On Ruled Surfaces According to Quasi-Frame in Euclidean 3-Space

2020·0.57 MB·

by  M. Khalifa Saad

Aust. J.Math. Anal. Appl.
Vol. 17(2020),No. 1,Art. 11,16pp.
AJMAA
ON RULED SURFACES ACCORDING TO QUASI-FRAME IN EUCLIDEAN
3-SPACE
M.KHALIFASAADANDR.A.ABDEL-BAKY
Received30January,2020;accepted14May,2020;published26May,2020.
DEPARTMENTOFMATHEMATICS,FACULTYOFSCIENCE,ISLAMICUNIVERSITYOFMADINAH,KSA.
DEPARTMENTOFMATHEMATICS,FACULTYOFSCIENCE,SOHAGUNIVERSITY,SOHAG,EGYPT.
[email protected], [email protected]
DEPARTMENTOFMATHEMATICS,FACULTYOFSCIENCE,ASSIUTUNIVERSITY,ASSIUT,EGYPT.
[email protected]
ABSTRACT. This paper aims to study the skew ruled surfaces by using the quasi-frame of
SmarandachecurvesintheEuclidean3-space. Also,werevealtherelationshipbetweenSerret-
Frenetandquasi-framesandgiveaparametricrepresentationofadirectionalruledsurfaceusing
thequasi-frame. Besides,somecomparativeexamplesaregivenandplottedwhichsupportour
methodandmainresults.
Keywordsandphrases: Smarandachecurves;Ruledsurfaces;Serret-Frenetframe;Quasi-frame.
2010MathematicsSubjectClassification. 53A04,53A05,53A35.
ISSN(electronic):1449-5910
(cid:13)c 2020AustralInternetPublishing.Allrightsreserved.
ThisresearchwassupportedbyIslamicUniversityofMadinah.WewouldliketothankourcolleaguesfromDeanshipofScientificResearch
whoprovidedinsightandexpertisethatgreatlyassistedtheresearch.
2 M.KHALIFASAADANDR.A.ABDEL-BAKY
1. INTRODUCTION
A ruled surface is a curved surface which can be generated by the continuous motion of
a straight line in space along a space curve called a directrix. This straight line is called a
generator, or ruling, of the surface. The importance of the ruled surface lies in the fact that it
is used in many practical applications in computer aided geometric design. The ruled surface
can be parameterized using the Serret-Frenet frame, however, this frame is undefined wherever
the curvature vanishes, such as at points of inflection or along straight sections of the curve
[8]. Thus, Bishop introduced a new frame along a space curve which is more suitable for
applications [6]. But, it is well known that Bishop frame calculations are not an easy task, see
[23, 24]. Therefore, various alternative methods have been proposed for computing the ruled
surfaces, see [13, 14]. Klok defined the sweep surfaces using rotation-minimizing frames [17].
A robust computation of the rotation minimizing frame for sweep surfaces was introduced by
Wang [23]. Inspired by the work of Coquillart [11], Mustafa introduced a new adapted frame
alongaspacecurveanddenotedthisthequasi-frame[20].
This paper is divided into three main parts. The first part is the introduction, which gives a
brief idea about this article. The second part deals with the basic concepts of the theory of the
classical differential geometry of curves in Euclidean 3-space. Also, the concepts of the quasi-
frame and quasi-equations are highlighted. The main results of this paper, which deal with the
study of skew ruled surfaces by using the quasi-frame of Smarandache curves are presented in
third part of this paper. Moreover, some computational examples are illustrated to confirm the
effectivenessoftheproposedmethod.
