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

Authors: Nidal Echabbi; Amina Ouazzani Chahdi
Journal: Abstract and Applied Analysis
 (2021)

---

## Abstract

In this work, we consider the Darboux frameT,V,U 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.


---

## Body

## 1. 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 theory 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 frameT,V,U 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.


## 2. 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 byE3 the Euclidean 3-space, with the usual metric(1)x,y=x1y1+x2y2+x3y3,where x=x1,x2,x3 and y=y1,y2,y3 are two vectors of E3.LetM be a regular surface and α=αs:I⊂ℝ⟶M be a unit speed curve on the surface M. The Darboux frame along the curve α is an orthonormal frame Ts,Vs,Us, 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:
(2)ddsTsVsUs=0kgskns−kgs0τgs−kns−τgs0TsVsUs,or
(3)dTds=Dns∧Ts,dVds=Drs∧Vs,dUds=Dos∧Us,where Dns,Drs, and Dos are the normal Darboux vectors field, the rectifying Darboux vector field, and the osculator Darboux vectors field, respectively, and are defined by
(4)Dns=−knsVs+kgsUs,Drs=τgsTs+kgsUs,Dos=τgsTs−knsVs,where kg,kn, and τg 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:(5)Ψs,v=βs+vXs,v∈ℝ,where β=βs is called the base curve of the ruled surface and Xs 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 curveα=αs lying on a regular surface, the following are well known:
(1)
αs is an asymptotic line if and only if the normal curvature kn vanishes(2)
αs is a geodesic curve if and only if the geodesic curvature kg vanishes(3)
αs is a principal line if and only if the geodesic torsion τg vanishesDefinition 2 [4].
Letα=αs:I⊂ℝ⟶E3 be a unit speed curve on the regular surface and Ts,Vs,Us the Darboux frame along the curve α. The integral curve γ of the vector field V is called V-direction curve of α, in other words γ′=V.We have(6)∂Ψ∂ss,v∧∂Ψ∂ss,v=β′s∧Xs+vX′s∧Xs.Then, the pointΨs0,v0 is said to be singular if β′s0∧Xs0+vX′s0∧Xs0=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(7)cs=βs−β′s,X′sX′s2Xs.Definition 3.
A ruled surfaceΨ is called developable if detβ′,X,X′=0.LettingI and II be the first and the second fundamental forms from the ruled surface Ψ, respectively, we have
(8)I=Eds2+2Fdsdv+Gdv2,II=Lds2+2Mdsdv+Ndv2,where
(9)E=Ψs,Ψs,F=Ψs,Ψv,G=Ψv,Ψv=1,(10)L=detΨs,Ψv,ΨssEG−F2,M=detΨs,Ψv,ΨsvEG−F2,N=detΨs,Ψv,ΨvvEG−F2=0.The mean curvatureH of a ruled surface is given as follows
(11)H=−2FM+L2E−F2.Definition 4.
A ruled surface is said to be minimal if its mean curvature vanishes identically.


## 3. Ruled Surfaces Defined byV-Direction Curves

Lettingα=αs:I⊂ℝ⟶E3 be a unit speed curve lying on a regular surface ,Ts,Vs,Us, the Darboux frames of α,kg,kn,τg are the geodesic curvature, the normal curvature, the geodesic torsion of α, and β=βs a V-direction curve, respectively.We consider the following ruled surface(12)Ψs,v=βs+vXs,v∈ℝ,where β=βs is the base curve and Xs the unit director vector of the straight line.


### 3.1. General Study

In this section, we propose to give some properties of the ruled surfaceΨ.Differentiating (12) with respect to s and v, we get
(13)Ψs=V+vX′,Ψv=X.Then, by using (10), we obtain the components of the first fundamental form E and F as follows:
(14)E=1+2vV,X′+v2X′2,F=V,X.Consequently, the condition of regularity of the ruled surface (12) is
(15)E−F2=1−V,X2+2vV,X′+v2X′2≠0.Differentiating (13) with respect to s and v, we get
(16)Ψss=V′+vX′′,Ψsv=X′,Ψvv=0,which allows, using formula (10), calculating the two components of the second fundamental form M and L, respectively, and we obtain
(17)M=detV,X,X′1−V,X2+2vV,X′+v2X′2,L=detV,X,V′+vdetV,X,X′′+detX′,X,V′+v2detX′,X,X′′1−V,X2+2vV,X′+v2X′2.Therefore, by using (11), the mean curvature of the ruled surface (12) is given as follows
(18)H=detV,X,V′−2detV,X,X′V,X+vdetV,X,X′′+detX′,X,V′+v2detX′,X,X′′21−V,X2+2vV,X′+v2X′23/2.On the other hand, the striction curve of the surface (12) is
(19)c=β−V,X′X′2X.


