Research ArticleSimultaneous Developability of Partner RuledSurfaces according to Darboux Frame in E3

Soukaina Ouarab

Hassan II University of Casablanca, Ben M’sik Faculty of Sciences, Department of Mathematics and Computer Sciences, Morocco

Correspondence should be addressed to Soukaina Ouarab; soukaina.ouarab.sma@gmail.com

Received 2 May 2021; Accepted 12 August 2021; Published 28 August 2021

Academic Editor: Gueo Grantcharov

Copyright © 2021 Soukaina Ouarab. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, we introduce original definitions of Partner ruled surfaces according to the Darboux frame of a curve lying on anarbitrary regular surface in E3. It concerns Tg Partner ruled surfaces, Tn Partner ruled surfaces, and gn Partner ruled surfaces. Weaim to study the simultaneous developability conditions of each couple of two Partner ruled surfaces. Finally, we give anillustrative example for our study.

1. Introduction

The theory of ruled surfaces forms an important and usefulclass of theories in differential geometry [1, 2]. This kind ofsurface is defined by the moving of a straight line along acurve. The various positions of the generating lines are calledthe rulings of the ruled surface. Such a surface, thus, has aparametric representation of the form

ϕ : s ; vð Þ ∈ I ×ℝ↦ c sð Þ + vX sð Þ, ð1Þ

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 ruledsurface is the property of developability. It defines ruled sur-faces that can be transformed into the plane without anydeformation and distortion; such surfaces form relativelysmall subsets that contain cylinders, cones, and tangentsurfaces. They are characterized with vanishing Gaussiancurvature [3–5].

Many geometers have studied some of the differentialgeometric concepts of the ruled surfaces by means of differ-ent moving frames, such as the Frenet-Serret frame, alterna-tive frame, and Bishop frame [6–8].

Another one of the most important moving frame of thedifferential geometry is the Darboux frame, which is a natu-

ral moving frame constructed on a surface that contains acurve. It is named after the French mathematician JeanGaston Darboux, in a four-volume collection of the studieshe published between 1887 and 1896. Since that time, therehave been many important repercussions of the Darbouxframe, having been examined for example in (Darboux,1896; O Neill, 1996). One can find studies of ruled surfaceswith the Darboux frame realized in Euclidean and non-Euclidean 3-space. For example, in [9], the authors con-structed the ruled surface whose rulings are constant linearcombinations of the Darboux frame vectors of its base curvealong a regular surface of reference; they studied the mostimportant properties of that ruled surface, characterized it,and presented examples with illustrations. Furthermore, in[10], the authors studied the characteristic properties of aruled surface with the Darboux frame and gave the relation-ship between the Darboux frame and the Frenet frame.Moreover, in [11], the authors defined the evolute offsetsof ruled surface with the Darboux frame and studied itscharacteristic properties in E3.

The main contribution of this work is to introduce newspecial couples of ruled surfaces defined by means of Dar-boux frame vectors of a regular curve lying on an arbitraryregular surface in E3. Our objective is to study the simulta-neous developability of such couples of surfaces. Throughour study, we are opening up some avenues for scientists

HindawiAbstract and Applied AnalysisVolume 2021, Article ID 3151501, 9 pageshttps://doi.org/10.1155/2021/3151501

to apply our approach in some areas such as architecturaldesign, medical science, surface modeling, engineering, andcomputer-aided geometric design [12–15].

The principle of this study is to consider a unit speedcurve cðsÞ lying on an arbitrary regular surface ϕ, associatethe Darboux frame fT , g, ng of cðsÞ on ϕ, and, then, definethree couples of ruled surfaces that are generated, recipro-cally, 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 developabil-ity of each couple of Partner ruled surfaces. Indeed, weinvestigate theorems that reply to our needs. Finally, wepresent an example with illustrations.

