Smarandache Ruled Surfaces according to Darboux Frame in E 3

(is paper presents a new approach of constructing special ruled surfaces and aims to study their developability and minimalist conditions. Our concept opens opportunities for application in engineering, surface modeling, and architectural design. (e principle of our study is to introduce the original definitions of Smarandache ruled surfaces according to Darboux frame of a curve lying on an arbitrary regular surface in E3. It concerns Tg-Smarandache ruled surface, Tn-smarandache ruled surface, and gn-Smarandache ruled surface. New theorems giving necessary and sufficient conditions for those surfaces to be developable and minimal are investigated. Finally, an example with illustrations is presented.


Introduction
In the classical differential geometry [1,2], a ruled surface is generated by a straight line moving along a curve c(s). e various positions of the generating line are called the rulings. Such a surface has the following parametric representation: ω: where I is an open interval of R, c(s) is a curve through which all rulings pass (it is called the base curve), and X → (s) are the rulings directors of the ruled surface.
Developability and minimalist notions are two of the most important properties that can be studied for ruled surfaces.
Developable ruled surfaces can be transformed into the plane without any deformation and distortion, and they form relatively small subsets that contain cylinders, cones, and the tangent surfaces. ey are characterized with vanishing Gaussian curvature [3][4][5].
Minimal surfaces are surfaces that locally minimize their area. ey refer to the fixed boundary curve of a surface area that is minimal with respect to other surfaces with the same boundary.
In light of the existing studies, several researchers were interested in the study of ruled surfaces with Darboux frame. Indeed, in [9], we constructed the ruled surface whose rulings are constant linear combinations of Darboux frame vectors of a regular curve lying on a regular surface of reference in E 3 . We made a comparative study between the two surfaces (the regular surface of reference and the new constructed ruled surface) along their common curve. We investigated properties of the constructed ruled surface and gave examples with illustrations. Furthermore, in [10], the authors showed that the rulings directors of any ruled surface can be expressed with the Darboux frame vectors in E 3 . Besides, they studied characteristic properties of that ruled surface and gave the relationship between Darboux frame and Frenet frame. Moreover, in [11], the authors defined the evolute offsets of ruled surface with Darboux frame and studied its characteristic properties in E 3 .
In curve theory, Smarandache curves were introduced for the first time in Minkowski space time [12]. It defines curves whose position vectors are composed by Frenet-Serret vectors on another regular curve. In [13], Smarandache curves according to Darboux frame were introduced in E 3 . It is a paper that represents an inspiration source for us to make the present work of ruled surface with Smarandache curves according to Darboux frame. is paper introduces three ruled surfaces generated by Tg-Smarandache curve, Tn-Smarandache curve, and gn-Smarandache curve, according to Darboux frame in E 3 . eorems giving necessary and sufficient conditions for those surfaces to be developable and minimal are investigated. Finally, an illustrative example is given.
e main contribution of our study is to introduce original notions of ruled surfaces that combine both notions of Smarandache curves and Darboux frame. Our main objective is to study developability and minimalist conditions of such surfaces. Our approach brings together an important significant value in theory and practical application. Indeed, the constructed form of those special ruled surfaces can be useful in applications to medical science, engineering, surface modeling, computer-aided geometric design, computer-aided manufacturing, and computer graphics. In addition, the developability and minimalist properties are the most important properties that are greatly needed and useful in contemporary architecture and architectural design. Hence, the concept of our study opens new avenues for software engineers and architects to some applications in their research fields. Here are some possibilities: (i) Developing language of geometrical communication for architecture and engineering. (ii) Constructing algorithms for control point of approximating surfaces. (iii) Facilitating reconstruction of special developable and minimal surfaces. (iv) Constructing algorithms for constructing special developable ruled surfaces. e outline of the paper is structured in the following manner. Some basic notions of ruled surface properties, Darboux frame, and Smarandache curves according to Darboux frame are presented in Section 2. In Section 3, the notions of new three Smarandache ruled surfaces according to Darboux frame are introduced, characterized in terms of developability and minimalist properties, and illustrated in an example. Finally, a summarized conclusion is given in Section 4.