2. PRELIMINARIES
Letα = α(s)beaunitspeedcurveinEuclidean3-spaceE3;byκ(s)andτ(s)wedenotethe
naturalcurvatureandtorsionofα =α(s),respectively. Weassumeα(cid:48)(cid:48)(s) (cid:54)=0foralls∈[0,L],
since this would give us a straight line. Throughout the paper, α(cid:48)(s) denotes the derivative of
α with respect to arc length parameter s. For each point of α(s), the set {t(s), n(s), b(s)} is
called the Serret-Frenet frame along α(s), where t(s) = α(cid:48)(s), n(s) = α(cid:48)(cid:48)(s)/(cid:13)(cid:13)α(cid:48)(cid:48)(s)(cid:13)(cid:13) and
b(s) = t(s)×n(s) are the unit tangent, principal normal, and binormal vectors of the curve at
the point α(s), respectively. The arc-length derivative of the Serret-Frenet frame is governed
bythefollowing:
 t(cid:48)(s)   0 κ(s) 0  t(s) 
(2.1)  n(cid:48)(s)  =  −κ(s) 0 τ(s)  n(s) .
b(cid:48)(s) 0 −τ(s) 0 b(s)
Although the Serret-Frenet frame can easily be computed, its rotation about the tangent of
a general space curve often leads to undesirable twist in motion design or sweeping surface
modeling. Also, the Serret-Frenet frame is not continuously defined for a C1-continuous space
curve, and even for a C2-continuous space curve the Serret-Frenet frame becomes undefined at
aninflectionpoint(i.e.,curvatureκ = 0),thiscausingunacceptablediscontinuitywhenusedfor
surface modeling. In order to resolve the above problem of the Serret–Frenet frame, Coquillart
andMustafaetal. introducedaquasi-frameofaspacecurveasfollows[11,19,20]:
Given a unit speed curve α = α(s), the quasi-frame (or simply Q-frame) along α(s) is given
by
t×ζ
(2.2) e (s)= t(s), e (s) = , e (s) = t×e ,
1 2 3 2
(cid:107)t×ζ(cid:107)
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RULEDSURFACESACCORDINGTOQUASI-FRAME 3
where ζ is called the projection vector. The relation between Serret-Frenet frame and quasi-
frameisgivenasfollows[19]:
    
e 1 0 0 t(s)
1
(2.3)  e2  =  0 cosϕ sinϕ  n(s) .
e 0 −sinϕ cosϕ b(s)
3
By taking the derivative of Eq. (2.3) with respect to s and using the inverse transformation, we
obtain
 e(cid:48)   0 κ κ  e 
1 1 2 1
(2.4)  e(cid:48)2  =  −κ1 0 κ3  e2 .
e(cid:48) −κ −κ 0 e
3 2 3 3
Thetriple(κ ,κ ,κ )iscalledtheQ-curvaturefunctionsofα(s)anddenotedby
1 2 3
κ = κcosϕ =< e(cid:48),e >, 
1 1 2 
(2.5) κ = −κsinϕ =< e(cid:48),e >,
2 1 3
κ = τ +ϕ(cid:48) =< e(cid:48),e >. 
3 2 3
In terms of these quantities, the Q-geodesics, Q-line of curvatures, and Q-asymptotic lines on
a smooth surface may be characterized as loci along which κ = 0, κ = 0, and κ = 0,
1 3 2
respectively. The Q-frame have many advantages over Serret-Frenet and Bishop frames. For
instance, the Q-frame can be defined even along a line (i.e., curvature κ = 0). However, the
Q-frame is singular in all cases where t and ζ are parallel. Thus, in these cases, where t and ζ
areparallel,theprojectionvectorζ canbechosenasζ =(0,1,0)orζ =(0,0,1)(fordetails,see
[19]).
Remark 1. If κ = 0, then the Q-frame {e (s), e (s), e (s)} turn out to the Bishop frame. In
3 1 2 3
thiscase,onecanshowthat:
(cid:16) (cid:17) 
κ1 = κcosϕ,κ2 = κsinϕ,ϕ = tan−1 κκ12 ; κ1 (cid:54)= 0 
(2.6) s
(cid:82)
ϕ−ϕ = − τds.
0 
s0
3. DIRECTIONAL RULED SURFACES
Theparameterizedsurface
(3.1) M : P(s,v) = α(s)+vL(s), v ∈ R,
iscalledaruledsurface;α(s)iscalledthebasecurve,andthelinepassingthroughα(s)which
is parallel to L(s) is called the ruling of the surface at α(s). The surface P(s,v) is regular, if
P ×P (cid:54)= 0 for all points, where P and P are the partial derivatives of P(s,v) with respect
s v s v
tosandv,respectively(see[2,12,15,16,18]).
