Some Special Ruled Surfaces Generated by a Direction Curve according to the Darboux Frame and their Characterizations

In this work, we consider the Darboux frame ðT , V ,UÞ of a curve lying on an arbitrary regular surface and we construct ruled surfaces having a base curve which is a V-direction curve. Subsequently, a detailed study of these surfaces is made in the case where the directing vector of their generatrices is a vector of the Darboux frame, a Darboux vector field. Finally, we give some examples for special curves such as the asymptotic line, geodesic curve, and principal line, with illustrations of the different cases studied.


Introduction
Ruled surfaces are well known as one of the most important surface families in the differential geometry of the surfaces. An important fact about these types of surfaces is that any ruled surface can always result from a continuous movement of a straight line along a curve.
Among the associative curves are the direction curves introduced by Choi and Kim [1] as integral curves of some vector fields generated by Frenet vectors of a given curve. This type of curves is included in several works. In terms of the Frenet frame, the direction curves are studied in [1,2]; this study has been extended to the alternative frame [3], Darboux frame [4], and Bishop frame [5].
Examen the ruled surfaces constructed by means of direction curves has recently attracted the attention of many differential geometers. In terms of the Frenet frame, Güven [6] defined two ruled surfaces such as normal and binormal surfaces by considering their base curves as the W-direction curve. He get some results about the developability and minimality of these surfaces and the conditions for which their base curve is an asymptotic line, a geodesic curve, or a principal line. In the same way, this two ruled surfaces were also defined with a base curve and adjoint of the base curve in [7]. Moreover, in [8], the authors improve the the-ory of the ruled surfaces in terms of principal-direction curves of a given curve; they obtained a new representation of these ruled surfaces by slant helices and principal elements of the ruled surface such as the pitch and angle of pitch. After that, in [9] and in terms of the Darboux frame, the authors used the direction curve to define a new ruled surface called the relatively osculating developable surface. Then, they have obtained some results about the existence, uniqueness, and singularity of such surface.
In this work, we consider the Darboux frame ðT, V, UÞ of a curve lying on an arbitrary regular surface and we construct ruled surfaces whose base curve is a V-direction curve. We give some results about the developability, minimality, and condition for which the striction curve is the base curve for the particular cases where the directing vector of the ruled surface is a vector of the Darboux frame, a Darboux vector field. Finally, we give some examples for special curves such as the asymptotic line, geodesic curve, and principal line, with illustrations of the different cases studied.

Preliminaries
In this section, we recall some basic concepts and properties on classical differential geometry of curves lying on a regular surface and of ruled surfaces, in the Euclidean 3-space.
(i) We denote by E 3 the Euclidean 3-space, with the usual metric where x = ðx 1 , x 2 , x 3 Þ and y = ðy 1 , y 2 , y 3 Þ are two vectors of E 3 : Let M be a regular surface and α = αðsÞ: I ⊂ ℝ ⟶ M be a unit speed curve on the surface M. The Darboux frame along the curve α is an orthonormal frame ðTðsÞ, VðsÞ, Uðs ÞÞ, where T is the unit tangent, U is the unit normal on the surface M, and V = U ∧ T: Then, the Darboux equations are given by the following relations: where D n ðsÞ,D r ðsÞ, and D o ðsÞ are the normal Darboux vectors field, the rectifying Darboux vector field, and the osculator Darboux vectors field, respectively, and are defined by where k g ,k n , and τ g are the geodesic curvature, the normal curvature, and the geodesic torsion of the curve α, respectively.
(i) A ruled surface [10] is generated by a one-parameter family of straight lines and has a parametric representation: where β = βðsÞ is called the base curve of the ruled surface and XðsÞ the unit vectors representing the direction of straight lines. If X is constant, then, the ruled surface is cylindrical; otherwise, the surface is said to be noncylindrical.
Definition 1 [10]. For a curve α = αðsÞ lying on a regular surface, the following are well known: We have Then, the point Ψðs 0 , v 0 Þ is said to be singular if β′ðs 0 Þ ∧ Xðs 0 Þ + vX ′ ðs 0 Þ ∧ Xðs 0 Þ = 0: If there exists a common perpendicular to two constructive rulings in the ruled surface, then, the foot of the common perpendicular on the main rulings is called a central point. The locus of the central point is called a striction curve [10].
The parametrization of the striction curve on the ruled surface is given by Definition 3. A ruled surface Ψ is called developable if det ð β′, X, X′Þ = 0: Letting I and II be the first and the second fundamental forms from the ruled surface Ψ, respectively, we have I = Eds 2 + 2Fdsdv + Gdv 2 , where Abstract and Applied Analysis The mean curvature H of a ruled surface is given as follows Definition 4. A ruled surface is said to be minimal if its mean curvature vanishes identically.

Ruled Surfaces Defined by V-Direction Curves
Letting α = αðsÞ: I ⊂ ℝ ⟶ E 3 be a unit speed curve lying on a regular surface ,ðTðsÞ, VðsÞ, UðsÞÞ, the Darboux frames of α,k g ,k n ,τ g are the geodesic curvature, the normal curvature, the geodesic torsion of α, and β = βðsÞ a V-direction curve, respectively. We consider the following ruled surface where β = βðsÞ is the base curve and XðsÞ the unit director vector of the straight line.

