# Simultaneous Developability of Partner Ruled Surfaces according to Darboux Frame inE3

Authors: Soukaina Ouarab
Journal: Abstract and Applied Analysis
 (2021)

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## Abstract

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 inE3. It concerns Tg Partner ruled surfaces, Tn Partner ruled surfaces, and gn 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.


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## Body

## 1. 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
(1)ϕ:s;v∈I×ℝ↦cs+vXs,where I is an open interval of ℝ, cs is called the base curve, and Xs 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 [3–5].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 [6–8].Another one of the most important moving frame of the differential geometry is the Darboux frame, which is a natural 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 E3.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 inE3. 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 [12–15].The principle of this study is to consider a unit speed curvecs lying on an arbitrary regular surface ϕ, associate the Darboux frame T,g,n of cs 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.


## 2. Preliminaries

Due to a unit speed curveα=αs that lies on a regular surface ϕ=ϕu,r,i.e.,αs=ϕus,rs, there exists the Darboux frame and it is denoted by T⟶s,g⟶s,n⟶s, where T⟶s=α′s is the unit tangent vector of the curve α=αs,n⟶s=ϕu×ϕr/ϕu×ϕrus,rs is the unit normal vector of the surface ϕ=ϕu,r along the curve α=αs, and g⟶s is the unit vector which is defined by g⟶s=n⟶s×T⟶s.The derivative formulae of the Darboux frame are given as follows:(2)T⟶′g⟶′n⟶′=0ρgρn−ρg0θg−ρn−θg0T⟶g⟶n⟶,where ρg is the geodesic curvature, ρn is the normal curvature, and θg is the geodesic torsion of the curve α=αs on the surface ϕ=ϕu,r.Definition 1 (see [16]).
The curveαs lying on a regular surface is as follows:
(i)
A geodesic curve if its geodesic curvatureρg vanishes(ii)
An asymptotic line if its normal curvatureρn vanishes(iii)
A principal line if its geodesic torsionθg vanishesLetφ:s,v↦cs+vX⟶s be a ruled surface in E3.Let denote byn⟶=n⟶s,v, the unit normal on the ruled surface φ at a regular point φs,v, we have
(3)n⟶=φs∧φvφs∧φv=c′+vX⟶′×X⟶c′+vX⟶′×X⟶,where φs=∂φs,v/∂s and φv=∂φs,v/∂v.The firstI and the second II fundamental forms of the ruled surface φ at a regular point φs,v are defined, respectively, by
(4)Iφsds+φvdv=Eds2+2Fdsdv+Gdv2,IIφsds+φvdv=eds2+2fdsdv+gdv2,where
(5)E=φs2,F=φs,φv,G=φv2,e=φss,n⟶,f=φsv,n⟶,g=φvv,n⟶=0.Definition 2.
The Gaussian curvatureK of the ruled surface φ at a regular point φs,v is given by
(6)K=−f2EG−F2.Proposition 3 (see [16]).
A ruled surface is developable if and only if its Gaussian curvature vanishes.