3.1. Directional Smarandache curves . It is known that Smarandache geometry is a geome-
trywhichhasatleastoneSmarandachedeniedaxiom[1]. AnaxiomissaidtobeSmarandache
denied,ifitbehavesinatleasttwodifferentwayswithinthesamespace. Smarandachegeome-
tries are connected with the theory of relativity and the parallel universes. Smarandache curves
aretheobjectsofSmarandachegeometry.
Consider a curve α = α(s) in E3 and let {e (s), e (s), e (s)} be a quasi-frame of α and κ ,
1 2 3 1
κ , and κ the curvature functions in arc length parameter s of α. So, we have the following
2 3
definition(formoredetails,see[3,4,5,6,7,8,9,10,13,21,22]).
Definition 3.1. A regular curve α(s) in E3, whose position vector is obtained by quasi-frame
vectorsofanotherregularcurver(s)iscalledSmarandachecurve.
AJMAA,Vol.17(2020),No.1,Art.11,16pp. AJMAA
4 M.KHALIFASAADANDR.A.ABDEL-BAKY
Inthefollowingexamples,weinvestigatesomespecialSmarandachecurvesofagivencurve
according to its quasi-frame called e e −, e e −,e e −Smarandache curves and then obtain
1 2 1 3 2 3
someoftheirdifferentialgeometricpropertieswhichrepresentthemainresults.
3.1.1. e e -Smarandache curves . Let r = r(s) be a space curve in E3with Q- frame {e (s),
1 2 1
e (s),e (s)}. Thenthee e −Smarandachecurvesofr aredefinedby
2 3 1 2
1
(3.2) α(s¯) = √ (e +e ).
1 2
2
Thecurvaturefunctionsκ andτ ofα canbecomputedasfollows:
α α
 1
κ2(2κ2 +(κ +κ )2)2 2
1 1 1 2 3
κα = λ  +(κ(cid:48)3κ1 +κ(cid:48)2κ1 −κ(cid:48)1(κ2 +κ3)−κ2(2κ21 +(κ2 +κ3)2))2  ;
1 +(κ(cid:48)κ +κ(cid:48)κ −κ(cid:48) (κ +κ )+κ (2κ2 +(κ +κ )2))2
3 1 2 1 1 2 3 3 1 2 3
(cid:18) (cid:19)
1
(3.3) λ = 2 κ2 + (κ +κ )2 3/2.
1 1 2 2 3
 (−κ(cid:48)κ −κ(cid:48)κ +κ(cid:48) (κ +κ )+κ (2κ2 +(κ +κ )2)) 
3 1 2 1 1 2 3 2 1 2 3
(κ(cid:48)κ +(3κ(cid:48) +2κ(cid:48))κ +κ (3κ(cid:48) +κ2 +κ2 +κ2)−κ(cid:48)(cid:48))
√  3 2 3 2 3 1 1 1 2 3 1 
2  −(κ(cid:48)κ +κ(cid:48)κ −κ(cid:48) (κ +κ )+κ (2κ2 +(κ +κ )2)) 
τ =  3 1 2 1 1 2 3 3 1 2 3 ;
α λ  ((2κ(cid:48) +3κ(cid:48))κ +κ(cid:48)κ −κ (−3κ(cid:48) +κ2 +κ2 +κ2)+κ(cid:48)(cid:48)) 
2  3 2 2 2 3 1 1 1 2 3 1 
 −κ (2κ2 +(κ +κ )2) 
1 1 2 3
(−κ(cid:48)κ +κ(cid:48)κ +2κ(cid:48) (κ −κ )+(κ +κ )(κ2 +κ2 +κ2)−κ(cid:48)(cid:48) −κ(cid:48)(cid:48))
3 1 2 1 1 2 3 2 3 1 2 3 3 2
 
κ2(2κ2 +(κ +κ )2)2
1 1 2 3
(3.4) λ2 =  +(κ(cid:48)3κ1 +κ(cid:48)2κ1 −κ(cid:48)1(κ2 +κ3)−κ2(2κ21 +(κ2 +κ3)2))2 .
