Smarandache Curves of a Spacelike Curve According to the Bishop Frame of Type-2

2016·0.19 MB·

by  Yasin Unluturk

International J.Math. Combin. Vol.4(2016), 29-43
Smarandache Curves of a
Spacelike Curve According to the Bishop Frame of Type-2
Yasin U¨nlu¨tu¨rk
DepartmentofMathematics,KırklareliUniversity,39100Kırklareli,Turkey
Su¨ha Yılmaz
DokuzEylu¨lUniversity,BucaFacultyofEducation
DepartmentofElementaryMathematicsEducation,35150,Buca-Izmir,Turkey
E-mail: [email protected],[email protected]
Abstract: In this study, we introduce new Smarandache curves of a spacelike curve ac-
cordingtotheBishop frameoftype-2inE3. Also, Smarandachebreadthcurvesaredefined
1
according to this frame in Minkowski 3-space. A third order vectorial differential equation
of position vector of Smarandache breadth curveshas been obtained in Minkowski 3-space.
Key Words: Smarandache curves, the Bishop frame of type-2, Smarandache breadth
curves, Minkowski 3-space.
AMS(2010): 53A05, 53B25, 53B30
§1. Introduction
Bishop frame extended to study canal and tubular surfaces [1]. Rotating camera orientations
relative to a stable forward-facing frame can be added by various techniques such as that of
Hanson and Ma [2]. This special frame also extended to height functions on a space curve [3].
The constructionofthe Bishopframe is due to L. R.Bishopandthe advantagesofBishop
frame,andcomparisonsofBishopframewiththeFrenetframeinEuclidean3-spaceweregiven
by Bishop [4] and Hanson [5]. That is why he defined this frame that curvature may vanish
at some points on the curve. That is, second derivative of the curve may be zero. In this
situation, an alternative frame is needed for non continously differentiable curves on which
Bishop (parallel transport frame) frame is well defined and constructed in Euclidean and its
ambient spaces [6,7,8].
A regular curve in Euclidean 3-space, whose position vector is composed of Frenet frame
vectors on another regular curve, is called a Smarandache curve. M. Turgut and S. Yılmaz
have defined a special case of such curves and call it Smarandache TB curves in the space E4
2 1
([9]) and Turgut also studied Smarandache breadth of pseudo null curves in E4 ([10]). A.T.Ali
1
hasintroducedsomespecialSmarandachecurvesintheEuclideanspace[11]. Moreover,special
1ReceivedApril3,2016,Accepted November6,2016.
30 YasinU¨nlu¨tu¨rkandSu¨haYılmaz
Smarandache curves have been investigated by using Bishop frame in Euclidean space [12].
Special Smarandache curves according to Sabban frame have been studied by [13]. Besides,
some special Smarandache curves have been obtained in E3 by [14].
1
Curvesof constantbreadthwereintroduced by L.Euler [15]. Some geometric propertiesof
planecurvesofconstantbreadthweregivenin[16]. And, inanotherwork[17],theseproperties
werestudiedintheEuclidean3-spaceE3. Moreover,MFujivara[18]hadobtainedaproblemto
determine whether there exist space curve of constant breadth or not, and he defined breadth
forspacecurvesonasurfaceofconstantbreadth. In[19],thesekindcurveswerestudiedinfour
dimensional Euclidean space E4. In [20], Yılmaz introduced a new version of Bishop frame in
E3 and called it Bishop frame of type-2 of regular curves by using common vector field as the
1
binormal vector of Serret-Frenet frame. Also, some characterizations of spacelike curves were
given according to the same frame by Yılmaz and U¨nlu¨tu¨rk [21]. A regular curve more than
2 breadths in Minkowski 3-space is called a Smarandache breadth curve. In the light of this
definition, westudy specialcasesofSmarandachecurvesaccordingto the new frameinE3.We
1
investigate position vector of simple closed spacelike curves and give some characterizations in
caseofconstantbreadthaccordingtotype-2BishopframeinE3. Thus,weextendthisclassical
1
topic in E3 into spacelike curves of constant breadth in E3, see [22] for details.
1
In this study, we introduce new Smarandache curves of a spacelike curve according to the
Bishop frame of type-2 in E3. Also, Smarandache breadth curves are defined accordingto this
1
frame in Minkowski 3-space. A third order vectorial differential equation of position vector of
Smarandache breadth curves has been obtained in Minkowski 3-space.
