Focal representation of k-slant helices in Em+1

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DOI: 10.1515/ausm-2015-0013

Focal representation of k-slant Helices

in E

m+1

G¨

unay ¨

Ozt¨

urk

Department of Mathematics

Kocaeli University, Turkey email: ogunay@kocaeli.edu.tr

Bet¨

ul Bulca

Uluda˘g University, Turkey email: bbulca@uludag.edu.tr

Beng¨

u Bayram

Department of Mathematics Balıkesir University, Turkey email: benguk@balikesir.edu.tr

Kadri Arslan

Uluda˘g University, Turkey email: arslan@uludag.edu.tr

Abstract. The focal representation of a generic regular curve γ in Em+1 consists of the centers of the osculating hyperplanes. A_{k-slant helix γ in} Em+1 _{is a (generic) regular curve whose unit normal vector}_V

k makes a constant angle with a ﬁxed direction−→_{U in E}m+1. In the present paper we proved that if_{γ is a k-slant helix in E}m+1_{, then the focal representation} Cγofγ in Em+1is an(m − k + 2)-slant helix in Em+1.

1 Introduction

Curves with constant slope, or so-called general helices (inclined curves), are well-known curves in the classical differential geometry of space curves. They are defined by the property that the tangent makes a constant angle with a fixed line (the axis of the general helix) (see, [1], [4], [7] and [8]). In [10], the definition is more restrictive: the fixed direction makes constant angle with these all the vectors of the Frenet frame. It is easy to check that the definition

2010 Mathematics Subject Classiﬁcation: 53A04, 53C42 Key words and phrases: Frenet curve, focal curve, slant helix

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only works in the odd dimensional case. Moreover, in the same reference, it is proven that the deﬁnition is equivalent to the fact the ratios κ2

κ1,

κ4

κ3, . . . , κibeing the curvatures, are constant. Further, J. Monterde has considered the

Frenet curves in Em which have constant curvature ratios (i.e., κ2

κ1,

κ3

κ2,

κ4

κ3. . . . are constant) [14]. The Frenet curves with constant curvature ratios are called ccr-curves. Obviously, ccr-curves are a subset of generalized helices in the sense of [10]. It is well known that curves with constant curvatures (W-curves) are well-known ccr-curves [12], [15].

Recently, Izumiya and Takeuchi have introduced the concept of slant helix in Euclidean 3-space E3 _{by requiring that the normal lines make a constant}

angle with a fixed direction [11]. Further in [3] Ali and Turgut considered the generalization of the concept of slant helix to Euclidean_{n-space E}n, and gave some characterizations for a non-degenerate slant helix. As a future work they remarked that it is possible to define a slant helix of type-k as a curve whose unit normal vector_V_k makes a constant angle with a fixed direction−→_{U [}9].

For a smooth curve (a source of light)_{γ in E}m+1, the caustic of_{γ (defined as} the envelope of the normal lines of_{γ) is a singular and stratified hypersurface.} The focal curve of_{γ , C}_γ , is defined as the singular stratum of dimension _{1 of} the caustic and it consists of the centers of the osculating hyperspheres of _{γ .} Since the center of any hypersphere tangent to_{γ at a point lies on the normal} plane to _{γ at that point, the focal curve of γ may be parametrized using the} Frenet frame(t, n₁_{, n}₂_{, dots, n}_m) of γ as follows:

Cγ(θ) = (γ + c1n1+ c2n2+ · · · + cmnm)(θ),

where the coeﬃcients _c₁_{, . . . , c}_m are smooth functions that are called focal curvatures of_{γ [}18].

This paper is organized as follows: Section _{2 gives some basic concepts of} the Frenet curves in Em+1. Section _{3 tells about the focal representation of} a generic curve given with a regular parametrization in Em+1_{. Further this}

section provides some basic properties of focal curves inEm+1and the structure of their curvatures. In the ﬁnal section we consider k-slant helices in Em+1. We prove that if_{γ is a k-slant helix in E}m+1 _{then the focal representation}_C

γ

of _{γ is an (m − k + 2)-slant helix in E}m+1_.

