers the relationships between singularities of these surfaces and classical geometric invariants of space curves have been studied. The notion of the distance-squared functions on space curves is useful for the study of singularities of focal developable [7, 10, 11]. For tangent developable, there are other techniques to study singularities [3{6, 8, 9]. The classical invariants of extrinsic differential geometry can be interpreted as \singularities" of these developable; however, the authors cannot find any article concerning singularities of the rectifying developable (i.e., the envelope of rectifying planes) of a space curve. The rectifying developable is an important surface in the following sense: the space curve γ is always a geodesic of the rectifying developable of itself (cf. [2, p. 308]). 1991 Mathematics Subject Classification: Primary 58C27; Secondary 53A04. The paper is in final form and no version of it will be published elsewhere. [137] 138 S. IZUMIYA ET AL. In this paper we introduce the notion of volumelike distance functions (or, binormal directed distance functions) on space curves. This function is quite useful for the study of generic singularities of rectifying developable of space curves. We also introduce the notion of volumelike height functions (or, tangential height functions) on space curves, which induce the notion of rectifying Gaussian surfaces and spherical Darboux images and these singularities are deeply related to the geometry of spherical tangential images of curves. As a consequence, we apply ordinary techniques of singularity theory for these func- tions and describe the relationships between the singularities of the above three subjects and differential geometric invariants of space curves. The main results in this paper are Theorems 2.1 and 2.2. We describe the geometric meaning of Theorem 2.2 in Section 4. The proof of Theorem 2.2 is given in Section 5. In Section 6 we consider generic properties of space curves. Since the calculations for the proof of Theorem 2.1 are terribly long and tedious, we only give a sketch of the proof in this paper. The basic techniques we used in this paper depend heavily on those in the book of Bruce and Giblin [1], so the authors are grateful to both of them. The authors also wish to thank Professors K. Tsukada and T. Fukui for valuable suggestions. All curves and maps considered here are of class C1 unless otherwise stated. 2. Basic notions and the main results. We now review some basic concepts on classical differential geometry of space curves in Euclidean space R3. For any two vectors x = (x1; x2; x3) and y = (y1; y2; y3), we denote by hx; yi the standard inner product. Let γ : I ! R3 be a curve withγ _ (t) 6= 0, whereγ _ (t) = dγ=dt(t). We also denote the norm of x by kxk. The arc-length of a curve γ, measured from γ(t0), t0 2 I, is Z t s(t) = kγ_ (t)k dt: t0 Then a parameter s is determined such that kγ0(s)k = 1, where γ0(s) = dγ=ds(s). So we say that a curve γ is parametrized by the arc-length if it satisfies kγ0(s)k = 1. We define t(s) = γ0(s) and call t(s) a unit tangent vector of γ at s. The curvature of γ is defined by κ(s) = pkγ00(s)k. If κ(s) 6= 0, then the unit principal normal vector n(s) of the curve γ at s is given by γ00(s) = κ(s)n(s). The unit vector b(s) = t(s) × n(s) is called a unit binormal vector of the curve γ at s. Then the following Frenet-Serret formula holds: 8 t0(s) = κ(s)n(s) < n0(s) = −κ(s)t(s) + τ(s)b(s) : b0(s) = −τ(s)n(s); where τ(s) is the torsion of the curve γ at s. It is well known that the point at where the