General Study.
In this section, we propose to give some properties of the ruled surface Ψ: Differentiating (12) with respect to s and v, we get Then, by using (10), we obtain the components of the first fundamental form E and F as follows: Consequently, the condition of regularity of the ruled surface (12) is Differentiating (13) with respect to s and v, we get which allows, using formula (10), calculating the two components of the second fundamental form M and L, respectively, and we obtain Therefore, by using (11), the mean curvature of the ruled surface (12) is given as follows On the other hand, the striction curve of the surface (12) is 3.2. Case Where X = T (Resp. V, U). Let us consider the ruled surfaces defined by 3 Abstract and Applied Analysis By using (2), we have If we substitute (21) in (15), we find Likewise, we have If we substitute (23) in (15), we find On the other hand, If we substitute (25) in (15), we find Corollary 5. The ruled surfaces Ψ 1 and Ψ 3 are regulars along the curve β, while Ψ 2 is singular along β: We denote by c 1 (resp. c 2 ,c 3 ) the striction curve of Ψ 1 ðresp:Ψ 2 , Ψ 3 Þ, and from (19), we have The result follows.

Corollary 6.
(1) β is the striction curve of the surface Ψ 1 if and only if α is a geodesic curve (2) β is the striction curve of the surface Ψ 2 (3) β is the striction curve of the surface Ψ 3 if and only if α is a principal line On the other hand, by using (2), we have (1) The ruled surface Ψ 1 (resp. Ψ 3 ) is developable if and only if α is an asymptotic line (2) The ruled surface Ψ 2 is developable Differentiating (2), we obtain By using (2) and (29), we have If we substitute (30) in (18), we find Likewise, we have Abstract and Applied Analysis If we substitute (32) in (18), we find where X r = D r /kD r k with D r = τ g T + k g U and ðτ g , k g Þ ≠ ð0 , 0Þ: Using (2), we get Hence, where We obtain If we substitute (38) in (15), we find The following corollaries follow: The ruled surface Ψ r is regular.

Corollary 10.
In the noncylindrical case (i.e., σ r ≠ 0), the curve β is the striction curve of the surface Ψ r .
On the other hand, we have det ðβ ′ , X r , X r ′ Þ = σ r ffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 2 g + τ 2 g q : Corollary 11. The ruled surface Ψ r is developable if and only if it is cylindrical.
According to (36), we have Therefore, by using (2) and (29), we obtain Then, If we substitute (42) in (18), we find Hence, we have the following corollary: Corollary 12. The ruled surface Ψ r is not minimal.

Case
Where X = X o . The ruled surface becomes where X o = ðD o /kD o kÞ with D o = τ g T − k n V and ðτ g , k n Þ ≠ ð0, 0Þ:

Abstract and Applied Analysis
Differentiating X o , we obtain Thus, where We obtain If we substitute (48) in (15), we get On the other hand, the striction curve of Ψ o is Hence, we have the following corollary: Corollary 13. In the noncylindrical case (i.e., σ o ≠ 0), we have the following: (1) Ψ o is singular along its striction curve Then, by using (2) and (29), we obtain Hence, If we substitute (53) in (18), we obtain Corollary 15. The surface Ψ o is not minimal.

3.3.3.
Case Where X = X n . The ruled surface becomes where X n = D n /kD n k with D n = −k n V + k g U and ðk g , k n Þ ≠ ð0, 0Þ: By using (2), we get Thus, where σ n = k g ′k n − k g k n ′ − τ g k 2 n + k 2 g k 2 n + k 2 Abstract and Applied Analysis It follows that If we substitute (59) in (15), we obtain On the other hand, the striction curve of Ψ n is Hence, we have the following corollary: Corollary 16. In the noncylindrical case (i.e., σ n ≠ 0), we have the following: (1) Ψ n is singular along its striction curve (2) β is the striction curve of the surface Ψ n if and only if α is a geodesic curve We have det ðβ ′ , X n , X n ′ Þ = 0: Hence, we have the following result.
Corollary 17. The ruled surface Ψ n is developable.

Examples
In this section, we reinforce the previous study by the given four examples. The first one corresponds to the general case, and the three others represent the particular cases where the initial curve is an asymptotic line, a geodesic curve, and a line of curvature.
In the examples which follow, the same notations as in the preceding paragraphs are retained. We denote by α the initial curve lying on a surface M defined by φ,ðT, V, UÞ the Darboux frame of α,k n ,k g ,τ g , the normal curvature, the geodesic curvature, and the geodesic torsion of the curve α, respectively; by β the V-direction curve; and by X r (resp., X o ,X n ) the unit rectifying (resp., osculator, normal) vector field. We give for each example the illustrations of the ruled surfaces denoted by Ψ 1 ,Ψ 2 , Ψ 3 ,Ψ r ,Ψ o ,Ψ n : Example 1. Let 1 αðsÞ = ððs/2Þ cos ð ffiffi ffi 2 p ln ðs/2ÞÞ, ðs/2Þ sin ð ffiffi ffi 2 p ln ðs/2ÞÞ, ðs/2ÞÞ be a curve lying on the surface 1 M given by the following parametrization: which can be seen in Figure 1. The Darboux frame of 1 α is   Abstract and Applied Analysis Then, The V-direction curve is given by where a 1 ,a 2 , a 3 are integration constants. Consequently, the surfaces

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