2 Basic concepts

Let_{γ = γ(s) : I → E}m+1 be a regular curve in Em+1, (i.e., γ(s) is nowhere zero) where I is an interval in R. Then γ is called a Frenet curve of osculating

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order _{d, (2 ≤ d ≤ m + 1) if γ}(s), γ(s),. . . ,γ(d)(s) are linearly independent

and _γ(s), γ(s),. . . ,γ(d+1)(s) are no longer linearly independent for all s in _{I [}18]. In this case, _{Im(γ) lies in a d-dimensional Euclidean subspace of} Em+1_{. To each Frenet curve of rank d there can be associated the orthonormal}

d-frame {t, n1, . . . , nd−1} along γ, the Frenet r-frame, and d − 1 functions

κ1, κ2, . . . , κd−1:I −→ R, the Frenet curvatures, such that

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ t n₁ n₂ . . . n_d−1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦= v ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 0 κ1 0 . . . 0 −κ1 0 κ2 . . . 0 0 −κ2 0 . . . 0 . . . κd−1 0 0 . . . −κd−1 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ t n1 n2 . . . nd−1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ (1)

where,_{v is the speed of γ. In fact, to obtain t, n}₁_{, . . . , n}_d−1 it is suﬃcient to ap-ply the Gram-Schmidt orthonormalization process to_γ(s), γ(s),. . . ,γ(d)(s). Moreover, the functions_κ₁_{, κ}₂_{, . . . , κ}_d−1 are easily obtained as by-product dur-ing this calculation. More precisely, _{t, n}₁_{, . . . , n}_d−1 and _κ₁_{, κ}₂_{, . . . , κ}_d−1 are determined by the following formulas:

v1(s) : = γ(s) ; t := _vv1(s) 1(s), vα(s) : = γ(α)(s) − α−1 i=1 < γ(α)(s), vi(s) > vi(s) vi(s)2 , (2) κα−1(s) : = _v vα(s) α−1(s) v1(s), nα−1 : = _vvα(s) α(s), where_{α ∈ {2, 3, . . . , d} (see, [}8])_.

A Frenet curve of rank _{d for which κ}₁_{, κ}₂_{, . . . , κ}_d−1 are constant is called (generalized) screw line or helix [6]. Since these curves are trajectories of the 1-parameter group of the Euclidean transformations, so, F. Klein and S. Lie called them _{W-curves [}12]. For more details see also [5]. _{γ is said to have} constant curvature ratios (that is to say, it is a ccr-curve) if all the quotients

κ2 κ1, κ3 κ2, κ4 κ3, . . . , κi

κi−1 (1 ≤ i ≤ m − 1) are constant [14], [15].

3 The focal representation of a curve in E

m+1

The hyperplane normal to_{γ at a point is the union of all lines normal to γ at} that point. The envelope of all hyperplanes normal toγ is thus a component

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of the focal set that we call the main component (the other component is the curve _{γ itself, but we will not consider it) [}16].

Deﬁnition 1 Given a generic curve (i.e., a Frenet curve of osculating order m + 1) γ : R → Em+1, let F : Em+1× R → R be the (m + 1)-parameter family

of real functions given by

F(q, θ) = 1

2q − γ(θ)

2

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The caustic of the family _{F is given by the set}

q ∈ Em+1: ∃θ ∈ R : Fq(θ) = 0 and Fq(θ) = 0

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[16].

Proposition 1 [17] The caustic of the family _{F(q, θ) =} 1₂q − γ(θ)2

coin-cides with the focal set of the curve_{γ : R → E}m+1_.