+(κ(cid:48)κ +κ(cid:48)κ −κ(cid:48) (κ +κ )+κ (2κ2 +(κ +κ )2))2
3 1 2 1 1 2 3 3 1 2 3
3.1.2. e e -Smarandache curves . Let r = r(s) be a space curve in E3with Q- frame {e (s),
1 3 1
e (s),e (s)}. Thenthee e −Smarandachecurvesofr aredefinedby
2 3 1 3
1
(3.5) β(s¯) = √ (e +e ).
1 3
2
Thecurvaturefunctionsκ andτ ofβ canbecomputedasfollows:
β β
 1
((−κ(cid:48) +κ(cid:48))κ +κ (2κ2 +(κ −κ )2)+κ(cid:48) (κ −κ ))2 2
1 1 3 2 1 2 1 3 2 1 3
κβ = µ  +κ22(2κ22 +(κ1 −κ3)2)2  ;
1 +((−κ(cid:48) +κ(cid:48))κ +κ(cid:48) (κ −κ )+(2κ2 +(κ −κ )2)κ )2
1 3 2 2 1 3 2 1 3 3
(cid:18) (cid:19)
1
(3.6) µ = 2 κ2 + (κ −κ )2 3/2.
1 2 2 1 3
AJMAA,Vol.17(2020),No.1,Art.11,16pp. AJMAA
RULEDSURFACESACCORDINGTOQUASI-FRAME 5
 κ (2κ2 +(κ −κ )2) 
2 2 1 3
((κ(cid:48) +κ(cid:48))κ +2κ(cid:48) (κ +κ )+(κ −κ )(κ2 +κ2 +κ2)−κ(cid:48)(cid:48) +κ(cid:48)(cid:48))
√  1 3 2 2 1 3 1 3 1 2 3 1 3 
2  −((−κ(cid:48) +κ(cid:48))κ +κ (2κ2 +(κ −κ )2)+κ(cid:48) (κ −κ )) 
τ =  1 3 2 1 2 1 3 2 1 3 ;
β µ  (−κ(cid:48) (κ −3κ )−2κ(cid:48)κ +κ (3κ(cid:48) +κ2 +κ2 +κ2)−κ(cid:48)(cid:48)) 
2  3 1 3 1 3 2 2 1 2 3 2 
 −((−κ(cid:48) +κ(cid:48))κ +κ(cid:48) (κ −κ )+(2κ2 +(κ −κ )2)κ ) 
1 3 2 2 1 3 2 1 3 3
(−2κ(cid:48)κ +κ(cid:48) (3κ −κ )−κ (−3κ(cid:48) +κ2 +κ2 +κ2)+κ(cid:48)(cid:48))
3 1 1 1 3 2 2 1 2 3 2
 
((−κ(cid:48) +κ(cid:48))κ +κ (2κ2 +(κ −κ )2)+κ(cid:48) (κ −κ ))2
1 3 2 1 2 1 3 2 1 3
(3.7) µ2 =  +κ22(2κ22 +(κ1 −κ3)2)2 .
+((−κ(cid:48) +κ(cid:48))κ +κ(cid:48) (κ −κ )+(2κ2 +(κ −κ )2)κ )2
1 3 2 2 1 3 2 1 3 3
3.1.3. e e -Smarandache curves . Let r = r(s) be a space curve in E3 with Q- frame {e (s),
2 3 1
e (s),e (s)}. Thenthee e −Smarandachecurvesofr aredefinedby
2 3 2 3
1
(3.8) γ(s¯) = √ (e +e ).
