Mannheim Partner Curve a Different View

2017·0.16 MB·

by  Suleyman Senyurt

International J.Math. Combin. Vol.2(2017), 84-91
Mannheim Partner Curve a Different View
Su¨leyman S¸enyurt, Yasin Altun and Ceyda Cevahir
(FacultyofArtsandSciences,DepartmentofMathematics,OrduUniversity,52200,Ordu/Turkey)
E-mail: [email protected],[email protected],[email protected]
Abstract: Inthisstudy,weinvestigated special Smarandachecurvesbelonging toSabban
frame drawn on the surface of the sphere by Darboux vector of Mannheim partner curve .
We created Sabban frame belonging to this curve. It were explained Smarandache curves
position vector is consisted by Sabban vectors belonging to this curve. Then, we calculated
geodesiccurvaturesofthisSmarandachecurves. Foundresultswereexpresseddependingon
theMannheim curve.
Key Words: Mannheim curvepair,Darboux vector,Smarandachecurves,Sabban frame,
geodesic curvature.
AMS(2010): 53A04.
§1. Introduction and Preliminaries
Let α : I E3 be a unit speed curve denote by T,N,B,p,q the moving Frenet apparatus.
→ { }
The Frenet formulae is given by ([5])
T′(s)=p(s)N(s)
N′(s)= p(s)T(s)+q(s)B(s) (1.1)
B′(s)= −q(s)N(s).
−

The Darboux vector defined by
W =qT+pB.
BytheunitDarbouxvector, wehave
q p
sinϕ= , cosϕ=
p2+q2 p2+q2
including p p
C =sinϕT +cosϕB
where ∠(W,B) = ϕ, ([4]). Let α and α be the C2-class differentiable two curves and T (s),
1 1
N (s), B (s) be the Frenet vectors of α . If the binormal vector of the curve α is linearly
1 1 1 1
dependentontheprincipalnormalvectorofthecurveα,then(α)isdefinedaMannheimcurve
1ReceivedJuly12,2016,AcceptedMay23,2017.
MannheimPartnerCurveaDifferentView 85
and (α ) a Mannheim partner curve of (α). The relations between the Frenet vectors we can
1
write [3, 6, 7]
T =cosσT sinσB
1 −
N =sinσT +cosσB (1.2)
B1 =N
1
and for the curvatures we get 
pσ′
p=
λq p2+q2 (1.3)
 p
q = p.
λq
Let (α,α ) be a curve pair in E3. For the vector C is the direction of the Mannheim
1 1
partner curve α we have [3,8]
1
p2+q2 σ′
C = C+ N. (1.4)
1 pp2+q2+σ′2 p2+q2+σ′2
p p
Let ω :I S2 be a unit speed spherical curve. We can write ([10])
→
ω(s)=ω(s), t(s)=ω′(s), d(s)=ω(s) t(s), (1.5)
∧
where, ω(s),t(s),d(s) frame is expressed the Sabban frame of ω on S2. Then we have equa-
{ }
tions ([10])
ω′(s)=t(s), t′(s)= ω(s)+κ (s)d(s), d′(s)= κ (s)t(s), (1.6)
g g
− −
where κ is expressed the geodesic curvature of the curve ω on S2 which is ([10])
g
κ (s)= t′(s),d(s) . (1.7)
g
h i
§2. Mannheim Partner Curve a Different View
Let (C ) be a unit speed spherical curve on S2. Then we can write
1
C =sinϕT +cosϕB ,
1 1 1
T =cosϕT sinϕB , (2.1)
CC1 T =N1 −, 1
1 ∧ C1 1
where ∠(C ,B ) = ϕ. Then fromthe equation (1.6) we have the following equations of (C )
1 1 1
are
C ′ =T
1 C1
 p2+q2
TC′1 =−C1 + p ϕp′2+qC21 ∧TC1 (2.2)
(C T )′ = T .