2. Preliminaries

Due to a unit speed curve α = αðsÞ that lies on a regularsurface 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 curveα = αðsÞ, n!ðsÞ = ððϕu × ϕrÞ/kϕu × ϕrkÞðuðsÞ, rðsÞÞ is the unitnormal vector of the surface ϕ = ϕðu, rÞ along the curveα = αðsÞ, and g!ðsÞ is the unit vector which is defined by

g!ðsÞ = n!ðsÞ × T!ðsÞ.

The derivative formulae of the Darboux frame are givenas follows:

T!′

g!′

n!′

26664

37775 =

0 ρg ρn

−ρg 0 θg

−ρn −θg 0

2664

3775

T!

g!

n!

26664

37775, ð2Þ

where ρg is the geodesic curvature, ρn is the normal curva-ture, and θg is the geodesic torsion of the curve α = αðsÞ onthe surface ϕ = ϕðu, rÞ.

Definition 1 (see [16]). The curve αðsÞ lying on a regularsurface is as follows:

(i) A geodesic curve if its geodesic curvature ρg vanishes

(ii) An asymptotic line if its normal curvature ρnvanishes

(iii) A principal line if its geodesic torsion θg vanishes

Let φ : ðs, vÞ↦ cðsÞ + vX!ðsÞ be a ruled surface in E3.

Let denote by n! = n!ðs, vÞ, the unit normal on the ruledsurface φ at a regular point φðs, vÞ, we have

n! = φs ∧ φv

φs ∧ φvk k =c′ + vX

!′� �

× X!

c′ + vX!′

� �× X

!��� ��� , ð3Þ

where φs = ∂φðs, vÞ/∂s and φv = ∂φðs, vÞ/∂v.

The first I and the second II fundamental forms ofthe ruled surface φ at a regular point φðs, vÞ are defined,respectively, by

I φsds + φvdvð Þ = Eds2 + 2Fdsdv +Gdv2,

II φsds + φvdvð Þ = eds2 + 2f dsdv + gdv2,ð4Þ

where

E = φsk k2, F = φs, φvh i, G = φvk k2,e = φss, n

!D E, f = φsv, n

!D E, g = φvv , n

!D E= 0:

ð5Þ

Definition 2. The Gaussian curvature K of the ruled surfaceφ at a regular point φðs, vÞ is given by

K = −f 2

EG − F2 : ð6Þ

Proposition 3 (see [16]). A ruled surface is developable if andonly if its Gaussian curvature vanishes.

3. Simultaneous Developability of PartnerRuled Surfaces according to the DarbouxFrame in E3

Definition 4. Let c : s ∈ I ↦ cðsÞ be a C2-class differentiableunit speed curve lying on a regular surface ϕ = ϕðu, rÞ. Letdenote by fTðsÞ, gðsÞ, nðsÞg the Darboux frame of c = cðsÞon ϕ = ϕðu, rÞ. The two ruled surfaces defined by

Tgφ : s, vð Þ ∈ I ×ℝ↦ T sð Þ + vg sð Þ,gTφ : s, vð Þ ∈ I ×ℝ↦ g sð Þ + vT sð Þ,

(ð7Þ

are called Tg Partner ruled surfaces according to theDarboux frame of the curve cðsÞ on the surface ϕ = ϕðu, rÞ.

Theorem 5. Tg Partner ruled surfaces (7) are simultaneouslydevelopable if and only if at least one of the following state-ments is verified:

(i) c = cðsÞ is a geodesic curve on the surface ϕ

(ii) c = cðsÞ is an asymptotic and a principal line on thesurface ϕ

Proof. By differentiating the first line of (7) with respect to sand v, respectively, and using Darboux derivative formulae(2), we get

Tgφs = −vρgT + ρgg + ρn + vθg� �

n,Tgφv = g:

(ð8Þ

2 Abstract and Applied Analysis

Then, by considering the cross product of both vectors in(8), we get the normal vector of the ruled surface Tgφ:

Tgφs × Tgφv = − ρn + vθg� �

T − vρgn, ð9Þ

which implies that under regularity condition, the unitnormal vector of the ruled surface Tgφ is given by

Tgφs × TgφvTgφs × Tgφv

�� �� = −1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ρn + vθg� �2 + v2ρ2g

q ρn + vθg� �

T + vρgnh i

:

ð10Þ

By applying the norms and the scalar product for bothvectors in (8), we get the components of the first fundamen-tal form of the ruled surface Tgφ:

TgE = 1 + vð Þρ2g + ρn + vθg� �2, ð11Þ

TgF = ρg, ð12ÞTgG = 1: ð13Þ

By differentiating the second line of (8) with respect to s,using Darboux derivative formulae (2) and making the sca-lar product with the unit normal (10), we get the secondcomponent of the second fundamental form of the ruledsurface Tgφ:

Tgf = −ρnρgffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ρn + vθg� �2 + v2ρ2g

q : ð14Þ

Thus, from (13) and (14), we get the Gaussian curvatureof the ruled surface Tgφ:

TgK = −ρnρg

ρn + vθg� �2 + v2ρ2g

!2

: ð15Þ

On another hand, by differentiating the second line of(7) with respect to s and v and using Darboux derivativeformulae (2), we get

gTφs = −ρgT + vρgg + θg + vρn� �

n,gTφv = T:

(ð16Þ

Then, the cross product of both vectors of (16) gives thenormal vector of the ruled surface gTφ:

gTφs × gTφv = θg + vρn� �

g − vρnn, ð17Þ

which implies that under regularity condition, the unitnormal vector of the ruled surface gTφ is given by

gTφs × gTφvgTφs × gTφv

�� �� = 1θg + vρn� �2 + v2ρ2g

θg + vρn� �

g − vρgnh i

:

ð18ÞBy applying the norms and the scalar product for both

vectors (16), we obtain the components of the first funda-mental form of the ruled surface gTφ:

gTE = ρ2g + v2ρ2g + θg + vρn� �2, ð19Þ

gT F = −ρg, ð20ÞgTG = 1: ð21Þ

By differentiating the second line of (16) with respect tos, using Darboux derivative formulae (2) and using the unitnormal (18), we get the second component of the secondfundamental form of the ruled surface gTφ:

gT f =θgρg

θg + vρn� �2 − v2ρ2g

: ð22Þ

Hence, from (21) and (22), we get the Gaussian curva-ture of the ruled surface gTφv:

gTK = −θgρg

θg + vρn� �2 − v2ρ2g

!2

: ð23Þ

Consequently, from (15) and (23), we deduce Tg Part-ner ruled surfaces

gTφ and gTφ are simultaneously developable if and onlyif TgK = gTK = 0, i:e:, ρnρg = θgρg = 0, which is equivalentto ρg = 0 or ρn = θg = 0. Thus, Theorem 5 is proved. ☐

Definition 6. Let c : s ∈ I ⟼ cðsÞ be a C2-class differentiableunit speed curve lying on a regular surface ϕ = ϕðu, rÞ. Letdenote by fTðsÞ, gðsÞ, nðsÞg the Darboux frame of c = cðsÞon ϕ = ϕðu, rÞ. The two ruled surfaces defined by

Tnφ : s, vð Þ ∈ I ×ℝ↦ T sð Þ + vn sð Þ,nTφ : s, vð Þ ∈ I ×ℝ↦ n sð Þ + vT sð Þ,

(ð24Þ

are called Tn Partner ruled surfaces according to the Dar-boux frame of the curve c = cðsÞ on the surface ϕ = ϕðu, rÞ.

Theorem 7. Tn Partner ruled surfaces (24) are simulta-neously developable if and only if at least one of the followingstatements is verified:

(i) c = cðsÞ is an asymptotic line on the surface ϕ

(ii) c = cðsÞ is a geodesic curve and a principal line on thesurface ϕ

3Abstract and Applied Analysis

Proof. Differentiating the first line of (24) with respect to sand v and using Darboux derivative formulae (2), we get

Tnφs = −vρnT + ρg − vθg� �

g + ρnn,Tnφv = n,

8<: ð25Þ

then, the cross product of both last vectors gives us thenormal vector of the ruled surface Tnφ:

Tnφs × Tnφv = ρg − vθg� �

T + vρng, ð26Þ

which implies that under regularity condition, the unit nor-mal vector of the ruled surface Tnφ takes the following form:

Tnφs × TnφvTnφs × Tnφv

�� �� = 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiρg − vθg� �2

+ v2ρ2n

r ρg − vθg� �

T + vρngh i

:

ð27Þ

By applying the norms and the scalar product for bothvectors (25), we get the components of the first fundamentalform of the ruled surface Tnφ:

TnE = v2ρ2n + ρg − vθg� �2

+ ρ2n, ð28Þ

TnF = ρn, ð29ÞTnG = 1: ð30Þ

By differentiating the second line of (25) with respectto s, using (2) and (27), we get the second component ofthe second fundamental form of the ruled surface Tnφ:

Tnf = −ρnρgffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ρg − vθg� �2

+ v2ρ2n

r : ð31Þ

Thus, from (30) and (31), we obtain the Gaussian cur-vature of the ruled surface Tnφ:

TnK = −ρnρg

ρg − vθg� �2

+ v2ρ2n

0B@

1CA

2

: ð32Þ

Let us now differentiate the second line of (24) withrespect to s and v, respectively, and using Darboux deriv-ative formulae (2), we get

nTφs = −ρnT + −θg + vρg� �

g + vρnn,nTφv = T:

8<: ð33Þ

The cross product of both vectors (33) gives the normalvector of the ruled surface nTφ:

nTφs × nTφs = vρng − −θg + vρg� �

n, ð34Þ

which gives, under regularity condition, the unit normalvector of the ruled surface nTφ as follows:

nTφs × nTφvnTφs × nTφv

�� �� = 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2ρ2n + −θg + vρg

� �2r vρng − −θg + vρg� �

nh i

:

ð35Þ

The norms and scalar product applied on both vectors in(33) give us the components of the first fundamental form ofthe ruled surface nTφ:

nTE = ρ2n + v2ρ2n� �

+ −θg + vρg� �2

, ð36Þ

nT F = −ρnTn , ð37Þ

G = 1: ð38ÞDifferentiating the second line of (33) with respect to s,

using Darboux derivative formulae (2), and using the unitnormal (35), we get the second component of the secondfundamental form of the ruled surface nTφ:

nT f =ρnθgffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

v2ρ2n + −θg + vρg� �2r : ð39Þ

Thus, from (38) and (39), we obtain the Gaussian curva-ture of the ruled surface nTφ as follows:

nTK = −ρnθg

v2ρ2n −θg + vρg� �2

0B@

1CA

2

: ð40Þ

Consequently, from (32) and (40), we deduce Tn Partnerruled surfaces

Tnφ and nTφ are simultaneously developable if and onlyif TnK = nTK = 0, i:e:, ρnρg = ρnθg = 0, which is equivalentto ρn = 0 or ρg = θg = 0. Thus, Theorem 7 is proved. ☐

Definition 8. Let c : s ∈ I ⟼ cðsÞ be a C2-class differentiableunit speed curve lying on a regular surface ϕ = ϕðu, rÞ. Letdenote by fTðsÞ, gðsÞ, nðsÞg the Darboux frame of c = cðsÞon ϕ = ϕðu, rÞ. The two ruled surfaces defined by

gnφ : s, vð Þ ∈ I ×ℝ↦ g sð Þ + vn sð Þ,ngφ : s, vð Þ ∈ I ×ℝ↦ n sð Þ + vg sð Þ,

(ð41Þ

4 Abstract and Applied Analysis

are called gn Partner ruled surfaces according to the Dar-boux frame of the curve c = cðsÞ on the surface ϕ = ϕðu, rÞ.

