Relatively normal-slant helices lying on a surface and their characterizations

Transcription

1 Hacettepe Journal of Mathematics and Statistics Volume , Relatively normal-slant helices lying on a surface and their characterizations Nesibe MAC T and Mustafa DÜLDÜL Abstract In this paper, we consider a regular curve on an oriented surface in Euclidean 3-space with the Darboux frame {T, V, U} along the curve, where T is the unit tangent vector eld of the curve, U is the surface normal restricted to the curve and V = U T. We dene a new curve on a surface by using the Darboux frame. This new curve whose vector eld V makes a constant angle with a xed direction is called as relatively normal-slant helix. We give some characterizations for such curves and obtain their axis. Besides we give some relations between some special curves general helices, integral curves, etc. and relatively normal-slant helices. Moreover, when a regular surface is given by its implicit or parametric equation, we introduce the method for generating the relatively normal-slant helix with the chosen direction and constant angle on the given surface. Keywords: Slant helix, generalized helix, Darboux frame, implicit surface, parametric surface, spherical indicatrix. 000 AMS Classication: 65L05, 53A04, 53A05 Received : Accepted : Doi : /HJMS Department of Mathematics, Faculty of Science and Arts, Yldz Technical University, stanbul, Turkey, ngurhan@yildiz.edu.tr Department of Mathematics, Faculty of Science and Arts, Yldz Technical University, stanbul, Turkey, mduldul@yildiz.edu.tr Corresponding Author.

2 Introduction Helical curves play important roles in not only CAD and CAGD but also in science and nature. In the eld of computer aided design and computer graphics, helices can be used for the tool path description, the simulation of kinematic motion or the design of highways, etc. [4]. They also arise in the structure of DNA, fractal geometry e.g. hyperhelices, etc. [5, 3]. In recent years, helical curves are studied widely in Euclidean spaces e.g. [1-3, 11, 15, 0] and in non-euclidean spaces e.g. [4, 8, 1]. A space curve whose tangent vector makes a constant angle with a xed direction is called a generalized helix and characterized by Lancret's theorem which says the ratio of torsion to curvature is constant [1]. Generalized helices are also studied in higher dimensional spaces [19, ]. In 004, Izumiya and Takeuchi dene a slant helix in E 3 by the property that the principal normal vector makes a constant angle with a xed direction. These curves are characterized by the constancy of geodesic curvature function of the principal normal indicatrix of the curve [10]. In 005 Kula and Yayl have studied spherical indicatrices of a slant helix and obtained that the spherical images are spherical helix [14]. Besides, slant helices according to Bishop frame are studied by Bukcu and Karacan in 009 [6]. Also, slant helices according to quaternionic frame [13] and slant helices in 3-dimensional Lie group [17] are investigated in 013. In [5], Zplar and Yayl introduce a Darboux helix which is dened as a curve whose Darboux vector ω = τt + B makes a constant angle with a xed direction, and give some characterizations of such curves in 01. Puig-Pey et al. introduce the method for generating the general helix for both implicit and parametric surfaces [0]. In a recent paper, Do an and Yayl study isophote curves and their characterizations in Euclidean 3-space [7]. An isophote curve is dened as a curve on a surface whose unit normal vector eld restricted to the curve makes a constant angle with a xed direction. They also obtain the axis of an isophote curve. In this study, we dene a relatively normal-slant helix on a surface by using the Darboux frame {T, V, U} along the curve whose vector eld V makes a constant angle with a xed direction. We give some characterizations for such curves and obtain their axis. Besides we give some relations between some special curves general helices, integral curves, etc. and relatively normal-slant helices. Moreover, when a regular surface is given by its implicit or parametric equation, we introduce the method for generating the relatively normal-slant helix with the chosen direction and constant angle on the given surface. This method is based on an initial-value problem of 1st order ordinary dierential equations. This paper is organized as follows: Section includes some basic denitions. We dene relatively normal-slant helices in section 3 and give some characterizations for such curves. The method for nding the relatively normal-slant helix on a given surface is introduced in section 4 for parametric and implicit surfaces and its application is presented in section 5. Section 6 includes some relations between the special curves and relatively normal-slant helices. Some further characterizations related with the subject are also given in section 7.. Preliminaries Let M be an oriented surface, and α : I R M be a regular curve with arc-length parametrization. If the Frenet frame along the curve is denoted by {T, N, B}, the Frenet