### 3.2. Case WhereX=T (Resp. V,U)

Let us consider the ruled surfaces defined by(20)Ψ1s,v=βs+vTs,Ψ2s,v=βs+vVs,Ψ3s,v=βs+vUs.By using (2), we have
(21)V,T=0,V,T′=kg,T′2=kg2+kn2.If we substitute (21) in (15), we find
(22)E1−F12=1+2vkg+v2kg2+kn2.Likewise, we have(23)V,V=1,V,V′=0,V′2=kg2+τg2.If we substitute (23) in (15), we find
(24)E2−F22=v2kg2+τg2.On the other hand,(25)V,U=0,V,U′=−τg,U′2=kn2+τg2.If we substitute (25) in (15), we find
(26)E3−F32=1−2vτg+v2kn2+τg2.Corollary 5.
The ruled surfacesΨ1 and Ψ3 are regulars along the curve β, while Ψ2 is singular along β.We denote byc1 (resp. c2,c3) the striction curve of Ψ1resp.Ψ2,Ψ3, and from (19), we have
(27)c1=β−kgkg2+kn2T,c2=β,c3=β+τgkn2+τg2U.The result follows.Corollary 6.
(1)
β is the striction curve of the surface Ψ1 if and only if α is a geodesic curve(2)
β is the striction curve of the surface Ψ2(3)
β is the striction curve of the surface Ψ3 if and only if α is a principal lineOn the other hand, by using (2), we have
(28)detV,T,T′=detV,U,U′=−kn,detV,V,V′=0.Corollary 7.
(1)
The ruled surfaceΨ1 (resp. Ψ3) is developable if and only if α is an asymptotic line(2)
The ruled surfaceΨ2 is developableDifferentiating (2), we obtain
(29)T′′=−kg2+kn2T−knτg−kg′V+kgτg+kn′U,V′′=−knτg+kg′T−kg2+τg2V−kgkn−τg′U,U′′=kgτg−kn′T−kgkn+τg′V−kn2+τg2U.By using (2) and (29), we have
(30)detV,T,V′−2detV,T,T′V,T=−τg,detV,T,T′′+detT′,T,V′=−kn′−2kgτg,detT′,T,T′′=kg′kn−kgkn′−τgkg2+kn2.If we substitute (30) in (18), we find
(31)H1=−τg−kn′+2kgτgv+kg′kn−kgkn′−τgkg2+kn2v221−2vkg+v2kg2+kn23/2.Likewise, we have(32)detV,U,V′−2detV,U,U′V,U=−kg,detV,U,U′′+detU′,U,V′=2kgτg−kn′,detU′,U,U′′=τgkn′−knτg′−kgkn2+τg2.If we substitute (32) in (18), we find
(33)H3=−kg+2kgτg−kn′v+τgkn′−knτg′−kgkn2+τg2v221−2vτg+v2kn2+τg23/2.Corollary 8.
The ruled surfaceΨ1resp.Ψ3 is minimal along its base curve if and only if α is a principal line (resp. a geodesic curve).


### 3.3. Case WhereX Is a Darboux Vector Field

#### 3.3.1. Case WhereX=Xr

The ruled surface becomes(34)Ψrs,v=βs+vXrs,where Xr=Dr/Dr with Dr=τgT+kgU and τg,kg≠0,0.Using (2), we get
(35)Xr′=τgT+kgUτg2+kg2′=kgτg′kg−kg′τg−knτg2+kg2τg2+kg23/2T−τgτg′kg−kg′τg−knτg2+kg2τg2+kg23/2U.Hence,(36)Xr′=kgσrT−τgσrU,where
(37)σr=τg′kg−kg′τg−knτg2+kg2τg2+kg23/2.We obtain(38)V,Xr=0,V,Xr′=0,Xr′2=σr2kg2+τg2.If we substitute (38) in (15), we find
(39)Er−Fr2=1+v2σr2kg2+τg2.The following corollaries follow:Corollary 9.
The ruled surfaceΨr is regular.Corollary 10.
In the noncylindrical case (i.e.,σr≠0), the curve β is the striction curve of the surface Ψr.On the other hand, we havedetβ′,Xr,Xr′=σrkg2+τg2.Corollary 11.
The ruled surfaceΨr is developable if and only if it is cylindrical.According to (36), we have
(40)Xr′=−σrV′.Therefore, by using (2) and (29), we obtain
(41)Xr′′=kgσr′+knτg+kg′σrT+σrkg2+τg2V+−τgσr′+kgkn−τg′σrU.Then,(42)detV,Xr,V′−2detV,Xr,Xr′=−kg2+τg2,detV,Xr,Xr′′+detXr′,Xr,V′=−σrkg2+τg2′,detXr′,Xr,Xr′′=−kg2+τg23/2σr2.If we substitute (42) in (18), we find
(43)Hr=−kg2+τg2+σrkg2+τg2′v+kg2+τg23/2σr2v221+v2σr2kg2+τg23/2.Hence, we have the following corollary:Corollary 12.
The ruled surfaceΨr is not minimal.