Deﬁnition 2 The center of the osculating hypersphere of γ at a point lies in

the hyperplane normal to the γ at that point. So we can write

Cγ = γ + c1n1+ c2n2+ · · · + cmnm, (5)

which is called the focal curve of _{γ, where c}₁_{, c}₂_{, . . . , c}_m are smooth functions

of the parameter of the curve_{γ. We call the function c}_i the_ith focal curvature

of _{γ. Moreover, the function c}₁ never vanishes and_c₁= _κ1

1 [18].

The focal curvatures of_{γ, parametrized by arc length s, satisfy the following} “scalar Frenet equations” for_c_m = 0 :

1 = κ1c1 c1 = κ2c2 c2 = −κ2c1+ κ3c3 . . . (6) cm−1 = −κm−1cm−2+ κmcm cm− (R 2 m´) 2cm = −κmcm−1

where_R_mis the radius of the osculating m-sphere. In particular_R2_m = C_γ−γ2 [18].

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Theorem 1 [16] Let _{γ : s → γ(s) ∈ E}m+1 be a regular generic curve. Write

κ1, κ2, . . . , κm for its Euclidean curvatures and{t, n1, n2, . . . , nm} for its Frenet

Frame. For each non-vertex_{γ(s) of γ, write ε(s) for the sign of (c}_m+c_m−1_κ_m)(s)

and _δ_α(s) for the sign of (−1)α_ε(s)κ_m(s), α = 1, . . . , m. Then the following holds:

a) The Frenet frame {T, N₁_{, N}₂_{, . . . , N}_m} of C_γ at _C_γ(s) is well-deﬁned and

its vectors are given by T = εnm, Nα = δαnm−l, for l = 1, . . . , m − 1, and

Nm= ±t. The sign in ±t is chosen in order to obtain a positive basis.

b) The Euclidean curvatures _K₁_{, K}₂_{, . . . , K}_m of the parametrized focal curve of γ, Cγ: s → Cγ(s), are related to those of γ by:

K1 |κm| = K2 κm−1 = · · · = |Km| κ1 = 1 |c m+ cm−1κm|, (7)

the sign of _K_m is equal to _δ_m times the sign chosen in±t.

That is the Frenet formulas of _C_γ at_C_γ(s) are T = 1 A|κm| N1 N1 = _A1 (− |κm| T + κm−1N2) N2 = _A1 (− |κm−1| N1+ κm−2N3) (8) . . . N_m−1 = 1 A(−κ2Nm−2∓δmκ1Nm) Nm = _A1∓δmκ1Nm−1 where_{A = |c}_m+ c_m−1_κ_m| .

Corollary 1 Let γ = γ(s) be a regular generic curve in Em+1 _and _C

γ : s →

Cγ(s) be the focal representation of γ. Then the Frenet frame of Cγ becomes

as follows; i) If _{m is even, then} T = nm N1 = −nm−1 N2 = nm−2 . . . (9) Nm−1 = −n1 Nm = t

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ii) If _{m is odd, then} T = nm N1 = −nm−1 N2 = nm−2 . . . (10) Nm−1 = n1 Nm = −t.

Proof. By the use of (7) with (8) we get the result.

4 k-Slant helices

Let _{γ = γ(s) : I → E}m+1 be a regular generic curve given with arclength parameter. Further, let −→_{U be a unit vector ﬁeld in E}m+1 such that for each s ∈ I the vector−→U is expressed as the linear combinations of the orthogonal basis{V₁(s), V₂(s), . . . , V_m+1(s)} with − → U = m+1 j=1 aj(s)Vj(s). (11)

where_a_j(s) are diﬀerentiable functions, 1 ≤ j ≤ m + 1.

Diﬀerentiating −→_{U and using the Frenet equations (}1), one can get d−→U ds = m+1 i=1 Pi(s)Vi(s), (12) where P1(s) = a 1− κ1a2, (13) Pi(s) = a

i+ κi−1ai−1− κiai+1, 2 ≤ i ≤ m,

Pm+1(s) = a

m+1+ κmam.