1 ∧ C1 − ϕ′ C1
 p
86 Su¨leymanS¸enyurt,YasinAltunandCeydaCevahir
From the equation (1.7), we have the following geodesic curvature of (C ) is
1
p2+q2
κ = T′ ,C T = κ = . (2.3)
g h C1 1 ∧ C1i ⇒ g p ϕ′
2.1 The β -Smarandache curve can be defined by
1
1
β (s)= (C +T ) (2.4)
1 √2 1 C1
or substituting the equation (2.1) into equation (2.4) we obtain
1
β (s)= (sinϕ+cosϕ)T +(cosϕ sinϕ)B . (2.5)
1 √2 1 − 1
(cid:16) (cid:17)
Differentiating (2.4) we can write
ϕ′(cosϕ sinϕ) p2+q2 ϕ′(cosϕ+sinϕ)
T = − T + N B . (2.6)
β1 2ϕ′2+p2+q2 1 2ϕp′2+p2+q2 1 − 2ϕ′2+p2+q2 1
q q q
Considering the equations (2.5) and (2.6) we get
p2+q2(cosϕ+sinϕ) ϕ′ p2+q2(cosϕ+sinϕ)
β1∧Tβ1 = p 2p2+2q2+4ϕ′2 T1 − 2p2+2q2+4ϕ′2N1 + p 2p2+2q2+4ϕ′2 B1. (2.7)
p p p
Differentiating (2.6) where coefficients
p2+q2 p2+q2 p2+q2
χ = 2 ( )2+( )′( )
1 − − ϕ′ ϕ′ ϕ′
p p p
p2+q2 p2+q2 p2+q2 p2+q2
χ = 2 3( )2 ( )4 ( )′( ) (2.8)
2 − − ϕ′ − ϕ′ − ϕ′ ϕ′
p p p p
p2+q2 p2+q2 p2+q2
χ = 2( )+( )3+( )′
3 ϕ′ ϕ′ ϕ′
p p p
including we can reach,
Tβ′1 = ϕ′4√2p(χ21+siqn2ϕ++ϕ′χ22c2osϕ)T1+ p2+χ3qϕ2′4+√ϕ2′2 2N1 + ϕ′4√2p(χ21+cqo2sϕ+−ϕ′χ222sinϕ)B1. (2.9)
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
From the equation (2.7) and (2.9) κβg1 geodesic curvature for Mannheim partner curve β1
is
κβ1 = T′ ,(C T )
g h β1 1 ∧ C1 β1i
1 p2+q2 p2+q2
= χ χ +2χ . (2.10)
2+ √pϕ2+′ q2 2)52(cid:16)p ϕ′ 1− p ϕ′ 2 3(cid:17)
(cid:0) (cid:0) (cid:1)
From the equation (1.2) and (1.3) Sabban apparatus of the β -Smarandache curve for
1
MannheimPartnerCurveaDifferentView 87
Mannheim curve are
(σ′+ p2+q2)cosσ σ′ p2+q2 (σ′+ p2+q2)sinσ
β (s) = T − N B,
1
2σ′2+2p2+q2 − 2σ′2+2p2+q2 − 2σ′2+2p2+q2
p p p
(σp′ p2+q2)ηcosσ pp2+q2+σ′2cosσ p η(σ′+ p2+q2)
T = − − T + N
β1 p p2+q2+σ′2p1+2η2 p2+q2+pσ′2 1+2η2
( p2+qp2 σ′)ηsinσ+p p2+q2+σ′2cosσ p p
+ − B,
p2+q2+σ′2 1+2η2
p p
p p
( p2+q2 σ′)cosσ+2η p2+q2+σ′2sinσ σ′+η p2+q2
β T = − T + N
1∧ β1 2+4η2 p2+q2+σ′2 2+4η2 p2+q2+σ′2
p p p
( p2+qp2+σ′)sinpσ 2η p2+q2+σ′2cosσ p p
− B,
− p 2+4η2 p2+pq2+σ′2
p p
(χ p2+q2+χ σ′)η4√2cosσ χ η4 2p2+q2+2σ′2sinσ
T′ = 1 2 − 3 T
β1 p (1+2η2)2 p2+q2+pσ′2
(χ σ′ χ p2+q2)η4√2p
+ 1 − 2 N
(1+2η2)2 p2+q2+σ′2
p
(χ σ′ χ pp2+q2)η4√2sinσ+χ η4 2p2+q2+2σ′2cosσ
+ 2 − 1 3 B
p (1+2η2)2 p2+q2+pσ′2
p
and
1 1 1
κβ1 = χ χ +2χ , (2.11)
g 2+ 1 52 η 1− η 2 3
η2 (cid:16) (cid:17)
where (cid:0) (cid:1)
1 (ϕ)′ p2+q2 ′λq p2+q2
= = (2.12)
η p2+q2 σ′2+p2+q2 σ′
p p
(cid:16) (cid:17)
and p p
χ = 2 1 + 1 1
1 − − η2 η′η
χ = 2 3 1 1 1 1 (2.13)
χ2 =−21 +− 1η2+−2η14 − η′η
3 η η3 η′

2.2 The β -Smarandache curve can be defined by