§2. Preliminaries
To meet the requirements in the next sections, here, the basic elements of the theory of curves
intheMinkowski3–spaceE3 arebrieflypresented. Thereexistsavastliteratureonthesubject
1
including several monographs,for example [23,24].
The three dimensional Minkowski space E3 is a real vector space R3 endowed with the
1
standard flat Lorentzian metric given by
, = dx2+dx2+dx2,
h iL − 1 2 3
where (x ,x ,x ) is a rectangular coordinate system of E3. This metric is an indefinite one.
1 2 3 1
Let u = (u ,u ,u ) and v = (v ,v ,v ) be arbitrary vectors in E3, the Lorentzian cross
1 2 3 1 2 3 1
product of u and v is defined as
i j k
−
u v = det u u u .
× − 1 2 3
 v v v 
 1 2 3 
 
Recall that a vector v E3 has one of three Lorentziancharacters: it is a spacelike vector
∈ 1
if v,v > 0 or v = 0; timelike v,v < 0 and null (lightlike) v,v = 0 for v = 0. Similarly, an
h i h i h i 6
SmarandacheCurvesofaSpacelikeCurveAccordingtotheBishopFrameofType-2 31
arbitrarycurveδ =δ(s) in E3 canlocally be spacelike,timelike or null(lightlike) if its velocity
1
vector α are ,respectively, spacelike, timelike or null (lightlike), for every s I R. The
′
∈ ⊂
pseudo-norm of an arbitrary vector a E3 is given by a = a,a . The curve α =α(s) is
∈ 1 k k |h i|
calledaunitspeedcurveifitsvelocityvectorα isunitonei.e., α =1.Forvectorsv,w E3,
′ pk ′k ∈ 1
they are said to be orthogonal each other if and only if v,w = 0. Denote by T,N,B the
h i { }
moving Serret-Frenet frame along the curve α=α(s) in the space E3.
1
For an arbitrary spacelike curve α = α(s) in E3, the Serret-Frenet formulae are given as
1
follows
T 0 κ 0 T
′
 N = γκ 0 τ . N , (2.1)
′
 B   0 τ 0   B 
 ′     
     
where γ = 1, and the functions κ and τ are, respectively, the first and second (torsion)
∓
T (s) det(α,α ,α )
′ ′ ′′ ′′′
curvature. T(s)=α(s),N(s)= ,B(s)=T(s) N(s) and τ(s)= .
′ κ(s) × κ2(s)
If γ = 1, then α(s) is a spacelike curve with spacelike principal normal N and timelike
−
binormal B, its Serret-Frenet invariants are given as
κ(s)= T (s),T (s) and τ(s)= N (s),B(s) .
h ′ ′ i −h ′ i
p
If γ = 1, then α(s) is a spacelike curve with timelike principal normal N and spacelike
binormal B, also we obtain its Serret-Frenet invariants as
κ(s)= T (s),T (s) and τ(s)= N (s),B(s) .
−h ′ ′ i h ′ i
p
The LorentziansphereS2 ofradiusr >0 andwith the centerinthe originofthe spaceE3
1 1
is defined by
S2(r)= p=(p ,p ,p ) E3 : p,p =r2 .
1 { 1 2 3 ∈ 1 h i }
Theorem 2.1 Let α= α(s) be a spacelike unit speed curve with a spacelike principal normal.
If Ω ,Ω ,B is an adapted frame, then we have
1 2
{ }
Ω 0 0 ξ Ω
′1 1 1
 Ω = 0 0 ξ . Ω  (2.2)
′2 − 2 2
 B   ξ ξ 0   B 
 ′   − 1 − 2   
     
Theorem 2.2 Let T,N,B and Ω ,Ω ,B be Frenet and Bishop frames, respectively. There
1 2
{ } { }
exists a relation between them as
T sinhθ(s) coshθ(s) 0 Ω
1
 N = coshθ(s) sinhθ(s) 0 . Ω , (2.3)
2
 B   0 0 1   B 
     
     
32 YasinU¨nlu¨tu¨rkandSu¨haYılmaz
where θ is the angle between the vectors N and Ω .
1
ξ =τ(s)coshθ(s),ξ =τ(s)sinhθ(s).
1 2
s
Theframe Ω ,Ω ,B isproperlyoriented,andτ andθ(s)= κ(s)dsarepolarcoordinates
1 2
{ }
0
R
for the curve α = α(s). We shall call the set Ω ,Ω ,B,ξ ,ξ as type-2 Bishop invariants
1 2 1 2
{ }
of the curve α=α(s) in E3.