#### 3.3.2. Case WhereX=Xo

The ruled surface becomes(44)Ψos,v=βs+vXos,where Xo=Do/Do with Do=τgT−knV and τg,kn≠0,0.DifferentiatingXo, we obtain
(45)Xo′=τgT−knVτg2+kn2′=knτg′kn−kn′τg+kgτg2+kn2τg2+kn23/2T+τgτg′kn−kn′τg+kgτg2+kn2τg2+kn23/2V.Thus,(46)Xo′=knσoT+τgσoV,where
(47)σo=τg′kn−kn′τg+kgτg2+kn2τg2+kn23/2.We obtain(48)V,Xo=−knkn2+τg2,V,Xo′=τgσo,Xo′2=σo2kn2+τg2.If we substitute (48) in (15), we get
(49)Eo−Fo2=τg+vτg2+kn2σo2τg2+kn2.On the other hand, the striction curve ofΨo is
(50)co=β−τgτg2+kn2σoXo.Hence, we have the following corollary:Corollary 13.
In the noncylindrical case (i.e.,σo≠0), we have the following:
(1)
Ψo is singular along its striction curve(2)
β is the striction curve of the surface Ψo if and only if α is a principal lineWe havedetβ′,Xo,Xo′=0. Hence, the result is as follows.Corollary 14.
The ruled surfaceΨo is developable.From (46), we have
(51)Xo′=−σoU′.Then, by using (2) and (29), we obtain
(52)Xo′′=knσo′−kgτg−kn′σoT+τgσo′+kgkn+τg′σoV+σokn2+τg2U.Hence,(53)detV,Xo,V′−2detV,Xo,Xo′V,Xo=−τg2k¯n2+τ¯g2detV,Xo,Xo′′+detXo′,Xo,V′=−2τgσokn2+τg2,detXo′,Xo,Xo′′=−kn2+τg23/2σo2.If we substitute (53) in (18), we obtain
(54)Ho=−τg2+kn22τg+vσokn2+τg2.Corollary 15.
The surfaceΨo is not minimal.


#### 3.3.3. Case WhereX=Xn

The ruled surface becomes(55)Ψns,v=βs+vXns,whereXn=Dn/Dn with Dn=−knV+kgU and kg,kn≠0,0.By using (2), we get
(56)Xn′=−knV+kgUkn2+kg2′=kgkg′kn−kgkn′−τgkn2+kg2kn2+kg23/2V+knkg′kn−kgkn′−τgkn2+kg2kn2+kg23/2U.Thus,(57)Xn′=kgσnV+knσnU,where
(58)σn=kg′kn−kgkn′−τgkn2+kg2kn2+kg23/2.It follows that
(59)V,Xn=−knkn2+kg2,V,Xn′=kgσn,Xn′2=σn2kn2+kg2.If we substitute (59) in (15), we obtain
(60)En−Fn2=kg+vkn2+kg2σn2kn2+kg2.On the other hand, the striction curve ofΨn is
(61)cn=β−kgkn2+kg2σnXn.Hence, we have the following corollary:Corollary 16.
In the noncylindrical case (i.e.,σn≠0), we have the following:
(1)
Ψn is singular along its striction curve(2)
β is the striction curve of the surface Ψn if and only if α is a geodesic curveWe havedetβ′,Xn,Xn′=0. Hence, we have the following result.Corollary 17.
The ruled surfaceΨn is developable.From (57), we have
(62)Xn′=σnT′.Then, by using (2) and (29), we obtain
(63)Xn′′=−σnkg2+kn2T+kgσn′−knτg−kg′σnV+knσn′+kgτg+kn′σnU.Then,(64)detV,Xn,V′−2detV,Xn,Xn′V,Xn=−kg2kg2+kn2detV,Xn,Xn′′+detXn′,Xn,V′=−2kgσnkg2+kn2,detXn′,Xn,Xn′′=−kg2+kn23/2σn2.If we substitute (64) in (18), we obtain
(65)Hn=−kg2+kn22kg+vσnkn2+kg2.Corollary 18.
The ruled surfaceΨn is not minimal.