torsion τ(s) vanishes is the point at where the curve has at least third order contact with the osculating plane. It corresponds to the degenerated singularities of the tangent developable of γ (cf. [3]). So we do not consider such a point for the study of singularities of rectifying developable of γ. We assume that τ(s) 6= 0 throughout this paper. For any unit speed curve γ : I −! R3, we call D(s) = τ(s)t(s) + κ(s)b(s) a Darboux vector of γ (cf. [7, Section 5.2]). By using the Darboux vector, the Frenet-Serret formula is rewritten RECTIFYING DEVELOPABLE 139 as follows: 8 t0(s) = D(s) × t(s) < n0(s) = D(s) × n(s) : b0(s) = D(s) × b(s): Thus the Darboux vector plays an important role for the study of space curves. We define a vector D~(s) = (τ/κ)(s)t(s) + b(s) and we call it a modified Darboux vector 2 D(s) along γ. We also define a spherical curve d : I −! S by d(s) = kD(s)k and the surfaces RG(γ) = fut(s) + b(s) j u 2 R; s 2 Ig; RD(γ) = fγ(s) + uD~(s) j u 2 Rg: We call the image of d the spherical Darboux image, the surface RG(γ) the rectifying Gaussian surface and the surface RD(γ) the rectifying developable of γ. Our main results are the following: 1 3 1 3 Theorem 2.1. Let Immr(S ; R ) be the space of regular curves γ : S −! R with τ 6= 0 and κ 6= 0 equipped with C1-topology. Then there exists a residual set O ⊂ 1 3 Immr(S ; R ) such that for any γ 2 O the following properties hold: (1) The number of the points s 2 S1 where (τ/κ)0(s) = 0 is finite. (2) There is no point s 2 S1 where (τ/κ)0(s) = (τ/κ)00(s) = 0. (3) The number of the points s 2 S1 where (τ/κ)00(s) = 0 is finite. (4) There is no point s 2 S1 where (τ/κ)00(s) = (τ/κ)000(s) = 0. Theorem 2.2. Let γ : I −! R3 be a unit speed curve. Then we have the following: (1) The spherical Darboux image is locally diffeomorphic to the ordinary cusp C at 0 00 d(s0) if and only if (τ/κ) (s0) = 0 and (τ/κ) (s0) 6= 0. (2) (a) The rectifying Gaussian surface is locally diffeomorphic to the cuspidal edge 0 C × R at u0t(s0) + b(s0) if and only if u0 = (τ/κ)(s0) and (τ/κ) (s0) 6= 0. (b) The rectifying Gaussian surface is locally diffeomorphic to the swallow tail 0 00 SW at u0t(s0) + b(s0) if and only if u0 = (τ/κ)(s0), (τ/κ) (s0) = 0 and (τ/κ) (s0) 6= 0. (3) (a) The rectifying developable is locally diffeomorphic to the cuspidal edge C × R ~ 0 00 1 at γ(s0) + u0D(s0) if and only if (τ/κ) (s0) 6= 0, (τ/κ) (s0) 6= 0 and u0 = 0 . (τ/κ) (s0) (b) The rectifying developable is locally diffeomorphic to the swallow tail SW at 0 00 000 γ(s0) + u0D~(s0) if and only if (τ/κ) (s0) 6= 0, (τ/κ) (s0) = 0, (τ/κ) (s0) 6= 0 and 1 u0 = 0 . (τ/κ) (s0) 2 3 Here, C = f(x1; x2) j x1 = x2 g is the ordinary cusp and SW = f(x1; x2; x3) j x1 = 4 2 3 3u + u v; x2 = 4u + 2uv; x3 = vg is the swallow tail. The geometric meanings of the singularities of Image d, RG(γ) and RD(γ) will be discussed in Section 4. 3. Families of smooth functions on a space curve. In this section we introduce two different families of functions on a space curve which are useful for the study of singularities of d, RG(γ) and RD(γ). Let γ : I ! R3 be a unit speed space curve with κ(s) 6= 0 and τ(s) 6= 0. 140 S. IZUMIYA ET AL. Volumelike distance functions. We now define a three-parameter family of smooth functions on I: 3 F : I × R ! R by F (s; u) = t(s) n(s) γ(s)−u = hγ(s)−u; b(s)i: Here, a b c denotes the determinant of the matrix a b c and ha; bi denotes the standard inner product of a and b. We call F a volumelike distance function (or a binormal directed distance function) on γ. We use 3 the notation fu(s) = F (s; u) for any u 2 R .