2 3
2
Thecurvaturefunctionsκ andτ ofγ canbecomputedasfollows:
γ γ
 1
((κ +κ )2κ +2κ 3)2 2
1 1 2 3 3
κγ = ν  +(κ(cid:48)3(κ1 +κ2)−(κ(cid:48)1 +κ(cid:48)2)κ3 +κ1((κ1 +κ2)2 +2κ23))2  ;
1 +(−κ(cid:48) (κ +κ )+(κ(cid:48) +κ(cid:48))κ +κ ((κ +κ )2 +2κ2))2
3 1 2 1 2 3 2 1 2 3
(cid:18) (cid:19)
1
(3.9) ν = 2 (κ +κ )2 +κ2 3/2.
1 2 1 2 3
 −(κ(cid:48) (κ +κ )−(κ(cid:48) +κ(cid:48))κ +κ ((κ +κ )2 +2κ2)) 
3 1 2 1 2 3 1 1 2 3
(2κ(cid:48)κ +κ(cid:48) (κ +3κ )+κ (3κ(cid:48) +κ2 +κ2 +κ2)−κ(cid:48)(cid:48))
√  1 2 2 1 2 3 3 1 2 3 3 
2  +(−κ(cid:48) (κ +κ )+(κ(cid:48) +κ(cid:48))κ +κ ((κ +κ )2 +2κ2)) 
τ =  3 1 2 1 2 3 2 1 2 3 ;
γ ν  (2κ(cid:48)κ +κ(cid:48) (3κ +κ )−κ (−3κ(cid:48) +κ2 +κ2 +κ2)+κ(cid:48)(cid:48)) 
2  2 1 1 1 2 3 3 1 2 3 3 
 +κ ((κ +κ )2 +2κ2) 
3 1 2 3
(2κ(cid:48) (κ −κ )+(κ(cid:48) −κ(cid:48))κ +(κ +κ )(κ2 +κ2 +κ2)−κ(cid:48)(cid:48) −κ(cid:48)(cid:48))
3 1 2 1 2 3 1 2 1 2 3 1 2
 
((κ +κ )2κ +2κ 3)2
1 2 3 3
(3.ν120) =  +(κ(cid:48)3(κ1 +κ2)−(κ(cid:48)1 +κ(cid:48)2)κ3 +κ1((κ1 +κ2)2 +2κ23))2 .
+(−κ(cid:48) (κ +κ )+(κ(cid:48) +κ(cid:48))κ +κ ((κ +κ )2 +2κ2))2
3 1 2 1 2 3 2 1 2 3
3.1.4. e e e -Smarandachecurves. Letr = r(s)beaspacecurveinE3 withQ-frame{e (s),
1 2 3 1
e (s),e (s)}. Thenthee e e −Smarandachecurvesofr aredefinedby
2 3 1 2 3
1
(3.11) δ(s¯) = √ (e +e +e ).
1 2 3
3
Thecurvaturefunctionsκ andτ ofδ canbecomputedasfollows:
δ δ
AJMAA,Vol.17(2020),No.1,Art.11,16pp. AJMAA
6 M.KHALIFASAADANDR.A.ABDEL-BAKY
 (cid:18) −κ(cid:48) (κ +κ )+κ(cid:48) (−κ +κ )+κ(cid:48)κ +κ (cid:19)2 21
3 1 2 2 1 3 1 2 3
 +2κ (κ2 +κ2 +κ (κ −κ )+κ κ +κ2) 
√  2 1 2 1 2 3 2 3 3 
3  (cid:18) κ(cid:48) (κ +κ )+κ(cid:48) (κ −κ )−κ(cid:48) (κ +κ ) (cid:19)2 
κ = √  + 3 1 2 2 1 3 1 2 3  ;
δ 2ξ  +2κ (κ2 +κ2 +κ (κ −κ )+κ κ +κ2) 
1  3 1 2 1 2 3 2 3 3 
 +(cid:18) κ(cid:48)3(κ1 +κ2)+κ(cid:48)2(κ1 −κ3)−κ(cid:48)1(κ2 +κ3) (cid:19)2 
+2κ (κ2 +κ κ +κ2 +(−κ +κ )κ +κ2)
1 1 1 2 2 1 2 3 3
(cid:0) (cid:0) (cid:1) (cid:1)
(3.12) ξ = 2 κ2 +κ2 +κ (κ −κ )+κ κ +κ2 3/2 .