## 3.1. General Study

In this section, we propose to give some properties of the ruled surfaceΨ.Differentiating (12) with respect to s and v, we get
(13)Ψs=V+vX′,Ψv=X.Then, by using (10), we obtain the components of the first fundamental form E and F as follows:
(14)E=1+2vV,X′+v2X′2,F=V,X.Consequently, the condition of regularity of the ruled surface (12) is
(15)E−F2=1−V,X2+2vV,X′+v2X′2≠0.Differentiating (13) with respect to s and v, we get
(16)Ψss=V′+vX′′,Ψsv=X′,Ψvv=0,which allows, using formula (10), calculating the two components of the second fundamental form M and L, respectively, and we obtain
(17)M=detV,X,X′1−V,X2+2vV,X′+v2X′2,L=detV,X,V′+vdetV,X,X′′+detX′,X,V′+v2detX′,X,X′′1−V,X2+2vV,X′+v2X′2.Therefore, by using (11), the mean curvature of the ruled surface (12) is given as follows
(18)H=detV,X,V′−2detV,X,X′V,X+vdetV,X,X′′+detX′,X,V′+v2detX′,X,X′′21−V,X2+2vV,X′+v2X′23/2.On the other hand, the striction curve of the surface (12) is
(19)c=β−V,X′X′2X.


## 3.2. Case WhereX=T (Resp. V,U)

Let us consider the ruled surfaces defined by(20)Ψ1s,v=βs+vTs,Ψ2s,v=βs+vVs,Ψ3s,v=βs+vUs.By using (2), we have
(21)V,T=0,V,T′=kg,T′2=kg2+kn2.If we substitute (21) in (15), we find
(22)E1−F12=1+2vkg+v2kg2+kn2.Likewise, we have(23)V,V=1,V,V′=0,V′2=kg2+τg2.If we substitute (23) in (15), we find
(24)E2−F22=v2kg2+τg2.On the other hand,(25)V,U=0,V,U′=−τg,U′2=kn2+τg2.If we substitute (25) in (15), we find
(26)E3−F32=1−2vτg+v2kn2+τg2.Corollary 5.
The ruled surfacesΨ1 and Ψ3 are regulars along the curve β, while Ψ2 is singular along β.We denote byc1 (resp. c2,c3) the striction curve of Ψ1resp.Ψ2,Ψ3, and from (19), we have
(27)c1=β−kgkg2+kn2T,c2=β,c3=β+τgkn2+τg2U.The result follows.Corollary 6.
(1)
β is the striction curve of the surface Ψ1 if and only if α is a geodesic curve(2)
β is the striction curve of the surface Ψ2(3)
β is the striction curve of the surface Ψ3 if and only if α is a principal lineOn the other hand, by using (2), we have
(28)detV,T,T′=detV,U,U′=−kn,detV,V,V′=0.Corollary 7.
(1)
The ruled surfaceΨ1 (resp. Ψ3) is developable if and only if α is an asymptotic line(2)
The ruled surfaceΨ2 is developableDifferentiating (2), we obtain
(29)T′′=−kg2+kn2T−knτg−kg′V+kgτg+kn′U,V′′=−knτg+kg′T−kg2+τg2V−kgkn−τg′U,U′′=kgτg−kn′T−kgkn+τg′V−kn2+τg2U.By using (2) and (29), we have
(30)detV,T,V′−2detV,T,T′V,T=−τg,detV,T,T′′+detT′,T,V′=−kn′−2kgτg,detT′,T,T′′=kg′kn−kgkn′−τgkg2+kn2.If we substitute (30) in (18), we find
(31)H1=−τg−kn′+2kgτgv+kg′kn−kgkn′−τgkg2+kn2v221−2vkg+v2kg2+kn23/2.Likewise, we have(32)detV,U,V′−2detV,U,U′V,U=−kg,detV,U,U′′+detU′,U,V′=2kgτg−kn′,detU′,U,U′′=τgkn′−knτg′−kgkn2+τg2.If we substitute (32) in (18), we find
(33)H3=−kg+2kgτg−kn′v+τgkn′−knτg′−kgkn2+τg2v221−2vτg+v2kn2+τg23/2.Corollary 8.
The ruled surfaceΨ1resp.Ψ3 is minimal along its base curve if and only if α is a principal line (resp. a geodesic curve).