2
1
β (s)= (C +C T ) (2.14)
2 √2 1 1 ∧ C1
88 Su¨leymanS¸enyurt,YasinAltunandCeydaCevahir
or from the equation (1.2), (1.4) and (2.1) we can write
p2+q2cosσ+ σ′2+p2+q2sinσ σ′
β (s) = T + N
2
2σ′2+2p2+q2 2σ′2+2p2+q2
p p
p p
σ′2+p2+q2cosσ p2+q2sinσ
+ − B. (2.15)
2σ′2+2p2+q2
p p
p
Differentiating (2.15) we can write
σ′cosσ p2+q2 σ′sinσ
T = T N B. (2.16)
β2 p2+q2+σ′2 − pp2+q2+σ′2 − p2+q2+σ′2
p p p
Considering the equations (2.15) and (2.16) it is easily seen
p2+q2+σ′2sinσ p2+q2cosσ σ′
β T = − T N
2∧ β2 2p2+q2+2σ′2 − 2p2+q2+2σ′2
p p
p p
p2+q2sinσ+ p2+q2+σ′2cosσ
+ B. (2.17)
p 2p2+pq2+2σ′2
p
Differentiating (2.16) we can write
2p2+q2+2σ′2sinσ η√2 p2+q2cosσ η√2σ′
T′ = − T + N
β2 p (η 1) p2+q2+pσ′2 (η 1) p2+q2+σ′2
− −
p p
η√2 p2+q2sinσ+ 2p2+q2+2σ′2cosσ
+ B, (2.18)
(η 1) p2+q2+σ′2
p p
−
p
where, κβg2 geodesic curvature for β2 is
1+η
κβ2 = . (2.19)
g η 1
−
2.3 The β -Smarandache curves can be defined by
3
1
β (s)= (T +C T ) (2.20)
3 √2 C1 1 ∧ C1
or from the equation (2.1), (1.4) and (1.2) we can write
σ′cosσ+ p2+q2+σ′2sinσ p2+q2
β (s) = T + N
3
2σp′2+2p2+q2 2σp′2+2p2+q2
p2p+q2+σ′2cosσ σ′sinσ p
+ − B. (2.21)
2σ′2+2p2+q2
p
p
MannheimPartnerCurveaDifferentView 89
Differentiating (2.21) we reach
p2+q2+σ′2sinσ (σ′+η p2+q2)cosσ p2+q2 ησ′
T = − T + − N
β3 2+η2 p2+q2+σ′2 2+η2 p2+q2+σ′2
p p p
(η p2+pq2+σ′)psinσ+ p2+q2+σ′2cosσ p p
+ B. (2.22)
2+η2 p2+q2+σ′2
p p
p p
Considering the equations (2.21) and (2.22) it is easily seen
(2 p2+q2 ησ′)cosσ+η p2+q2+σ′2sinσ
β T = − T
3∧ β3 p 4+2η2 p2+pq2+σ′2
2σ′ pη p2+qp2
+ − N
4+2η2 pp2+q2+σ′2
p(ησ′ 2 pp2+q2)sinσ+η p2+q2+σ′2cosσ
+ − B. (2.23)
4+2η2 p2+q2+σ′2
p p
p p
Differentiating (2.22) where
δ = 1 +2 1 +21 1
1 η η3 η′η
δ = 1 3 1 2 1 1 (2.24)
δ2 =− 1− 2η21−+η14 − η′
3 −η2 − η4 η′

including we have
(δ p2+q2 δ σ′)η4√2cosσ+δ η4 2p2+q2+2σ′2sinσ
T′ = 2 − 1 3 T
β3 p (2+η2)2 σ′2+p2+pq2
η4√2(δ σ′+δ p2+q2)p
2 1
+ N
(2+η2)2 σ′2+p2+q2
p
(δ σ′ δ pp2+q2)η4√2sinσ+δ η4 2p2+q2+2σ′2cosσ
1 2 3
+ − B, (2.25)
(2+η2)2 σ′2+p2+q2
p p
p
where, κβg3 geodesic curvature for Mannheim curve β3 is
1
κβ3 = 2η5δ η4δ +η4δ . (2.26)
g (2+η2)52 1− 2 3
(cid:0) (cid:1)
2.4 The β -Smarandache curves can be defined by
4
1
β (s)= (C +T +C T ) (2.27)
4 √3 1 C1 1 ∧ C1
90 Su¨leymanS¸enyurt,YasinAltunandCeydaCevahir
or from the equation (1.2), (1.4) and (2.1) we can write
(σ′+ p2+q2)cosσ+ σ′2+p2+q2sinσ σ′ p2+q2
β (s) = T + − N
4
3σ′2+3p2+q2 3σ′2+3p2+q2
p p p
σ′2+p2+pq2cosσ (σ′+ p2+q2)sinσ p
+ − B. (2.28)
3σ′2+3p2+q2
p p
p
Differentiating (2.28), we reach
(η 1)σ′ η p2+q2 cosσ+ p2+q2+σ′2sinσ