Preliminaries
In the Euclidean 3-space E 3 , we consider the usual metric given by where ( It is said to be non-cylindrical, if its rulings directors are not constant, i.e., ‖X → ′ ‖ ≠ 0. Let us denote by m → � m → (s, v) the unit normal on the ruled surface Ψ at a regular point, and we have where Ψ s � (zΨ(s, v)/zs) and Ψ v � (zΨ(s, v)/zv).
Definition 1 (see [14]). e distribution parameter λ � λ(s) of a non-cylindrical ruled surface is given by Definition 2 (see [14]). A ruled surface is developable if its distribution parameter vanishes.
e first, I, and the second, II, fundamental forms of ruled surface Ψ at a regular point are defined, respectively, by where Definition 3. e Gaussian curvature K and the mean curvature H of the ruled surface Ψ at a regular point are given, respectively, by Proposition 1 (see [14]). A ruled surface is developable if and only if its Gaussian curvature vanishes.

Proposition 2 (see [14]). A regular surface is minimal if and only if its mean curvature vanishes.
Due to a unit speed curve α � α(s) that lies on a regular surface ϕ � ϕ(u, r), there exists the Darboux frame and it is is the unit normal vector of the surface ϕ � ϕ(u, r) along α � α(s), and g → (s) is the unit vector which is defined by g → (s) � n → (s) × T → (s). e derivative formula of the Darboux frame is given as follows: Journal of Mathematics where ρ g is the geodesic curvature, ρ n is the normal curvature, and θ g is the geodesic torsion of the curve α � α(s) on the surface ϕ � ϕ(u, r).
(i) α(s) is a geodesic curve on ϕ if its geodesic curvature ρ g vanishes. (ii) α(s) is an asymptotic line on ϕ if its normal curvature ρ n vanishes. (iii) α(s) is a principal line on ϕ if its geodesic torsion θ g vanishes.
Definition 4 (see [13]). Tg-Smarandache curves according to Darboux frame of the curve α � α(s) on the surface ϕ are given by Definition 5 (see [13]). Tn-Smarandache curves according to Darboux frame of the curve α � α(s) on the surface ϕ are given by Definition 6 (see [13]). gn-Smarandache curves according to Darboux frame of the curve α � α(s) on the surface ϕ are given by

Smarandache Ruled Surfaces according to Darboux Frame in E 3
We start our section by giving original definitions of Smarandache ruled surfaces according to Darboux frame of a regular curve lying on a regular surface in E 3 .
ese ruled surfaces are called Tg-Smarandache ruled surface, Tn-Smarandache ruled surface, and gn− Smarandache ruled surface, according to Darboux frame of the curve α � α(s) on the surface ϕ, respectively.
In the following, we investigate theorems that give necessary and sufficient conditions for Smarandache ruled surfaces (12) to be developable and minimal.

Theorem 2. Tg-Smarandache ruled surface of (12) is minimal if and only if
Proof. Differentiating the first line of (12) with respect to s and v, respectively, and using (8), we get e cross product of these two vectors gives the normal vector on Tg-Smarandache ruled surface of (12): So, under regularity condition, the unit normal takes the following form:

Journal of Mathematics
Applying the norms and the scalar product for (13), we get the components of the first fundamental form of Tg-Smarandache ruled surface of (12), as follows: Differentiating (13) with respect to s and v, respectively, and using (8), we get Hence, from (15) and (17), we get the components of the second fundamental form of Tg− Smarandache ruled surface of (12), as follows: Consequently, from (16) and (18), we obtain the Gaussian curvature and the mean curvature of Tg-Smarandache ruled surface of (12): which replies to both above theorems.

Theorem 3. Tn-Smarandache ruled surface of (12) is developable if and only if c(s)
is an asymptotic line on the surface ϕ orθ g � ρ g .

Theorem 4. Tn-Smarandache ruled surface of (12) is minimal if and only if
Proof. Differentiating the second line of (12) with respect to s and v, respectively, and using (8), we get which implies that the normal vector on Tn− Smarandache ruled surface of (12) is (21) So, under regularity condition, the unit normal is given by Applying the norms and the scalar product for (20), we get the components of the first fundamental form of Tn-Smarandache ruled surface of (12): Differentiating (20) with respect to s and v, respectively, and using (8), we get Hence, from (22) and (24), we get the components of the second fundamental form of Tn-Smarandache ruled surface of (12): (ρ n ′ + ρ g θ g )]+ 2θ g (ρ g + ρ n ) 2 � 0.