## 3.3. Case WhereX Is a Darboux Vector Field

### 3.3.1. Case WhereX=Xr

The ruled surface becomes(34)Ψrs,v=βs+vXrs,where Xr=Dr/Dr with Dr=τgT+kgU and τg,kg≠0,0.Using (2), we get
(35)Xr′=τgT+kgUτg2+kg2′=kgτg′kg−kg′τg−knτg2+kg2τg2+kg23/2T−τgτg′kg−kg′τg−knτg2+kg2τg2+kg23/2U.Hence,(36)Xr′=kgσrT−τgσrU,where
(37)σr=τg′kg−kg′τg−knτg2+kg2τg2+kg23/2.We obtain(38)V,Xr=0,V,Xr′=0,Xr′2=σr2kg2+τg2.If we substitute (38) in (15), we find
(39)Er−Fr2=1+v2σr2kg2+τg2.The following corollaries follow:Corollary 9.
The ruled surfaceΨr is regular.Corollary 10.
In the noncylindrical case (i.e.,σr≠0), the curve β is the striction curve of the surface Ψr.On the other hand, we havedetβ′,Xr,Xr′=σrkg2+τg2.Corollary 11.
The ruled surfaceΨr is developable if and only if it is cylindrical.According to (36), we have
(40)Xr′=−σrV′.Therefore, by using (2) and (29), we obtain
(41)Xr′′=kgσr′+knτg+kg′σrT+σrkg2+τg2V+−τgσr′+kgkn−τg′σrU.Then,(42)detV,Xr,V′−2detV,Xr,Xr′=−kg2+τg2,detV,Xr,Xr′′+detXr′,Xr,V′=−σrkg2+τg2′,detXr′,Xr,Xr′′=−kg2+τg23/2σr2.If we substitute (42) in (18), we find
(43)Hr=−kg2+τg2+σrkg2+τg2′v+kg2+τg23/2σr2v221+v2σr2kg2+τg23/2.Hence, we have the following corollary:Corollary 12.
The ruled surfaceΨr is not minimal.


### 3.3.2. Case WhereX=Xo

The ruled surface becomes(44)Ψos,v=βs+vXos,where Xo=Do/Do with Do=τgT−knV and τg,kn≠0,0.DifferentiatingXo, we obtain
(45)Xo′=τgT−knVτg2+kn2′=knτg′kn−kn′τg+kgτg2+kn2τg2+kn23/2T+τgτg′kn−kn′τg+kgτg2+kn2τg2+kn23/2V.Thus,(46)Xo′=knσoT+τgσoV,where
(47)σo=τg′kn−kn′τg+kgτg2+kn2τg2+kn23/2.We obtain(48)V,Xo=−knkn2+τg2,V,Xo′=τgσo,Xo′2=σo2kn2+τg2.If we substitute (48) in (15), we get
(49)Eo−Fo2=τg+vτg2+kn2σo2τg2+kn2.On the other hand, the striction curve ofΨo is
(50)co=β−τgτg2+kn2σoXo.Hence, we have the following corollary:Corollary 13.
In the noncylindrical case (i.e.,σo≠0), we have the following:
(1)
Ψo is singular along its striction curve(2)
β is the striction curve of the surface Ψo if and only if α is a principal lineWe havedetβ′,Xo,Xo′=0. Hence, the result is as follows.Corollary 14.
The ruled surfaceΨo is developable.From (46), we have
(51)Xo′=−σoU′.Then, by using (2) and (29), we obtain
(52)Xo′′=knσo′−kgτg−kn′σoT+τgσo′+kgkn+τg′σoV+σokn2+τg2U.Hence,(53)detV,Xo,V′−2detV,Xo,Xo′V,Xo=−τg2k¯n2+τ¯g2detV,Xo,Xo′′+detXo′,Xo,V′=−2τgσokn2+τg2,detXo′,Xo,Xo′′=−kn2+τg23/2σo2.If we substitute (53) in (18), we obtain
(54)Ho=−τg2+kn22τg+vσokn2+τg2.Corollary 15.
The surfaceΨo is not minimal.