1
§3. Smarandache Curves of a Spacelike Curve
Inthissection,wewillcharacterizealltypesofSmarandachecurvesofspacelikecurveα=α(s)
according to type-2 Bishop frame in Minkowski 3-space E3.
1
3.1 Ω Ω Smarandache Curves
1 2
−
Definition 3.1 Let α = α(s) be a unit speed regular curve in E3 and Ωα,Ωα,B be its
1 { 1 2 α}
moving Bishop frame. ΩαΩα Smarandache curves are defined by
1 2−
β(s )= 1 (Ωα+Ωα). (3.1)
∗ √2 1 2
Now we can investigate Bishop invariants of ΩαΩα Smarandachecurves of the curveα=
1 2−
α(s). Differentiating (3.1) with respect to s gives
dβ ds 1
β˙ = . ∗ = (ξα ξα)B , (3.2)
ds ds √2 1 − 2 α
and
ds 1
T . ∗ = (ξα ξα)B ,
β ds √2 1 − 2 α
where
ds 1
∗ = ξα ξα . (3.3)
ds √2| 1 − 2|
The tangent vector of the curve β can be written as follows
T =β . (3.4)
β α
Differentiating (3.4) with respect to s, we obtain
dT ds
β. ∗ = (ξαΩα+ξαΩα). (3.5)
ds ds − 1 1 2 2
∗
Substituting (3.3) into (3.5) gives
√2
T = (ξαΩα+ξαΩα).
β′ − ξα ξα 1 1 2 2
| 1 − 2|
SmarandacheCurvesofaSpacelikeCurveAccordingtotheBishopFrameofType-2 33
Then the first curvature and the principal normal vector field of β are, respectively, com-
puted as
√2
T =κ = (ξα)2+(ξα)2,
β′ β ξα ξα − 1 2
(cid:13) (cid:13) | 1 − 2|
(cid:13) (cid:13) p
and (cid:13) (cid:13)
1
N = − (ξαΩα+ξαΩα).
β (ξα)2+(ξα)2 1 1 2 2
− 1 2
On the other hand, we expresps
Ωα Ωα β
1 − 1 2 α
Bβ = (ξα−)2+(ξα)2 (cid:12)(cid:12) 0 0 1 (cid:12)(cid:12),
− 1 2 (cid:12)(cid:12) ξα ξα 0 (cid:12)(cid:12)
p (cid:12) 1 2 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
So the binormal vector of β is computed as fo(cid:12)llows (cid:12)
1
B = − (ξαΩα+ξαΩα).
β (ξα)2+(ξα)2 2 1 1 2
− 1 2
p
Differentiating (3.2) with respect to s in order to calculate the torsion of the curve β, we
obtain
1
β¨= [ (ξα+ξα)ξαΩα (ξα+ξα)ξαΩα+(ξ˙α+ξ˙α)B ],
√2 − 1 2 1 1 − 1 2 2 2 1 2 α
and similarly
...
β = 1 [( 3ξαξ˙α 2ξαξ˙α ξ˙αξα ξαξ˙α)Ωα
√2 − 1 1 − 1 2 − 1 2 − 1 1 1
+( 2ξ˙αξα 2(ξ˙α)2 ξαξ˙α ξαξ˙α)Ωα
− 1 2 − 2 − 1 2 − 2 2 2
+(ξ¨α+ξ¨α (ξα)3 (ξα)2ξα ξα(ξα)2+(ξα)2)B ].
1 2 − 1 − 1 2 − 1 2 2 α
The torsion of the curve β is found
1 (ξα ξα)2
τ = 1 − 2 [(ξα+ξα)K (s) (ξα+ξα)ξαK (s)],
β 4√2(ξα)2+(ξα)2 1 2 2 − 1 2 2 1
1 2
where
K (s)= 3ξαξ˙α 2ξαξ˙α ξ˙αξα ξαξ˙α,
1 − 1 1 − 1 2 − 1 2 − 1 1
K (s)= 2ξ˙αξα 2(ξ˙α)2 ξαξ˙α ξαξ˙α,
2 − 1 2 − 2 − 1 2 − 2 2
K (s)=ξ¨α+ξ¨α (ξα)3 (ξα)2ξα ξα(ξα)2+(ξα)2.