3 399 formulas are given by T = N, N = T + τb, B = τn, where T is unit tangent vector, N is principal normal vector, B is the binormal vector; and τ are the curvature and the torsion of α, respectively. On the other hand, if we denote the Darboux frame along the curve α by {T, V, U}, we have the derivative formulae of the Darboux frame as T = gv + nu, V = gt + τ gu, U = nt τ gv, where T is the unit tangent vector of the curve, U is the unit normal vector of the surface restricted to the curve, V is the unit vector given by V = U T, and g, n, τ g denote the geodesic curvature, normal curvature, geodesic torsion of the curve, respectively [18]. The relations between geodesic curvature, normal curvature, geodesic torsion and, τ are given as follows: g = sin ϕ, n = cos ϕ, τ g = τ + dϕ ds, where ϕ is the angle between the vectors N and U. If the surface M is given by its parametric equation Xu, v = xu, v, yu, v, zu, v, we may write αs = Xus, vs for the curve α. Thus, the tangent vector of the curve α at the point αs becomes dα.1 ds = du Xu ds + dv Xv ds, where X u and X v denote the partial dierentiation of X with respect to u and v, respectively. Hence, since αs is a unit-speed curve, we have du. E + F du dv dv ds ds ds + G = 1, ds where E = X u, X u, F = X u, X v and G = X v, X v are the rst fundamental form coecients of the surface. On the other hand, if the surface M is given by its implicit equation fx, y, z = 0, then the unit speed curve αs = xs, ys, zs lying on M satises dx.3 f x ds + dy fy ds + dz fz ds = 0 and dx dy dz = 1, ds ds ds where f a = f a..1. Denition Slant helix. A unit speed curve is called a slant helix if its unit principal normal vector makes a constant angle with a xed direction [10]... Theorem. Let α be a unit speed curve with 0. Then α is a slant helix if and only if τ σs = s + τ 3/ is a constant function [10].

4 Denition V-direction curve. Let γ be a unit speed curve on an oriented surface M and {T, V, U} be the Darboux frame along γ. The curve γ lying on M is called a V-direction curve of γ if it is the integral curve of the vector eld V. In other words, if γ is the V-direction curve of γ, then Vs = γ s [16]..4. Denition Darboux vector elds. Let M be an oriented surface, and α : I R M be a regular curve with arc-length parametrization. The vector elds D ns, D rs, D os along α given by D ns = nsvs + gsus, D rs = τ gsts + gsus, D os = τ gsts nsvs are called the normal Darboux vector eld, the rectifying Darboux vector eld and the osculating Darboux vector eld along α, respectively [9]. 3. Relatively normal-slant helix and its axis In this section, we introduce a new kind of helix which is lying on a surface. Let M be an oriented surface in E 3 and γ be a regular curve lying on M. Let us denote the Darboux frame along γ with {T, V, U}, where T is the unit tangent vector eld of γ, U is the unit normal vector eld of the surface which is restricted to the curve γ and V = U T Denition Relatively normal-indicatrix. Let γ be a unit speed curve with arclength parameter s on an oriented surface M and {T, V, U} be the Darboux frame along γ. We dene the curve which Vs draws on the unit sphere S as the relatively normalindicatrix of γ, and denote it by γ v. Thus, γ vs = Vs. 3.. Denition Relatively normal-slant helix. Let γ be a unit speed curve on an oriented surface M and {T, V, U} be the Darboux frame along γ. The curve γ is called a relatively normal-slant helix if the vector eld V of γ makes a constant angle with a xed direction, i.e. there exists a xed unit vector d and a constant angle θ such that V, d = cosθ Theorem. A unit speed curve γ on a surface M with τ gs, gs 0, 0 is a relatively normal-slant helix if and only if 3.1 σ vs = 1 g + τg 3 τ g g gτ g n g + τ g s is a constant function. Proof. Let the unit speed curve γ on a surface M with τ gs, gs 0, 0 be a relatively normal-slant helix. Then, by the denition, the vector eld V of γ makes a constant angle with a xed direction. Thus, the relatively normal-indicatrix of the curve γ is part of a circle, i.e. it has constant curvature and zero torsion. For the relatively normal-indicatrix of the curve γ, we may write γ vs = Vs which, by dierentiating with respect to s, yields γ vs = gt + τ gu γ v s = g nτ g T g + τ g V + g n + τ g U.