Theorem 9. gn Partner ruled surfaces are simultaneouslydevelopable if and only if at least one of the following state-ments is verified:

(i) c = cðsÞ is a principal line on the surface ϕ

(ii) c = cðsÞ is a geodesic curve and an asymptotic line onthe surface ϕ

Proof. Differentiating the first line of (41) with respect to sand v, respectively, and using Darboux derivative formulae(2), we get

gnφs = − ρg + vρn� �

T − vθgg + θgn,ngφv = n:

8<: ð42Þ

We get the normal vector of the ruled surface gnφ byrealizing the cross product of both vectors (42):

gnφs × gnφv = −vθgT + ρg + vρn� �

g, ð43Þ

which implies that under regularity condition, the unit nor-mal vector of the ruled surface gnφ is given by

gnφs × gnφvgnφs × gnφv

�� �� = 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2θ2g + ρg + vρn

� �2r −vθgT + ρg + vρn� �

gh i

:

ð44Þ

By applying the norms and the scalar product for bothvectors of (42), we obtain

gnE = ρg + vρn� �2

+ v2θ2g + θ2g, ð45Þ

gnF = θg, ð46ÞgnG = 1: ð47Þ

By differentiating the second line of (42) with respect tos, using (2) and (44), we obtain

gn f = −θgρgffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

v2θ2g + ρg + vρn� �2r : ð48Þ

Hence, from (47) and (48), we obtain the Gaussiancurvature of ruled surface gnφ:

gnK = −θgρg

v2θ2g + ρg + vρn� �2

0B@

1CA

2

: ð49Þ

Let us now differentiate the second line of (41) withrespect to s and v, respectively, and use Darboux derivativeformulae (2), we get

ngφs = − ρn + vρg� �

T − θgg + vθgn,ngφv = g:

8<: ð50Þ

By realizing the cross product of both vectors of (50),we get the normal vector of the ruled surface ngφv:

ngφs × ngφv = −vθgT − ρn + vρg� �

n, ð51Þ

which implies that under regularity condition, the unit nor-mal vector of the ruled surface ngφv is given by

ngφs × ngφvngφs × ngφv

�� �� = −1

v2θ2g + ρn + vρg� �2 vθgT + ρn + vρg

� �n

h i:

ð52Þ

By applying the norms and the scalar product for (50),we obtain

ngE = ρn + vρg� �2

+ θ2g + v2θ2g, ð53Þ

ngF = −θg, ð54Þ

ngG = 1: ð55Þ

By differentiating the second line of (50) with respect tos, using (2) and (52), we get

ng f = −θgρn

v2θ2g + ρn + vρg� �2 : ð56Þ

Hence, from (55) and (56), we obtain the Gaussiancurvature of the ruled surface ngφv as follows:

ngK = −θgρn

v2θ2g + ρn + vρg� �2

0B@

1CA: ð57Þ

Consequently, from (49) and (57), we deduce that gnPartner ruled surfaces gnφ and ngφ are simultaneously devel-opable if and only if gnK = ngK = 0, i:e:, θgρg = θgρn = 0,which is equivalent to θg − 0 or ρg = ρn = 0. Thus, Theorem9 is proved. ☐

Here follows, we give an example of our study and presentcorresponding illustrations.

5Abstract and Applied Analysis

Example 1. Let us consider the regular surface parameter-ized by

eφ u, rð Þ = 2 cos u2 −

rffiffiffi2

p sin u2 , 2 sin

u2 −

rffiffiffi2

p cos u2 ,

rffiffiffi2

p�

:

ð58Þ

It is easy to see that the curve ecðsÞ = ð2 cos ðs/2Þ,2 sin ðs/2Þ, 0Þ lies on the regular surface eφ; indeed, we haveecðsÞ = eφðs, 0Þ.

The Darboux frame vectors and the Darboux invariantsof ecðsÞ on eφ are given, respectively, by

T sð Þ = −sin s2� �

, cos s2� �

, 0� �

,

g sð Þ = 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 + sin2 sð Þ

p −sin sð Þ cos s2� �

,− sin sð Þ sin s2� �

, 1� �

,

n sð Þ = 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 + sin2 sð Þ

p cos s2� �

, sin s2� �

, sin sð Þ� �

,

ρg sð Þ = sin sð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 + sin2 sð Þ

p ,

ρn sð Þ = −1

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 + sin2 sð Þ

p ,

θg sð Þ = −cos sð Þ

1 + sin2 sð Þ : ð59Þ

Hence, Tg Partner ruled surfaces, Tn Partner ruledsurfaces, and gn Partner ruled surfaces according to the

−4−3

−2−1

01

23

4

−4−3

−2−1

01

23

4−5

0

5

Figure 1: Ruled surface Tgφ of (60).