1 1 2 1 2 3 2 3 3
 (cid:18) (cid:19) 
κ(cid:48) (κ +κ )+κ(cid:48) (κ −κ )−κ(cid:48) (κ +κ )
− 3 1 2 2 1 3 1 2 3
 −2κ2(κ21 +κ22 +κ1(κ2 −κ3)+κ2κ3 +κ23) 
 (cid:18) (cid:19) 
 κ(cid:48) (3κ +κ )+2κ(cid:48) (κ +κ )+κ(cid:48) (κ +3κ ) 
 1 1 2 2 1 3 3 2 3 
 +(κ1 −κ3)(κ21 +κ22 +κ23)−κ(cid:48)1(cid:48) +κ(cid:48)3(cid:48) 
 (cid:18) (cid:19) 
√  + κ(cid:48)3(κ1 +κ2)+κ(cid:48)2(κ1 −κ3)−κ(cid:48)1(κ2 +κ3) 
τδ = ξ3  (cid:18) 2κ(cid:48)+(κ2κ−3(κκ21)++κκ22(cid:48)+(−κ31κ(κ2+−κκ)3)−+κκ(cid:48) 2(κ3κ3 ++κκ23)) (cid:19) ;
2  3 1 2 1 1 3 2 2 3 
 +(κ1 +κ2)(κ21 +κ22 +κ23)−κ(cid:48)1(cid:48) −κ(cid:48)2(cid:48) 
 (cid:18) (cid:19) 
 κ(cid:48) (κ +κ )+κ(cid:48) (κ −κ )−κ(cid:48) (κ +κ ) 
 − 3 1 2 2 1 3 1 2 3 
 +2κ1(κ21 +κ1κ2 +κ22 +(−κ1 +κ2)κ3 +κ23) 
 (cid:18) (cid:19) 
 κ(cid:48)2(κ1 +3κ2)−κ(cid:48)3(κ1 −3κ3)+2κ(cid:48)1(κ2 −κ3) 
+(κ +κ )(κ2 +κ2 +κ2)−κ(cid:48)(cid:48) −κ(cid:48)(cid:48)
2 3 1 2 3 3 2
 (cid:18) −κ(cid:48) (κ +κ )+κ(cid:48) (−κ +κ )+κ(cid:48) (κ +κ ) (cid:19)2 
3 1 2 2 1 3 1 2 3
 +2κ (κ2 +κ2 +κ (κ −κ )+κ κ +κ2) 
 2 1 2 1 2 3 2 3 3 
 (cid:18) κ(cid:48) (κ +κ )+κ(cid:48) (κ −κ )−κ(cid:48) (κ +κ ) (cid:19)2 
(3.13) ξ =  + 3 1 2 2 1 3 1 2 3 .
2  +2κ (κ2 +κ2 +κ (κ −κ )+κ κ +κ2) 
 3 1 2 1 2 3 2 3 3 
 +(cid:18) κ(cid:48)3(κ1 +κ2)+κ(cid:48)2(κ1 −κ3)−κ(cid:48)1(κ2 +κ3) (cid:19)2 
+2κ (κ2 +κ κ +κ2 +(−κ +κ )κ +κ2)
1 1 1 2 2 1 2 3 3
3.2. Examples.
Example3.1. Assumethatthecurver(s)isgivenby
r = (sin(es),sin(es),es).
ItisverifythattheSerret–Frenetcurvatureandtorsionofthiscurveareobtainedby
√
2sin(es)
κ = and τ = 0,
(2+cos(2es))3/2
respectively.