5 401 The curvature v and the torsion τ v of γ v are obtained as vs = γ v γ v = 1 + σ γ v vs, 3 3. τ vs = 1 g s+τ g s 11 σ v s 1+σ vs, 1 where σ vs = τ g g +τ g 3 g gτ g n g + τg. Hence, since γ vs has constant curvature and zero torsion, we have σ vs = constant. Suppose that σ vs = constant. Then, by using 3. the relatively normalindicatrix of γ has constant curvature and zero torsion, i.e. relatively normal-indicatrix is part of a circle which means V makes constant angle with a xed direction. Thus, γ is a relatively normal-slant helix Remark. The constant function σ vs is also the geodesic curvature of the relatively normal-indicatrix of γ Remark. The invariant σ vs is equal to δrs, where δ g +τg r is given in [9]. We may give the following corollaries for the special cases of the relatively normal-slant helices Corollary. i Let γ be an asymptotic curve on M with gs 0. Then, γ is a relatively normal-slant helix on M if and only if 3.3 g τg g + τg 3 s = constant. g ii If γ is an asymptotic curve on M with gs 0, we have = g, τ = τ g. Then, γ is a relatively normal-slant helix on M if and only if γ is a slant helix Corollary. Let γ be a geodesic curve on M with τ gs 0. Then, γ is a relatively normal-slant helix on M if and only if 3.4 ns τ gs = constant, i.e. γ is a relatively normal-slant helix on M if and only if γ is a generalized helix on M Corollary. Let γ be a line of curvature on M with gs 0. Then, γ is a relatively normal-slant helix on M if and only if 3.5 ns gs = constant. Now, it is usual to ask that how can we nd the axis of a relatively normal-slant helix. Our goal is now to obtain the axis of our new dened slant helix. Let a unit speed curve γ lying on an oriented surface M be a relatively normal-slant helix. Then, by the denition, V makes a constant angle θ with the xed unit vector d = at+bv+cu, i.e. V, d = cos θ = b. By dierentiating the last equation with respect to s gives V, d = 0 gt + τ gu, d = 0. If we dierentiate d = at + bv + cu with respect to s, we obtain d = a b g c nt + a g cτ gv + c + a n + bτ gu = 0,

6 40 i.e. 3.6 a b g c n = 0, a g cτ g = 0, c + a n + bτ g = 0. If we substitute c = g τ g a, τ gs 0, into the rst and third equations and multiply the third one with g τ g, we obtain a rst order homogeneous linear dierential equation as g 1 + a + g g a = 0 τ g τ g τ g which has the general solution τ g a = λ, g + τg where λ is the constant of integration. Substituting this solution into the second equation of 3.6, we obtain g c = λ g + τg On the other hand, since d is unit length, i.e. a + b + c = 1, we nd λ = ± sin θ. Therefore, d can be written as τ g g 3.7 d = ± sin θt + cos θv ± sin θu. g + τg g + τg We need to determine the constant angle θ to complete the axis. If we dierentiate V, d = 0 with respect to s along the curve γ, we obtain V τ g g gτ g n g + τg, d = ± sin θ g + τg g + τg cos θ = 0. As a result we have 3.8 cot θ = ± τ g g gτ g n g + τg. g + τg 3 Hence, nding θ from 3.8 and substituting the result into 3.7 gives us the axis of the relatively normal-slant helix By using 3.8, it is easy to see that d is a constant vector, i.e. d = Calculating a relatively normal-slant helix on a surface In this section we introduce the methods for nding the relatively normal-slant helix on a given surface. We discuss the methods separately for parametric and implicit surfaces Relatively normal-slant helix on a parametric surface. Let M be a regular oriented surface in E 3 with the parametrization X = Xu, v. Our goal is now, when a xed unit direction d and a constant angle θ are given, to give the method which enables us to nd the relatively normal-slant helix if exists lying on M which accepts d as an axis and θ as the constant angle. Let γs = Xus, vs be the unit speed relatively normal-slant helix lying on M with axis d, constant angle θ, and Darboux frame eld {T, V, U}. Thus, we have V, d = cos θ. We need to nd us, vs to obtain γ. Since γ = T = X u du ds + Xv dv ds and U = Xu Xv X u X v,