−5

0

5

−5

0

50.7

0.75

0.8

0.85

0.9

0.95

1

Figure 2: Ruled surface gTφ of (60).

6 Abstract and Applied Analysis

−6−4

−20

24

6

−6−4

−20

24

6−1

−0.5

0

0.5

1

Figure 4: Ruled surface nTφ of (61).

−6−4

−20

24

6

−6−4

−20

24

6−5

0

5

Figure 5: Ruled surface gnφ of (62).

−6−4

−20

24

6

−6−4

−20

24

6−4

−2

0

2

4

Figure 3: Ruled surface Tnφ of (61).

7Abstract and Applied Analysis

Darboux frame of ecðsÞ on eφ are given, respectively, asfollows:

where RðsÞ = 1/ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 + sin2ðsÞp

:Here follows, we present the illustrations of the Tg Part-

ner ruled surface (60) represented by Figures 1 and 2, the TnPartner ruled surfaces (61) represented by Figures 3 and 4,and the gn Partner ruled surfaces (62) represented byFigures 5 and 6, respectively.

4. Conclusion

In this paper, we presented a novel method to constructthree special couples of ruled surfaces according to the Dar-boux frame of a regular curve cðsÞ on a regular surface in E3.These three couples of surfaces were called Tg Partner ruledsurfaces, Tn Partner ruled surfaces, and gn Partner ruledsurfaces, respectively. We investigated theorems that givenecessary and sufficient conditions for each couple of twoPartner ruled surfaces to be simultaneously developable.The obtained results reveal that the simultaneous develop-ability conditions are related to the properties of the curvecðsÞ on the surface. Finally, we presented an illustrativeexample.

On the other hand, our approach can also provideexcellent support for architectural design, surface model-ing, computer-aided geometric design, and engineeringapplication.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest.

References

[1] D. J. Struik, Lectures on Classical Differential Geometry,Addison Wesley, Dover, 2nd edition, 1988.

[2] M. P. Do Carmo,Differential Geometry of Curves and Surfaces,Prentice Hall, Englewood Cliffs, NJ, 1976.

[3] G. Hu, H. Cao, J. Wu, and G. Wei, “Construction ofdevelopable surfaces using generalized C-Bézier bases withshape parameters,” Computational and Applied Mathematics,vol. 39, no. 3, 2020.

−4 −3 −2 −1 0 1 2 3 4

−4−3−2

−10

12

34−10

−50510

Figure 6: Ruled surface ngφ of (62).

Tgφ = −sin s2� �

, cos s2� �

, 0� �

+ vR sð Þ −sin sð Þ cos s2� �

,− sin sð Þ sin s2� �

, 1� �

,

gTφ = R sð Þ −sin sð Þ cos s2� �

,− sin sð Þ sin s2� �

, 1� �

+ v −sin s2� �

, cos s2� �

, 0� �

,

8><>: ð60Þ

Tnφ = −sin s2� �

, cos s2� �

, 0� �

+ vR sð Þ cos s2� �

, sin s2� �

, sin sð Þ� �

,

nTφ = R sð Þ cos s2� �

, sin s2� �

, sin sð Þ� �

+ v −sin s2� �

, cos s2� �

, 0� �

,

8><>: ð61Þ

gnφ = R sð Þ −sin sð Þ cos s2� �

,− sin sð Þ sin s2� �

, 1� �

+ v cos s2� �

, sin s2� �

, sin sð Þ� �h i

,

ngφ = R sð Þ cos s2� �

, sin s2� �

, sin sð Þ� �

+ v −sin sð Þ cos s2� �

,− sin sð Þ sin s2� �

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8 Abstract and Applied Analysis

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9Abstract and Applied Analysis