## 3. Simultaneous Developability of Partner Ruled Surfaces according to the Darboux Frame inE3

Definition 4.
Letc:s∈I↦cs be a C2-class differentiable unit speed curve lying on a regular surface ϕ=ϕu,r. Let denote by Ts,gs,ns the Darboux frame of c=cs on ϕ=ϕu,r. The two ruled surfaces defined by
(7)φTg:s,v∈I×ℝ↦Ts+vgs,φgT:s,v∈I×ℝ↦gs+vTs,are called Tg Partner ruled surfaces according to the Darboux frame of the curve cs on the surface ϕ=ϕu,r.Theorem 5.
Tg Partner ruled surfaces (7) are simultaneously developable if and only if at least one of the following statements is verified:
(i)
c=cs is a geodesic curve on the surface ϕ(ii)
c=cs is an asymptotic and a principal line on the surface ϕProof.
By differentiating the first line of (7) with respect to s and v, respectively, and using Darboux derivative formulae (2), we get
(8)φTgs=−vρgT+ρgg+ρn+vθgn,φTgv=g.
Then, by considering the cross product of both vectors in (8), we get the normal vector of the ruled surface φTg:
(9)φTgs×φTgv=−ρn+vθgT−vρgn,which implies that under regularity condition, the unit normal vector of the ruled surface φTg is given by
(10)φTgs×φTgvφTgs×φTgv=−1ρn+vθg2+v2ρg2ρn+vθgT+vρgn.
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 φTg:
(11)ETg=1+vρg2+ρn+vθg2,(12)FTg=ρg,(13)GTg=1.
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 φTg:
(14)fTg=−ρnρgρn+vθg2+v2ρg2.
Thus, from (13) and (14), we get the Gaussian curvature of the ruled surface φTg:
(15)KTg=−ρnρgρn+vθg2+v2ρg22.
On another hand, by differentiating the second line of (7) with respect to s and v and using Darboux derivative formulae (2), we get
(16)φgTs=−ρgT+vρgg+θg+vρnn,φgTv=T.
Then, the cross product of both vectors of (16) gives the normal vector of the ruled surface φgT:
(17)φgTs×φgTv=θg+vρng−vρnn,which implies that under regularity condition, the unit normal vector of the ruled surface φgT is given by
(18)φgTs×φgTvφgTs×φgTv=1θg+vρn2+v2ρg2θg+vρng−vρgn.
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 φgT:
(19)EgT=ρg2+v2ρg2+θg+vρn2,(20)FgT=−ρg,(21)GgT=1.
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 φgT:
(22)fgT=θgρgθg+vρn2−v2ρg2.
Hence, from (21) and (22), we get the Gaussian curvature of the ruled surface φgTv:
(23)KgT=−θgρgθg+vρn2−v2ρg22.
Consequently, from (15) and (23), we deduce Tg Partner ruled surfaces
φgT and φgT are simultaneously developable if and only if TgK=KgT=0,i.e.,ρnρg=θgρg=0, which is equivalent to ρg=0 or ρn=θg=0. Thus, Theorem 5 is proved.☐Definition 6.
Letc:s∈I⟼cs be a C2-class differentiable unit speed curve lying on a regular surface ϕ=ϕu,r. Let denote by Ts,gs,ns the Darboux frame of c=cs on ϕ=ϕu,r. The two ruled surfaces defined by
(24)φTn:s,v∈I×ℝ↦Ts+vns,φnT:s,v∈I×ℝ↦ns+vTs,are called Tn Partner ruled surfaces according to the Darboux frame of the curve c=cs on the surface ϕ=ϕu,r.Theorem 7.
Tn Partner ruled surfaces (24) are simultaneously developable if and only if at least one of the following statements is verified:
(i)
c=cs is an asymptotic line on the surface ϕ(ii)
c=cs is a geodesic curve and a principal line on the surface ϕProof.
Differentiating the first line of (24) with respect to s and v and using Darboux derivative formulae (2), we get
(25)φTns=−vρnT+ρg−vθgg+ρnn,φTnv=n,then, the cross product of both last vectors gives us the normal vector of the ruled surface φTn:
(26)φTns×φTnv=ρg−vθgT+vρng,which implies that under regularity condition, the unit normal vector of the ruled surface φTn takes the following form:
(27)φTns×φTnvφTns×φTnv=1ρg−vθg2+v2ρn2ρg−vθgT+vρng.
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 φTn:
(28)ETn=v2ρn2+ρg−vθg2+ρn2,(29)FTn=ρn,(30)GTn=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 φTn:
(31)fTn=−ρnρgρg−vθg2+v2ρn2.
Thus, from (30) and (31), we obtain the Gaussian curvature of the ruled surface φTn:
(32)KTn=−ρnρgρg−vθg2+v2ρn22.
Let us now differentiate the second line of (24) with respect to s and v, respectively, and using Darboux derivative formulae (2), we get
(33)φnTs=−ρnT+−θg+vρgg+vρnn,φnTv=T.
The cross product of both vectors (33) gives the normal vector of the ruled surface φnT:
(34)φnTs×φnTs=vρng−−θg+vρgn,which gives, under regularity condition, the unit normal vector of the ruled surface φnT as follows:
(35)φnTs×φnTvφnTs×φnTv=1v2ρn2+−θg+vρg2vρng−−θg+vρgn.
The norms and scalar product applied on both vectors in (33) give us the components of the first fundamental form of the ruled surface φnT:
(36)EnT=ρn2+v2ρn2+−θg+vρg2,(37)FnT=−ρnnT,(38)G=1.