If the vector ﬁeld −→_{U is constant then the following system of ordinary} dif-ferential equations are obtained

0 = a1− κ1a2,

0 = a2+ κ1a1− κ2a3, (14)

0 = ai+ κi−1ai−1− κiai+1, 3 ≤ i ≤ m,

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Deﬁnition 3 Recall that a unit speed generic curve γ = γ(s) : I → Em+1 _is

called a _{k-type slant helix if the vector ﬁeld V}_k (_{1 ≤ k ≤ m + 1) makes a}

constant angle _θ_k with the ﬁxed direction −→_{U in E}m+1, that is

<−→U , Vk>= cos θk, θk= π

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A 1-type slant helix is known as cylindrical helix [2] or generalized helix [13], [4]. For the characterization of generalized helices in(n + 2)-dimensional Lorentzian spaceLn+2 see [19].

We give the following result;

Theorem 2 Let γ = γ(s) be a regular generic curve in Em+1_{. If C}

γ : s →

Cγ(s) is the focal representation of γ then the following statements are valid;

i) If _{γ is a 1-slant helix then the focal representation C}_γ of _{γ is an (m +}

1)-slant helix in Em+1_.

ii) If _{γ is an (m + 1)-slant helix then the focal representation C}_γ of _{γ is a}

1-slant helix in Em+1.

iii) If_{γ is a k-slant helix (2 < k < m) then the focal representation C}_γ of _γ

is an (m − k + 2)-slant helix in Em+1_.

Proof. i) Suppose γ is a 1-slant helix in Em+1_{. Then by Deﬁnition 3 the vector}

field_V₁ makes a constant angle_θ₁ with the fixed direction−→_{U defined in (}11), that is

<−→U , V1>= cos θ1, θ1= π

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For the focal representation _C_γ(s) of γ, we can choose the orthogonal basis {V1(s) = t, V2(s) = n1, . . . , Vm+1(s) = nm}

such that the equalities (9) or (10) is hold_{. Hence, we get,}

<−→U , V1>=<−→U , t >=<−→U , ±Nm>= cons. (17)

where{T, N₁_{, N}₂_{, . . . , N}_m} is the Frenet frame of C_γ at point _C_γ(s). From the equality (17) it is easy to see that_C_γ is an (m+1)-slant helix ofEm+1_.

ii) Suppose _{γ is an (m + 1)-slant helix in E}m+1_{. Then by Deﬁnition 3 the}

vector field _V_m+1 makes a constant angle _θ_m+1 with the fixed direction −→_U defined in (11), that is

<−→U , Vm+1>= cos θm+1, θm+1= π

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For the focal representation _C_γ(s) of γ, one can get

<−→U , Vm+1>=<−→U , nm>=<−→U , T >= cons. (19)

where{V₁ = t, V₂ = n₁_{, . . . , V}_m+1= n_m} and {T, N₁_{, N}₂_{, . . . , N}_m} are the Frenet frame of _{γ and C}_γ, respectively. From the equality (19) it is easy to see that Cγ is a 1-slant helix of Em+1.

iii) Suppose _{γ is a k-slant helix in E}m+1 (2 ≤ k ≤ m). Then by Definition 3 the vector field Vk makes a constant angle θk with the fixed direction −→U

deﬁned in (11), that is

<−→U , Vk>= cos θk, θk= π

2,2 ≤ k ≤ m. (20)

Let _C_γ(s) be the focal representation of γ. Then using the equalities (9) or (10) we get <−→U , Vk>=<−→U , nk−1>=<−→U , Nm−k+1>= cons., 2 ≤ k ≤ m (21) where {V1 = t, V2 = n1, . . . , Vm+1= nm} and V1 = T, V2= N1, . . . , Vm−k+2= Nm−k+1, . . . , Vm+1 = Nm

are the Frenet frame of _{γ and C}_γ, respectively. From the equality (21) it is easy to see that _C_γ is an (_{m − k + 2)-slant helix of E}m+1_.

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