Hence τ = 0; the angle between the rotation minimizing frame (Bishop frame) and the Serret-
Frenet frame is constant, therefore the Bishop frame is also not suitable for this example. The
quasi-framecanbecalculatedby
(cid:32) (cid:33)
1 cos(es)
e = − ,0, ,
1 (cid:112) (cid:112)
1+cos2(es) 1+cos2(es)
(cid:32) (cid:33)
1 cos(es)
e = − ,0, ,
2 (cid:112) (cid:112)
1+cos2(es) 1+cos2(es)
AJMAA,Vol.17(2020),No.1,Art.11,16pp. AJMAA
RULEDSURFACESACCORDINGTOQUASI-FRAME 7
 
cos2(es) 1 cos(es)
e3 = (cid:112) (cid:112) ,−(cid:113) , (cid:112) (cid:112) .
1+cos2(es) 2+cos(2es) 2− 2 1+cos2(es) 2+cos(2es)
3+cos(2es)
Also,thequasi-curvaturefunctionsareobtainedasfollows
es(sin(es)+cos(es)sin(2es))
κ = ,
1 (cid:112)1+cos2(es)(2+cos(2es))3/2
escos2(es)sin(es) essin(es)
κ = − +
2 (cid:112)1+cos2(es)(2+cos(2es))2 (2+cos(2es))3/2(cid:113)2− 2
3+cos(2es)
escos(es)sin(2es)
+ ,
(cid:112)1+cos2(es)(2+cos(2es))2
essin(2es)
κ = − .
3 (cid:112)
2+cos(2es)(3+cos(2es))
Now, we can construct the following Smarandache curves according to quasi-frame vectors
asfollows:
1. e e −Smarandachecurve
1 2
Thee e −Smarandachecurveα isgivenby
1 2
e +e
α = 1√ 2,
2
andcanbecalculatedasfollows
 (cid:18) (cid:19) 
√1 −√ 1 + √cos(es)
 2 1+cos2(es) 2+cos(2es) 
α =  √ √cos(es) .
 2 2+cos(2es) 
 (cid:18) (cid:19) 
 √1 √cos(es) + √ 1 
2 1+cos2(es) 2+cos(2es)
UsingEqs. (3.3)and(3.4),thecurvaturesofα arecomputedasfollows
(cid:118)
(cid:117) e6s(12+9cos(2es)+cos(4es))
(cid:117)
(cid:116) (cid:15)2sin6(es)
1
(6+5cos(2es)+cos2(2es))6
κ = √ √ ;
(cid:18) (cid:16) √ (cid:17) (cid:19)3/2
e2s 37+28cos(2es)−4 2cos(es) 2+cos(2es) 3+cos(2es)+3cos(4es) sin2(es)
4
(13+10cos(2es)+cos(4es))2
√
(cid:18) (cid:112) (cid:112) (cid:19)
37+28cos(2es)−4 2cos(es) 2+cos(2es) 3+cos(2es)
(cid:15) = ,
1 +3cos(4es)
τ = 0.
Theruledsurfacegeneratedbythebasecurver andgeneratorα isgivenby
M : Ψ (s,t) = β +tα;t ∈ R,
1 1
AJMAA,Vol.17(2020),No.1,Art.11,16pp. AJMAA
8 M.KHALIFASAADANDR.A.ABDEL-BAKY
Figure1: Thequasi-sphericalcurveα.
whichiscalculatedasfollows
 √tcos(es) − √ t +sin(es) 
√
2 2+cos(2es) 3+cos(2es)
Ψ (s,t) =  √ √tcos(es) +sin(es) .
1  2 2+cos(2es) 
 es + √ t + √tcos(es) 
√
2 2+cos(2es) 3+cos(2es)
TheruledsurfacesaccordingtoSerret-FrenetandquasiframesareshowninFigures2aand2b,
respectively.
(A) (B)
Figure2: (A)Ψ withFrenetvectors. (B)Ψ withquasi-normalvectors.
1 1
2. e e −Smarandachecurves
1 3
Thee e −Smarandachecurveβ isgivenby
1 3
e +e
β = 1√ 3,
2
AJMAA,Vol.17(2020),No.1,Art.11,16pp. AJMAA
RULEDSURFACESACCORDINGTOQUASI-FRAME 9
andcanbecalculatedasfollows
√
 (cid:16) (cid:17) 
cos(es) cos(es)+ 1+cos2(es)
√ √
 2+cos(2es) 3+cos(2es) 
 (cid:18) (cid:19) 
β =  √1 √cos(es) − 1 .