### 3.3.3. Case WhereX=Xn

The ruled surface becomes(55)Ψns,v=βs+vXns,whereXn=Dn/Dn with Dn=−knV+kgU and kg,kn≠0,0.By using (2), we get
(56)Xn′=−knV+kgUkn2+kg2′=kgkg′kn−kgkn′−τgkn2+kg2kn2+kg23/2V+knkg′kn−kgkn′−τgkn2+kg2kn2+kg23/2U.Thus,(57)Xn′=kgσnV+knσnU,where
(58)σn=kg′kn−kgkn′−τgkn2+kg2kn2+kg23/2.It follows that
(59)V,Xn=−knkn2+kg2,V,Xn′=kgσn,Xn′2=σn2kn2+kg2.If we substitute (59) in (15), we obtain
(60)En−Fn2=kg+vkn2+kg2σn2kn2+kg2.On the other hand, the striction curve ofΨn is
(61)cn=β−kgkn2+kg2σnXn.Hence, we have the following corollary:Corollary 16.
In the noncylindrical case (i.e.,σn≠0), we have the following:
(1)
Ψn is singular along its striction curve(2)
β is the striction curve of the surface Ψn if and only if α is a geodesic curveWe havedetβ′,Xn,Xn′=0. Hence, we have the following result.Corollary 17.
The ruled surfaceΨn is developable.From (57), we have
(62)Xn′=σnT′.Then, by using (2) and (29), we obtain
(63)Xn′′=−σnkg2+kn2T+kgσn′−knτg−kg′σnV+knσn′+kgτg+kn′σnU.Then,(64)detV,Xn,V′−2detV,Xn,Xn′V,Xn=−kg2kg2+kn2detV,Xn,Xn′′+detXn′,Xn,V′=−2kgσnkg2+kn2,detXn′,Xn,Xn′′=−kg2+kn23/2σn2.If we substitute (64) in (18), we obtain
(65)Hn=−kg2+kn22kg+vσnkn2+kg2.Corollary 18.
The ruled surfaceΨn is not minimal.


## 3.3.1. Case WhereX=Xr

The ruled surface becomes(34)Ψrs,v=βs+vXrs,where Xr=Dr/Dr with Dr=τgT+kgU and τg,kg≠0,0.Using (2), we get
(35)Xr′=τgT+kgUτg2+kg2′=kgτg′kg−kg′τg−knτg2+kg2τg2+kg23/2T−τgτg′kg−kg′τg−knτg2+kg2τg2+kg23/2U.Hence,(36)Xr′=kgσrT−τgσrU,where
(37)σr=τg′kg−kg′τg−knτg2+kg2τg2+kg23/2.We obtain(38)V,Xr=0,V,Xr′=0,Xr′2=σr2kg2+τg2.If we substitute (38) in (15), we find
(39)Er−Fr2=1+v2σr2kg2+τg2.The following corollaries follow:Corollary 9.
The ruled surfaceΨr is regular.Corollary 10.
In the noncylindrical case (i.e.,σr≠0), the curve β is the striction curve of the surface Ψr.On the other hand, we havedetβ′,Xr,Xr′=σrkg2+τg2.Corollary 11.
The ruled surfaceΨr is developable if and only if it is cylindrical.According to (36), we have
(40)Xr′=−σrV′.Therefore, by using (2) and (29), we obtain
(41)Xr′′=kgσr′+knτg+kg′σrT+σrkg2+τg2V+−τgσr′+kgkn−τg′σrU.Then,(42)detV,Xr,V′−2detV,Xr,Xr′=−kg2+τg2,detV,Xr,Xr′′+detXr′,Xr,V′=−σrkg2+τg2′,detXr′,Xr,Xr′′=−kg2+τg23/2σr2.If we substitute (42) in (18), we find
(43)Hr=−kg2+τg2+σrkg2+τg2′v+kg2+τg23/2σr2v221+v2σr2kg2+τg23/2.Hence, we have the following corollary:Corollary 12.
The ruled surfaceΨr is not minimal.


