Simultaneous Developability of Partner Ruled Surfaces according to Darboux Frame in 
 
 
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<jats:p>In this paper, we introduce original definitions of Partner ruled surfaces according to the Darboux frame of a curve lying on an arbitrary regular surface in <jats:inline-formula>
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                  </jats:inline-formula>. It concerns <jats:inline-formula>
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                  </jats:inline-formula> Partner ruled surfaces, <jats:inline-formula>
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                  </jats:inline-formula> Partner ruled surfaces, and <jats:inline-formula>
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                        <mi>g</mi>
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                  </jats:inline-formula> Partner ruled surfaces. We aim to study the simultaneous developability conditions of each couple of two Partner ruled surfaces. Finally, we give an illustrative example for our study.</jats:p>


Introduction
The theory of ruled surfaces forms an important and useful class of theories in differential geometry [1,2]. This kind of surface is defined by the moving of a straight line along a curve. The various positions of the generating lines are called the rulings of the ruled surface. Such a surface, thus, has a parametric representation of the form where I is an open interval of ℝ, cðsÞ is called the base curve, and XðsÞ are the ruling directors. One of the most interesting properties related to ruled surface is the property of developability. It defines ruled surfaces that can be transformed into the plane without any deformation and distortion; such surfaces form relatively small subsets that contain cylinders, cones, and tangent surfaces. They are characterized with vanishing Gaussian curvature [3][4][5].
Many geometers have studied some of the differential geometric concepts of the ruled surfaces by means of different moving frames, such as the Frenet-Serret frame, alternative frame, and Bishop frame [6][7][8].
Another one of the most important moving frame of the differential geometry is the Darboux frame, which is a natu-ral moving frame constructed on a surface that contains a curve. It is named after the French mathematician Jean Gaston Darboux, in a four-volume collection of the studies he published between 1887 and 1896. Since that time, there have been many important repercussions of the Darboux frame, having been examined for example in (Darboux, 1896; O Neill, 1996). One can find studies of ruled surfaces with the Darboux frame realized in Euclidean and non-Euclidean 3-space. For example, in [9], the authors constructed the ruled surface whose rulings are constant linear combinations of the Darboux frame vectors of its base curve along a regular surface of reference; they studied the most important properties of that ruled surface, characterized it, and presented examples with illustrations. Furthermore, in [10], the authors studied the characteristic properties of a ruled surface with the Darboux frame and gave the relationship between the Darboux frame and the Frenet frame. Moreover, in [11], the authors defined the evolute offsets of ruled surface with the Darboux frame and studied its characteristic properties in E 3 .
The main contribution of this work is to introduce new special couples of ruled surfaces defined by means of Darboux frame vectors of a regular curve lying on an arbitrary regular surface in E 3 . Our objective is to study the simultaneous developability of such couples of surfaces. Through our study, we are opening up some avenues for scientists to apply our approach in some areas such as architectural design, medical science, surface modeling, engineering, and computer-aided geometric design [12][13][14][15].
The principle of this study is to consider a unit speed curve cðsÞ lying on an arbitrary regular surface ϕ, associate the Darboux frame fT, g, ng of cðsÞ on ϕ, and, then, define three couples of ruled surfaces that are generated, reciprocally, by T, g, and n. We call them Tg Partner ruled surfaces, Tn Partner ruled surfaces, and gn Partner ruled surfaces, respectively. We aim to study the simultaneous developability of each couple of Partner ruled surfaces. Indeed, we investigate theorems that reply to our needs. Finally, we present an example with illustrations.