(cid:113)
 2 2+cos(2es) 2− 2 
 √ 3+cos(2es) 
 √cos(es)+ √1+cos2(es) 
2+cos(2es) 3+cos(2es)
FromEqs. (3.6)and(3.7),thecurvaturesofβ arecomputedas
(cid:118)
(cid:117) e6s(cid:15)2
(cid:117) 2
(cid:116) (81+79cos(2es)+15cos(4es)+cos(6es))sin6(es)
(2+cos(2es))6(3+cos(2es))7
κ = √ ;
(cid:18) (cid:16) √ (cid:17) (cid:19)3/2
e2s 15+10cos(2es)+ 2 3+cos(2es)(5cos(es)+cos(3es))+cos(4es) sin2(es)
(6+5cos(2es)+cos2(2es))2
(cid:16) √ (cid:112) (cid:17)
(cid:15) = 15+10cos(2es)+ 2 3+cos(2es)(5cos(es)+cos(3es))+cos(4es) ,
2
τ = 0.
Figure3: Thequasi-sphericalcurveβ.
Theruledsurfacegeneratedbythebasecurver andgeneratorβ isgivenby
M : Ψ (s,t) = r+tβ;t ∈ R,
2 2
whichiscalculatedasfollows
√
 (cid:16) (cid:17) 
tcos(es) cos(es)+ 1+cos2(es)
√ √ +sin(es)
 2+cos(2es)√3+cos(2es) 
 (cid:16)√ (cid:17) 
 t 2cos(es)− 3+cos(2es) 
Ψ (s,t) =  √ +sin(es) .
2
 2 2+cos(2es)√ 
 (cid:16) (cid:17) 
 t cos(es)+ 1+cos2(es) 
es + √ √
2+cos(2es) 3+cos(2es)
TheruledsurfacesaccordingtoSerret–FrenetandquasiframesareshowninFigures4aand4b,
respectively.
AJMAA,Vol.17(2020),No.1,Art.11,16pp. AJMAA
10 M.KHALIFASAADANDR.A.ABDEL-BAKY
(A) (B)
Figure4: (A)Ψ withFrenetvectors. (B)Ψ withquasi-normalvectors.
2 2
3. e e −Smarandachecurves
2 3
Thee e −Smarandachecurveγ isgivenby
2 3
e +e
γ = 2√ 3,
2
andcanbecalculatedasfollows
√
 
√cos2(es)− √2+cos(2es)
 2+cos(2es) 3+cos(2es) 
 − 1 
γ =  (cid:16)2(cid:113)√23++ccooss((22eess)) (cid:17) .
 cos(es) 1+ 2+cos(2es) 
√ √
2+cos(2es) 3+cos(2es)
AccordingtoEqs. (3.9)and(3.10),thecurvaturesofγ arecomputedas
√ (cid:113)
2 e6s(12+9cos(2es)+cos(4es))(cid:15)3sin6(es)
(13+10cos(2es)+cos(4es))6
κ = √ √ ;
(cid:18) (cid:16) (cid:16) (cid:17) (cid:17) (cid:19)3/2
e2s 29+12 2+cos(2es)+4cos(2es) 6+ 2+cos(2es) +3cos(4es) sin2(es)
(13+10cos(2es)+cos(4es))2
 (cid:112) 
2971+1584 2+cos(2es)+8cos(2es)
(cid:16) (cid:17)
(cid:112)
 453+205 2+cos(2es) 
 
(cid:15)3 =  +4(cid:16)263+84(cid:112)2+cos(2es)(cid:17)cos(4es) ,
 
 (cid:16) (cid:17) 
(cid:112)
+8 19+3 2+cos(2es) cos(6es)+9cos(8es)
τ = 0.
Theruledsurfacegeneratedbythebasecurver andgeneratorγ isgivenby
M : Ψ (s,t) = r+tγ;t ∈ R,
3 3
AJMAA,Vol.17(2020),No.1,Art.11,16pp. AJMAA