3 1 2 − 1 − 1 2 − 1 2 2
3.2 Ω B Smarandache Curves
1
−
Definition 3.2 Let α = α(s) be a unit speed regular curve in E3 and Ωα,Ωα,B be its
1 { 1 2 α}
moving Bishop frame. ΩαB Smarandache curves are defined by
1 −
1
β(s∗)= √2(Ωα1 +Bα). (3.6)
34 YasinU¨nlu¨tu¨rkandSu¨haYılmaz
Now we can investigate Bishop invariants of ΠαB Smarandache curves of the curve
1 α−
α=α(s). Differentiating (3.6) with respect to s, we have
dβ ds 1
β˙ = . ∗ = (ξαB ξαΩα ξαΩα), (3.7)
ds ds √2 1 α− 1 1 − 2 2
and
ds 1
T . ∗ = (ξαB ξαΩα ξαΩα),
β ds √2 1 α− 1 1 − 2 2
where
ds ξα
∗ = 2 . (3.8)
ds 2
The tangent vector of the curve β can be written as follows
√2
T = ( ξαΩα ξαΩα+ξαB ). (3.9)
β ξα − 1 1 − 2 2 1 α
2
Differentiating (3.9) with respect to s gives
dT ds ξα
β. ∗ = 2 (L (s)Ωα+L (s)Ωα+L (s)B ), (3.10)
ds ds √2 1 1 2 2 3 α
∗
where
ξαξ˙α
L (s)= ξα (ξα)2+ 1 2 ,L (s)=ξαξα,
1 − 1 − 1 ξα 2 1 2
2
ξαξ˙α
L (s)= (ξα)2+(ξα)2+ξ˙α 1 2 .
3 − 1 2 1 − ξα
2
Substituting (3.8) into (3.10) gives
2√2
T = (L (s)Ωα+L (s)Ωα+L (s)B ),
β′ (ξα)2 1 1 2 2 3 α
2
then the first curvature and the principal normal vector field of β are, respectively,
2√2
T =κ = L2(s)+L2(s) L2(s),
β′ β (ξα)2 1 2 − 3
2
(cid:13) (cid:13) p
(cid:13) (cid:13)
and (cid:13) (cid:13)
1
N = − (L (s)Ωα+L (s)Ωα+L (s)B ).
β L2(s)+L2(s) L2(s) 1 1 2 2 3 α
1 2 − 3
On the other hanpd, we have
√2
B = [(ξαL (s)+ξαL (s))Ωα
β ξα L2(s)+L2(s) L2(s) 1 2 2 3 1
2 1 2 − 3
+(ξαL (s)+ξαL (s))Ωα+(ξαL (s) ξαL (s))B ].
p1 1 1 3 2 2 1 − 1 2 α
Differentiating (3.7) with respect to s in order to calculate the torsion of the curve β, we
SmarandacheCurvesofaSpacelikeCurveAccordingtotheBishopFrameofType-2 35
find
1
β¨= [ ((ξα)2+ξ˙α)Ωα+( ξαξ˙α ξ˙α)Ωα (ξ¨α (ξα)2+(ξα)2)B ],
√2 − 1 1 1 − 1 2 − 2 2 − 1 − 1 2 α
and similarly
... 1
β = [( 2ξαξ˙α ξ¨α)Ωα+( ξ˙αξ˙α ξ¨α ξαξ¨α)Ωα
√2 − 1 1 − 1 1 − 1 2 − 2 − 1 2 2
+( (ξα)3 ξαξ˙α+ξαξ˙αξα ξαξ˙α)B ].
− 1 − 1 1 1 2 2 − 2 2 α
The torsion of the curve β is
(εα)4
τ = 2 [ ξαM (s) ξαM (s) ξαM (s)],
β 16√2 − 1 1 − 2 2 − 1 3
where
M (s)= 3ξαξ˙α 2ξαξ˙α ξ˙αξα ξαξ˙α,
1 − 1 1 − 1 2 − 1 2 − 1 1
M (s)= 2ξ˙αξα 2(ξ˙α)2 ξαξ˙α ξαξ˙α,
2 − 1 2 − 2 − 1 2 − 2 2
M (s)=ξ¨α+ξ¨α (ξα)3 (ξα)2ξα ξα(ξα)2+(ξα)2.