## 3.3.2. Case WhereX=Xo

The ruled surface becomes(44)Ψos,v=βs+vXos,where Xo=Do/Do with Do=τgT−knV and τg,kn≠0,0.DifferentiatingXo, we obtain
(45)Xo′=τgT−knVτg2+kn2′=knτg′kn−kn′τg+kgτg2+kn2τg2+kn23/2T+τgτg′kn−kn′τg+kgτg2+kn2τg2+kn23/2V.Thus,(46)Xo′=knσoT+τgσoV,where
(47)σo=τg′kn−kn′τg+kgτg2+kn2τg2+kn23/2.We obtain(48)V,Xo=−knkn2+τg2,V,Xo′=τgσo,Xo′2=σo2kn2+τg2.If we substitute (48) in (15), we get
(49)Eo−Fo2=τg+vτg2+kn2σo2τg2+kn2.On the other hand, the striction curve ofΨo is
(50)co=β−τgτg2+kn2σoXo.Hence, we have the following corollary:Corollary 13.
In the noncylindrical case (i.e.,σo≠0), we have the following:
(1)
Ψo is singular along its striction curve(2)
β is the striction curve of the surface Ψo if and only if α is a principal lineWe havedetβ′,Xo,Xo′=0. Hence, the result is as follows.Corollary 14.
The ruled surfaceΨo is developable.From (46), we have
(51)Xo′=−σoU′.Then, by using (2) and (29), we obtain
(52)Xo′′=knσo′−kgτg−kn′σoT+τgσo′+kgkn+τg′σoV+σokn2+τg2U.Hence,(53)detV,Xo,V′−2detV,Xo,Xo′V,Xo=−τg2k¯n2+τ¯g2detV,Xo,Xo′′+detXo′,Xo,V′=−2τgσokn2+τg2,detXo′,Xo,Xo′′=−kn2+τg23/2σo2.If we substitute (53) in (18), we obtain
(54)Ho=−τg2+kn22τg+vσokn2+τg2.Corollary 15.
The surfaceΨo is not minimal.


## 3.3.3. Case WhereX=Xn

The ruled surface becomes(55)Ψns,v=βs+vXns,whereXn=Dn/Dn with Dn=−knV+kgU and kg,kn≠0,0.By using (2), we get
(56)Xn′=−knV+kgUkn2+kg2′=kgkg′kn−kgkn′−τgkn2+kg2kn2+kg23/2V+knkg′kn−kgkn′−τgkn2+kg2kn2+kg23/2U.Thus,(57)Xn′=kgσnV+knσnU,where
(58)σn=kg′kn−kgkn′−τgkn2+kg2kn2+kg23/2.It follows that
(59)V,Xn=−knkn2+kg2,V,Xn′=kgσn,Xn′2=σn2kn2+kg2.If we substitute (59) in (15), we obtain
(60)En−Fn2=kg+vkn2+kg2σn2kn2+kg2.On the other hand, the striction curve ofΨn is
(61)cn=β−kgkn2+kg2σnXn.Hence, we have the following corollary:Corollary 16.
In the noncylindrical case (i.e.,σn≠0), we have the following:
(1)
Ψn is singular along its striction curve(2)
β is the striction curve of the surface Ψn if and only if α is a geodesic curveWe havedetβ′,Xn,Xn′=0. Hence, we have the following result.Corollary 17.
The ruled surfaceΨn is developable.From (57), we have
(62)Xn′=σnT′.Then, by using (2) and (29), we obtain
(63)Xn′′=−σnkg2+kn2T+kgσn′−knτg−kg′σnV+knσn′+kgτg+kn′σnU.Then,(64)detV,Xn,V′−2detV,Xn,Xn′V,Xn=−kg2kg2+kn2detV,Xn,Xn′′+detXn′,Xn,V′=−2kgσnkg2+kn2,detXn′,Xn,Xn′′=−kg2+kn23/2σn2.If we substitute (64) in (18), we obtain
(65)Hn=−kg2+kn22kg+vσnkn2+kg2.Corollary 18.
The ruled surfaceΨn is not minimal.