T = − − T
β4 (cid:0) 2p(1 η+η(cid:1)2) p2+pq2+σ′2
−
ησ′ p(1 η) p2+qp2
+ − − N
2(1 η+η2) p2+q2+σ′2
p
−
pη p2+q2+(1p η)σ′ sinσ+ p2+q2+σ′2cosσ
+ − B. (2.29)
(cid:0) p 2(1 η+η(cid:1)2) p2+pq2+σ′2
−
p p
Considering the equations (2.28) and (2.29) it is easily seen
((2 η) p2+q2 (1+η)σ′)cosσ+(2η1) p2+q2+σ′2sinσ
β T = − − T
4∧ β4 6 6η+6η2 p2+q2+σ′2
p p
−
(2 η)σ′+(1p+η) p2+q2 p
+ − N (2.30)
6 6η+6η2 p2+q2+σ′2
p
−
p((η 2) p2+pq2 (1+η)σ′)sinσ+(2η 1) p2+q2+σ′2cosσ
+ − − − B.
p 6 6η+6η2 p2+q2+σp′2
−
p p
Differentiating (2.29) where
ρ = 2+41 4 1 +2 1 +21 21 1
1 − η − η2 η3 η′ η −
ρρ23 ==−2η12−+42ηη112−+44η1η213+−22η1η314−+2η1η1′4(cid:0)2−−η1′η1(cid:0)1(cid:1)+ η1(cid:1) (2.31)
 (cid:0) (cid:1)
including we can write
(ρ p2+q2+ρ σ′)η4√3cosσ+ρ η4 3p2+q2+3σ′2sinσ
T′ = 1 2 3 T
β4 4(1 η+η2)2 σ′2+p2+q2
p p
−
+ (ρ1σ′−ρ2 p2+q2)η4√3pN (2.32)
4(1 η+η2p)2 σ′2+p2+q2
−
+ρ3η4 3p2+q2p+3σ′2cosσ−(ρ1 p2+q2+ρ2σ′)η4√3sinσB,
p 4(1 η+η2)2 σp′2+p2+q2
−
p
MannheimPartnerCurveaDifferentView 91
where, κβg4 geodesic curvature for Mannheim curve β4(sβ4) is
(2η4 η5)ρ (η5+η4)ρ +(2η5 η4)ρ
κβ4 = − 1− 2 − 3. (2.33)
g 4√2(1 η+η2)52
−
References
[1] Ali A.T., Special Smarandache curves in the Euclidian space, International Journal of
Mathematical Combinatorics, 2, 30-36,2010.
[2] C¸alı¸skan A. and S¸enyurt S., Smarandache curves in terms of Sabban frame of spherical
indicatrix curves, Gen. Math. Notes, 31(2), 1-15, 2015.
[3] C¸alı¸skan A. and S¸enyurt S., N∗C∗-Smarandache curves of Mannheim curve couple ac-
cording to Frenet frame, International Journal of Mathematical Combinatorics, 1, 1-13,
2015.
[4] Fenchel, W.,On the Differential Geometry of closed space curves, Bulletinofthe American
Mathematical Society, 57, 44-54, 1951.
[5] HacısalihoˇgluH.H.,Differential Geometry(in Turkish),AcademicPressInc. Ankara,1994.
[6] Liu H. and Wang, F., Mannheim partner curves in 3-space,Journal of Geometry, 88(1-2),
120-126,2008.
[7] OrbayK. and KasapE., On Mannheim partner curvesE3, International Journal of Phys-
ical Sciences, 4(5), 261-264,2009.
[8] S¸enyurt S., Natural lifts and the geodesic sprays for the spherical indicatrices of the
Mannheim partner curves in E3, International Journal of the Physical Sciences, 7(16),
2414-2421,2012.
[9] Turgut M. and Yılmaz S., Smarandache curves in Minkowski space-time, International
Journal of Mathematical Combinatorics, 3, 51-55, 2008.
[10] Ta¸sk¨opru¨K.andTosunM.,SmarandachecurvesonS2, Boletim da Sociedade Paranaense
de Matematica 3 Srie., 32(1), 51-59,2014.