7 403 we have i.e. V = U T = V = Xu Xv X u X v du X u ds + dv Xv ds [ 1 EX v F X du ] dv u + F Xv GXu. X u X v ds ds Substituting the last equation into V, d = cos θ gives us 4.1 E X v, d F X u, d du dv + F Xv, d G Xu, d = cos θ Xu Xv. ds ds Combining Eqs.. and 4.1, we obtain du = cos θeg F 3 Xv,d ± ds AEG F 4. 3 X u,d dv = cos θeg F ds AEG F where A,, and are given by A = E X v, d F X u, d X v, d + G X u, d, = 4 cos θeg F [ X v, d EG F AG ] +4AEG F [F X v, d G X u, d ], = 4 cos θeg F [ X u, d EG F AE ] +4AEG F [E X v, d F X u, d ]. If we solve the system 4. together with the initial point { u0 = u0 4.3 v0 = v 0, we obtain the desired relatively normal-slant helix on M by substituting us, vs into Xu, v Remark. i If and/or is negative, it means there does not exist a relatively normal-slant helix with the given axis and angle. ii If 0 and 0, then we have two relatively normal-slant helices on the surface. 4.. Relatively normal-slant helix on an implicit surface. Let M be a surface given in implicit form by fx, y, z = 0. Let us now nd the relatively normal-slant helix γs which makes the given constant angle θ with the given axis d = a, b, c and lying on M. Let γs = xs, ys, zs and {T, V, U} be its Darboux frame eld. We need to nd xs, ys, zs to obtain γs We assume that s is the arc-length parameter. Since the unit normal vector eld of the surface is U = V = f f T = 1 f f, we obtain f dz f y ds dy fz ds, dx fz ds dz fx ds, dy fx ds dx fy ds If we substitute the last equation into V, d = cos θ, we get 4.4 bf z cf dx dy dz y + cfx afz + afy bfx = f cos θ. ds ds ds If we consider 4.4 and.3, we obtain dx 4.5 dx ds = 1 Ω, dy ds ds dz by means of as ds [ f yaf y bf x f zcf x af dz z fy f cos θ ds ],.

8 dy ds = 1 [ f zbf z cf y f xaf y bf dz ] x + fx f cos θ, Ω ds where Ω = cf x af xf z bf yf z + cf y 0. Substituting 4.5 and 4.6 into.4 give us a quadratic equation with respect to dz where dz dz q 1 + q + q3 = 0, ds ds ds as q 1 = 1 Ω [b f 4 x + a f 4 y + a + b f 4 z abf xf 3 y acf xf 3 z bcf yf 3 z abf yf 3 x + a + b f xf y + c + b f xf z + a + c f y f z 4abf xf yf z ] + 1, q = 1 ] [ f cos θbf xf Ω z af yfx + bf xfy af yfz + bfx 3 afy 3, q 3 = 1 Ω [ From this equation we have 4.7 dz ds = q ± ] f cos θfx + fy 1. q 4q 1q 3 q 1. If we substitute 4.7 into 4.5 and 4.6, we obtain an explicit 1st order ordinary dierential equation system. Thus, together with the initial point x0 = x y0 = y 0 z0 = z 0 we have an initial value problem. The solution of this problem gives us the relatively normal-slant helix on M. 4.. Remark. i If q 4q 1q 3 < 0 at the point x 0, y 0, z 0, then there does not exist any relatively normal-slant helix with the given direction d and angle θ. ii If q 4q 1q 3 > 0 at the point x 0, y 0, z 0, then we have two relatively normal-slant helices passing through the initial point. 5. Examples 5.1. Example. Let M be the surface given by Xu, v = u cos v, u sin v, u. Applying our method, the relatively normal-slant helix on M which makes the constant angle θ = π 3 with the chosen direction d = 0, 0, 1 and starting from the inital point P = 1, 0, 1 is given in Figure 1. Another application is given in Figure in which we obtain two relatively normal-slant helices lying on the surface x + y z + x +y 1 = 0 with the initial point 13, 0, 3 in each gure the general helices are obtained by the method given in [0].