Preliminaries
Due to a unit speed curve α = αðsÞ that lies on a regular surface ϕ = ϕðu, rÞ, i:e:, αðsÞ = ϕðuðsÞ, rðsÞÞ, there exists the Darboux frame and it is denoted by fT ! ðsÞ, g ! ðsÞ, n ! ðsÞg, where T ! ðsÞ = α ′ ðsÞ is the unit tangent vector of the curve α = αðsÞ, n ! ðsÞ = ððϕ u × ϕ r Þ/kϕ u × ϕ r kÞðuðsÞ, rðsÞÞ is the unit normal vector of the surface ϕ = ϕðu, rÞ along the curve α = αðsÞ, and g ! ðsÞ is the unit vector which is defined by where ρ g is the geodesic curvature, ρ n is the normal curvature, and θ g is the geodesic torsion of the curve α = αðsÞ on the surface ϕ = ϕðu, rÞ.
The first I and the second II fundamental forms of the ruled surface φ at a regular point φðs, vÞ are defined, respectively, by where Definition 2. The Gaussian curvature K of the ruled surface φ at a regular point φðs, vÞ is given by Proposition 3 (see [16]). A ruled surface is developable if and only if its Gaussian curvature vanishes.

Simultaneous Developability of Partner Ruled Surfaces according to the Darboux Frame in E 3
Definition 4. Let c : s ∈ I ↦ cðsÞ be a C 2 -class differentiable unit speed curve lying on a regular surface ϕ = ϕðu, rÞ. Let denote by fTðsÞ, gðsÞ, nðsÞg the Darboux frame of c = cðsÞ on ϕ = ϕðu, rÞ. The two ruled surfaces defined by Proof. By differentiating the first line of (7) with respect to s and v, respectively, and using Darboux derivative formulae (2), we get Abstract and Applied Analysis Then, by considering the cross product of both vectors in (8), we get the normal vector of the ruled surface Tg φ: which implies that under regularity condition, the unit normal vector of the ruled surface Tg φ is given by By applying the norms and the scalar product for both vectors in (8), we get the components of the first fundamental form of the ruled surface Tg φ: By differentiating the second line of (8) with respect to s, using Darboux derivative formulae (2) and making the scalar product with the unit normal (10), we get the second component of the second fundamental form of the ruled surface Tg φ: Thus, from (13) and (14), we get the Gaussian curvature of the ruled surface Tg φ: On another hand, by differentiating the second line of (7) with respect to s and v and using Darboux derivative formulae (2), we get Then, the cross product of both vectors of (16) gives the normal vector of the ruled surface gT φ: which implies that under regularity condition, the unit normal vector of the ruled surface gT φ is given by By applying the norms and the scalar product for both vectors (16), we obtain the components of the first fundamental form of the ruled surface gT φ: By differentiating the second line of (16) with respect to s, using Darboux derivative formulae (2) and using the unit normal (18), we get the second component of the second fundamental form of the ruled surface gT φ: Hence, from (21) and (22), we get the Gaussian curvature of the ruled surface gT φ v : Consequently, from (15) and (23), we deduce Tg Partner ruled surfaces gT φ and gT φ are simultaneously developable if and only if Tg K = gT K = 0, i:e:, ρ n ρ g = θ g ρ g = 0, which is equivalent to ρ g = 0 or ρ n = θ g = 0. Thus, Theorem 5 is proved. ☐ Definition 6. Let c : s ∈ I ⟼ cðsÞ be a C 2 -class differentiable unit speed curve lying on a regular surface ϕ = ϕðu, rÞ. Let denote by fTðsÞ, gðsÞ, nðsÞg the Darboux frame of c = cðsÞ on ϕ = ϕðu, rÞ. The two ruled surfaces defined by Proof. Differentiating the first line of (24) with respect to s and v and using Darboux derivative formulae (2), we get Tn φ s = −vρ n T + ρ g − vθ g g + ρ n n, then, the cross product of both last vectors gives us the normal vector of the ruled surface Tn φ: which implies that under regularity condition, the unit normal vector of the ruled surface Tn φ takes the following form: By applying the norms and the scalar product for both vectors (25), we get the components of the first fundamental form of the ruled surface Tn φ: Tn G = 1: By differentiating the second line of (25) with respect to s, using (2) and (27), we get the second component of the second fundamental form of the ruled surface Tn φ: Thus, from (30) and (31), we obtain the Gaussian curvature of the ruled surface Tn φ: Let us now differentiate