## 4. 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 byα the initial curve lying on a surface M defined by φ,T,V,U the Darboux frame of α,kn,kg,τg, the normal curvature, the geodesic curvature, and the geodesic torsion of the curve α, respectively; by β the V-direction curve; and by Xr (resp., Xo,Xn) the unit rectifying (resp., osculator, normal) vector field. We give for each example the illustrations of the ruled surfaces denoted by Ψ1,Ψ2, Ψ3,Ψr,Ψo,Ψn.Example 1.
Let1αs=s/2cos2lns/2,s/2sin2lns/2,s/2 be a curve lying on the surface M1 given by the following parametrization:
(66)1φu,v=ucosv,usinv,u,which can be seen in Figure 1. The Darboux frame of α1 is
(67)T1=12cos2lns2−12sin2lns212sin2lns2+12cos2lns212,V1=−12sin2lns2−12cos2lns212cos2lns2−12sin2lns2−12,U1=−12cos2lns2−12sin2lns212.
Then,(68)k1n=12s,k1g=1s,τ1g=12s.
It follows that(69)X1r=−123cos2lns2−16sin2lns2−123sin2lns2+16cos2lns232,X1o=12cos2lns212sin2lns212,X1n=−123cos2lns2+16sin2lns2−123sin2lns2−16cos2lns232.
TheV-direction curve is given by
(70)1β=−2s3sin2lns2+s6cos2lns2+a12s3cos2lns2+s6sin2lns2+a2−s2+a3,
wherea1,a2, a3 are integration constants. Consequently, the surfaces 1Ψ1, 1Ψ2, 1Ψ3, 1Ψr, 1Ψo, 1Ψn are given as illustrate in Figures 2–7.Example 2.
Letα2s=coss/2,sins/2,s/2 be the curve lying on the surface 2M given by the following parametrization:
(71)2φs,v=coss2−v2sins2,sins2+v2coss2,s2+v2,
which can be seen in Figure8. The Darboux frame of 2α is
(72)T2s=−12sins2,12coss2,12,V2s=coss2,sins2,0,U2s=−12sins2,12coss2,−12.
Then,k2n=0,k2g=−1/2, and τ2g=1/2 and
(73)X2r=0,0,1,X2o=−12sins2,12coss2,−12,X2n=12sins2,−12coss2,12.
Let2βs=2sins/2+b1,−2coss/2+b2,b3, where b1,b2,b3 are integration costants. Consequently, the surfaces 2Ψ1, 2Ψ2, 2Ψ3, 2Ψr, 2Ψo, 2Ψn are given as illustrated in Figures 9–13.Example 3.
We consider the curve3αs=1/2coss,1/2sins,s/2, lying on the surface 3M defined by
(74)3φs,v=12coss,12sins,s2−2v,which can be seen in Figure 14. The Darboux frame of 3α is
(75)T3s=−12sins,12coss,12,V3s=−12sins,12coss,−12,U3s=−coss,−sins,0.
Then,k3n=1/2, k3g=0,, and τ3g=1/2 and
(76)X3r=−12sins,12coss,12,X3o=0,0,1,X3n=12sins,−12coss,12.
Let3βs=1/2coss+c1,1/2sins+c2,−s/2+c3, where c1,c2,c3 are integration costants. Consequently, the surfaces 3Ψ1, 3Ψ2, 3Ψ3, 3Ψr, , 3Ψn are given as illustrated in Figures 15–19.Example 4.
We consider the curve4αs=coss,sins,0 lying on the surface M4 given by the following parametrization:
(77)4φs,v=coss−v2coss,sins−v2sins,v2,
which can be seen in Figure20. The Darboux frame of 4α is
(78)T4s=−sins,coss,0,V4s=−12coss,−12sins,12,U4s=12coss,12sins,12.
Then,k4n=−1/2, k4g=1/2, and τ4g=0 and
(79)X4r=12coss,12sins,12,X4o=1−2coss,−12sins,12X4n=0,0,1.
Let4βs=−1/2sins+d1,1/2coss+d2,s/2+d3, where d1,d2,d3 are integration costants. Consequently, the surfaces 4Ψ1, 4Ψ2, 4Ψ3, 4Ψr, 4Ψo, 4Ψn are given as illustrated in Figures 21–24.Figure 1
M1.Figure 2
Ψ11.Figure 3
Ψ12.Figure 4
Ψ13.Figure 5
Ψ1r.Figure 6
Ψ1o.Figure 7
Ψ1n.Figure 8
M2.Figure 9
Ψ21 and Ψ2o.Figure 10
Ψ22.Figure 11
Ψ23.Figure 12
Ψ2r.Figure 13
Ψ2n.Figure 14
M3.Figure 15
Ψ31 and Ψ3r.Figure 16
Ψ32.Figure 17
Ψ33.Figure 18
Ψ3o.Figure 19
Ψ3n.Figure 20
M4.Figure 21
Ψ41.Figure 22
Ψ42 and Ψ4o.Figure 23
Ψ43 and Ψ4r.Figure 24
Ψ4n.



---

*Source: 8624794-2021-09-11.xml*