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 φnT:
(39)fnT=ρnθgv2ρn2+−θg+vρg2.
Thus, from (38) and (39), we obtain the Gaussian curvature of the ruled surface φnT as follows:
(40)KnT=−ρnθgv2ρn2−θg+vρg22.
Consequently, from (32) and (40), we deduce Tn Partner ruled surfaces
φTn and φnT are simultaneously developable if and only if TnK=KnT=0,i.e.,ρnρg=ρnθg=0, which is equivalent to ρn=0 or ρg=θg=0. Thus, Theorem 7 is proved.☐Definition 8.
Letc:s∈I⟼cs be a C2-class differentiable unit speed curve lying on a regular surface ϕ=ϕu,r. Let denote by Ts,gs,ns the Darboux frame of c=cs on ϕ=ϕu,r. The two ruled surfaces defined by
(41)φgn:s,v∈I×ℝ↦gs+vns,φng:s,v∈I×ℝ↦ns+vgs,are called gn Partner ruled surfaces according to the Darboux frame of the curve c=cs on the surface ϕ=ϕu,r.Theorem 9.
gn Partner ruled surfaces are simultaneously developable if and only if at least one of the following statements is verified:
(i)
c=cs is a principal line on the surface ϕ(ii)
c=cs is a geodesic curve and an asymptotic line on the surface ϕProof.
Differentiating the first line of (41) with respect to s and v, respectively, and using Darboux derivative formulae (2), we get
(42)φgns=−ρg+vρnT−vθgg+θgn,φngv=n.
We get the normal vector of the ruled surfaceφgn by realizing the cross product of both vectors (42):
(43)φgns×φgnv=−vθgT+ρg+vρng,which implies that under regularity condition, the unit normal vector of the ruled surface φgn is given by
(44)φgns×φgnvφgns×φgnv=1v2θg2+ρg+vρn2−vθgT+ρg+vρng.
By applying the norms and the scalar product for both vectors of (42), we obtain
(45)Egn=ρg+vρn2+v2θg2+θg2,(46)Fgn=θg,(47)Ggn=1.
By differentiating the second line of (42) with respect to s, using (2) and (44), we obtain
(48)fgn=−θgρgv2θg2+ρg+vρn2.
Hence, from (47) and (48), we obtain the Gaussian curvature of ruled surface φgn:
(49)Kgn=−θgρgv2θg2+ρg+vρn22.
Let us now differentiate the second line of (41) with respect to s and v, respectively, and use Darboux derivative formulae (2), we get
(50)φngs=−ρn+vρgT−θgg+vθgn,φngv=g.
By realizing the cross product of both vectors of (50), we get the normal vector of the ruled surface φngv:
(51)φngs×φngv=−vθgT−ρn+vρgn,which implies that under regularity condition, the unit normal vector of the ruled surface φngv is given by
(52)φngs×φngvφngs×φngv=−1v2θg2+ρn+vρg2vθgT+ρn+vρgn.
By applying the norms and the scalar product for (50), we obtain
(53)Eng=ρn+vρg2+θg2+v2θg2,(54)Fng=−θg,(55)Gng=1.
By differentiating the second line of (50) with respect to s, using (2) and (52), we get
(56)fng=−θgρnv2θg2+ρn+vρg2.
Hence, from (55) and (56), we obtain the Gaussian curvature of the ruled surface φngv as follows:
(57)Kng=−θgρnv2θg2+ρn+vρg2.
Consequently, from (49) and (57), we deduce that gn Partner ruled surfaces φgn and φng are simultaneously developable if and only if gnK=Kng=0,i.e.,θgρg=θgρn=0, which is equivalent to θg−0 or ρg=ρn=0. Thus, Theorem 9 is proved.☐Here follows, we give an example of our study and present corresponding illustrations.Example 1.
Let us consider the regular surface parameterized by(58)φeu,r=2cosu2−r2sinu2,2sinu2−r2cosu2,r2.
It is easy to see that the curveces=2coss/2,2sins/2,0 lies on the regular surface φe; indeed, we have ces=φes,0.
The Darboux frame vectors and the Darboux invariants ofces on φe are given, respectively, by
(59)Ts=−sins2,coss2,0,gs=11+sin2s−sinscoss2,−sinssins2,1,ns=11+sin2scoss2,sins2,sins,ρgs=sins1+sin2s,ρns=−121+sin2s,θgs=−coss1+sin2s.
Hence,Tg Partner ruled surfaces, Tn Partner ruled surfaces, and gn Partner ruled surfaces according to the Darboux frame of ecs on φe are given, respectively, as follows:
(60)φTg=−sins2,coss2,0+vRs−sinscoss2,−sinssins2,1,φgT=Rs−sinscoss2,−sinssins2,1+v−sins2,coss2,0,(61)φTn=−sins2,coss2,0+vRscoss2,sins2,sins,φnT=Rscoss2,sins2,sins+v−sins2,coss2,0,(62)φgn=Rs−sinscoss2,−sinssins2,1+vcoss2,sins2,sins,φng=Rscoss2,sins2,sins+v−sinscoss2,−sinssins2,1,where Rs=1/1+sin2s.
Here follows, we present the illustrations of theTg 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.Figure 1
Ruled surfaceφTg of (60).Figure 2
Ruled surfaceφgT of (60).Figure 3
Ruled surfaceφTn of (61).Figure 4
Ruled surfaceφnT of (61).Figure 5
Ruled surfaceφgn of (62).Figure 6
Ruled surfaceφng of (62).


## 4. 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 curvecs on a regular surface in E3. 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 cs 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.



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