the second line of (24) with respect to s and v, respectively, and using Darboux derivative formulae (2), we get nT φ s = −ρ n T + −θ g + vρ g g + vρ n n, The cross product of both vectors (33) gives the normal vector of the ruled surface nT φ: which gives, under regularity condition, the unit normal vector of the ruled surface nT φ as follows: The norms and scalar product applied on both vectors in (33) give us the components of the first fundamental form of the ruled surface nT φ: nT F = −ρ nT n , ð37Þ Differentiating the second line of (33) with respect to s, using Darboux derivative formulae (2), and using the unit normal (35), we get the second component of the second fundamental form of the ruled surface nT φ: Thus, from (38) and (39), we obtain the Gaussian curvature of the ruled surface nT φ as follows: Consequently, from (32) and (40), we deduce Tn Partner ruled surfaces Tn φ and nT φ are simultaneously developable if and only if Tn K = nT K = 0, i:e:, ρ n ρ g = ρ n θ g = 0, which is equivalent to ρ n = 0 or ρ g = θ g = 0. Thus, Theorem 7 is proved. Proof. Differentiating the first line of (41) with respect to s and v, respectively, and using Darboux derivative formulae (2), we get gn φ s = − ρ g + vρ n T − vθ g g + θ g n, We get the normal vector of the ruled surface gn φ by realizing the cross product of both vectors (42): which implies that under regularity condition, the unit normal vector of the ruled surface gn φ is given by By applying the norms and the scalar product for both vectors of (42), we obtain By differentiating the second line of (42) with respect to s, using (2) and (44), we obtain Hence, from (47) and (48), we obtain the Gaussian curvature of ruled surface gn φ: Let us now differentiate the second line of (41) with respect to s and v, respectively, and use Darboux derivative formulae (2), we get ng φ s = − ρ n + vρ g T − θ g g + vθ g n, By realizing the cross product of both vectors of (50), we get the normal vector of the ruled surface ng φ v : which implies that under regularity condition, the unit normal vector of the ruled surface ng φ v is given by By applying the norms and the scalar product for (50), we obtain By differentiating the second line of (50) with respect to s, using (2) and (52), we get Hence, from (55) and (56), we obtain the Gaussian curvature of the ruled surface ng φ v as follows: Consequently, from (49) and (57), we deduce that gn Partner ruled surfaces gn φ and ng φ are simultaneously developable if and only if gn K = ng K = 0, i:e:, θ g ρ g = θ g ρ n = 0, which is equivalent to θ g − 0 or ρ g = ρ n = 0. Thus, Theorem 9 is proved. ☐ Here follows, we give an example of our study and present corresponding illustrations.   Figure 2: Ruled surface gT φ of (60). 6 Abstract and Applied Analysis

Abstract and Applied Analysis
Darboux frame of e cðsÞ on e φ are given, respectively, as follows: where RðsÞ = 1/ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 + sin 2 ðsÞ p : Here follows, we present the illustrations of the Tg Partner ruled surface (60) represented by Figures 1 and 2, the Tn Partner ruled surfaces (61) represented by Figures 3 and 4, and the gn Partner ruled surfaces (62) represented by Figures 5 and 6, respectively.

Conclusion
In this paper, we presented a novel method to construct three special couples of ruled surfaces according to the Darboux frame of a regular curve cðsÞ on a regular surface in E 3 . These three couples of surfaces were called Tg Partner ruled surfaces, Tn Partner ruled surfaces, and gn Partner ruled surfaces, respectively. We investigated theorems that give necessary and sufficient conditions for each couple of two Partner ruled surfaces to be simultaneously developable. The obtained results reveal that the simultaneous developability conditions are related to the properties of the curve cðsÞ on the surface. Finally, we presented an illustrative example.
On the other hand, our approach can also provide excellent support for architectural design, surface modeling, computer-aided geometric design, and engineering application.

Data Availability
No data were used to support this study.

Conflicts of Interest
The author declares that there are no conflicts of interest.