9 405 Figure 1. General helix colored with blue and relatively normalslant helix colored with black with same axis d = 0, 0, 1 and angle θ = π/3 6. Some special curves related with a relatively normal-slant helix 6.1. Theorem. Let M be an oriented surface in E 3, γ be a unit speed curve on M, and γ be the V-direction curve of γ. Then γ is a relatively normal-slant helix if and only if γ is a general helix. Proof. Since γ is the V-direction curve of γ, the relations between the curvature and the torsion τ of γ and the geodesic curvature, normal curvature and geodesic torsion of γ are given by, [16], 6.1 = g + τ g, τ = n + gτ g gτ g. g + τg Thus, we have τ = 1 τ g g + τg 3 g gτ g n g + τg. It means that γ is a relatively normal-slant helix if and only if γ is a general helix. 6.. Theorem. Let M be an oriented surface in E 3, γ : I R M be a regular curve with arc-length parametrization and D r be the rectifying Darboux vector eld of γ. Then γ is a relatively normal-slant helix if and only if the integral curve of the rectifying Darboux vector eld D r is a circular helix. Proof. The rectifying Darboux vector eld D r of γ is given by 6. D rs = τ gsts + gsus. Let β be the integral curve of D r. So we have D r = β = τ gt + gu.

10 406 Figure. General helix colored with blue θ = π/3 and relatively normal-slant helices colored with black θ = π/3 and green θ = π/4 with same axis d = 0, 0, 1 By dierentiating this equation with respect to s we obtain β = τ g g nt + g + nτ gu, β = τ g g n T + τ g g n gv + nu + g + nτ g U + g + nτ g nt τ gv. Therefore, since β β = n g + τg + τ g g τ g g V, the curvature β of β is obtained as β = β β 1 = τ g β 3 g + τg 3 g gτ g n g + τg = σ vs and the torsion τ β of β is obtained as τ β = β β, β β β = 1. Therefore τ β β = 1 σ vs. It means that γ is a relatively normal-slant helix if and only if the integral curve of the rectifying Darboux vector eld D r is a circular helix Theorem. Let γ : I M E 3 be a unit speed curve with gs, τ gs 0, 0. Then the following conditions are equivalent. i γ is a relatively normal-slant helix,

11 407 ii The relatively normal-indicatrix of γ is part of a circle in S, iii σ vs = 1 g +τ g 3 τ g g gτ g n g + τ g iv There exists a unit vector d such that v The V-direction curve of γ is a general helix, vi The D r-integral curve of γ is a circular helix. 7. Further characterizations s is constant, D d, r is constant, D r Similar to the previous sections, we can also give the following characterizations for the general helices and isophote curves Theorem. Let M be an oriented surface in E 3, γ : I R M be a regular curve with arc-length parametrization and D n be the normal Darboux vector eld of γ. Then γ is a general helix if and only if the integral curve of the normal Darboux vector eld D n is a circular helix. 7.. Theorem. Let M be an oriented surface in E 3, γ : I R M be a regular curve with arc-length parametrization and D o be the osculating Darboux vector eld of γ. Then γ is an isophote curve if and only if the integral curve of the osculating Darboux vector eld D o is a circular helix. Acknowledgment. The authors would like to thank the reviewer for the valuable comments and suggestions. References [1] Ali AT. Position vectors of slant helices in Euclidean Space, Journal of the Egyptian Mathematical Society 01; 0: 1-6. [] Ali AT. New special curves and their spherical indicatrices, Global Journal of Advanced Research on Classical and Modern Geometries 01; 1: [3] Ali AT, Turgut M. Some characterizations of slant helices in the Euclidean space E n, Hacettepe Journal of Mathematics and Statistics 010; 393, [4] Ali AT, Turgut M. Position vector of a time-like slant helix in Minkowski 3-space, Journal of Mathematical Analysis and Applications 010; 365, [5] Biton YY, Coleman BD, Swigon D. On bifurcations of equilibria of intrinsically curved, electrically charged, rod-like structures that model DNA molecules in solution, Journal of Elasticity 007; 87, [6] Bukcu B, Karacan MK. The slant helices according to Bishop frame, International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering 009; 311, [7] Do an F, Yayl Y. On isophote curves and their characterizations, Turkish Journal of Mathematics 015; 39: , Journal of Mathematical Analysis and Ap- [8] Gökçelik F, Gök. Null W-slant helices in E1 3 plications 014; 40 1, -41. [9] Hananoi S, Ito N, Izumiya S. Spherical Darboux images of curves on surfaces, Beitr Algebra Geom. 015; 56: [10] Izumiya S, Takeuchi N. New special curves and developable surfaces, Turkish Journal of Mathematics 004; 8: [11] Izumiya S, Takeuchi N. Generic properties of helices and Bertrand curves, Journal of Geometry 00; 74: [1] Kahraman F, Gök, Hacisaliho lu HH. On the quaternionic B slant helices in the semi- Euclidean space E 4, Applied Mathematics and Computation 01; 18 11,

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