3 1 2 − 1 − 1 2 − 1 2 2
3.3 Ω B Smarandache Curves
2
−
Definition 3.3 Let α = α(s) be a unit speed regular curve in E3 and Ωα,Ωα,B be its
1 { 1 2 α}
moving Bishop frame. ΩαB Smarandache curves are defined by
2 −
1
β(s )= (Ωα+B ). (3.11)
∗ √2 2 α
Now we can investigate Bishop invariants of ΩαB Smarandache curves of the curve
1 α−
α=α(s). Differentiating (3.11) with respect to s, we have
dβ ds 1
β˙ = . ∗ = ( ξαB ξαΩα ξαΩα), (3.12)
ds ds √2 − 2 α− 1 1 − 2 2
and
ds 1
T . ∗ = ( ξαΩα ξαΩα ξαB ),
β ds √2 − 1 1 − 2 2 − 2 α
where
ds 2(ξα)2 (ξα)2
∗ = 2 − 1 . (3.13)
ds 2
r
The tangent vector of the curve β can be written as follows
ξαΩα ξαΩα ξαB
Tβ = − 1 1 − 2 2 − 2 α. (3.14)
2(ξα)2 (ξα)2
2 − 1
p
Differentiating (3.14) with respect to s gives
dT ds
β. ∗ =(N (s)Ωα+N (s)Ωα+N (s)B ), (3.15)
ds ds 1 1 2 2 3 α
∗
36 YasinU¨nlu¨tu¨rkandSu¨haYılmaz
where
N1(s)= 12(4ξ2αξ˙2α−2ξ1αξ˙1α)(2(ξ2α)2−(ξ1α)2)−23ξ1α
−(2(ξ2α)2−(ξ1α)2)−21ξ˙1α+(2(ξ2α)2−(ξ1α)2)−21ξ1αξ2α,
N2(s)= 12(4ξ2αξ˙2α−2ξ1αξ˙1α)(2(ξ2α)2−(ξ1α)2)−23ξ1α
−(2(ξ2α)2−(ξ1α)2)−21ξ˙1α+(2(ξ2α)2−(ξ1α)2)(ξ2α)2,
N3(s)= 12(4ξ2αξ˙2α−2ξ1αξ˙1α)(2(ξ2α)2−(ξ1α)2)−23
−(2(ξ2α)2−(ξ1α)2)−21((ξ2α)2−(ξ1α)2−ξ˙2α).
Substituting (3.13) into (3.15) gives
2
T = (N (s)Ωα+N (s)Ωα+N (s)B ),
β′ 2(ξα)2 (ξα)2 1 1 2 2 3 α
r 2 − 1
then the first curvature and the principal normal vector field of β are, respectively, found as
follows
2
κ = T = N2(s)+N2(s)+N2(s),
β β′ 2(ξα)2 (ξα)2 1 2 3
(cid:13) (cid:13) r 2 − 1 p
(cid:13) (cid:13)
and (cid:13) (cid:13)
1
Nβ = N2(s)+N−2(s)+N2(s)(N1(s)Ωα1 +N2(s)Ωα2 +L3(s)Bα). (3.16)
1 2 3
On the other hapnd, we have
1
B = [( ξαN (s)+ξαN (s))Ωα
β 2(ξ2α)2−(ξ1α)2 N12(s)+N22(s)+N32(s) − 2 3 2 2 1 (3.17)
+( ξαN (s)+ξαN (s))Ωα+(ξαN ξαN (s))B ].
p − 2 3 p 2 1 2 1 2− 2 1 α
Differentiating (3.12) with respect to s in order to calculate the torsionof the curve β, we
obtain
1
β¨= [(ξαξα+ξ˙α)Ωα+((ξα)2 ξ˙α)Ωα+( ξ˙α+ξα (ξα)2)B ],
√2 2 1 1 1 2 − 2 2 − 2 2 − 1 α
and similarly
... 1
β = [(2ξ˙αξα+ξαξ˙α ξ¨α ξαξα+(ξα)3)Ωα
√2 2 1 2 1 − 1 − 2 1 1 1
+(3ξαξ˙α ξ¨α (ξα)2 (ξα)2ξα)Ωα
2 2 − 2 − 2 − 1 2 2
+((ξα)2ξα (ξα)3+ξαξ˙α ξ¨α+ξ˙α 3ξαξ˙α)B ].
1 2 − 2 2 2 − 2 2 − 1 1 α
The torsion of the curve β is
2(ξα)2 (ξα)2
τ = 2 − 1 [P (s)((ξα)2 ξ˙α) P (s)( ξ˙α+ξα (ξα)2)]ξα
β 4√2 { 3 2 − 2 − 2 − 2 2 − 1 1
+[P (s)(ξαξα ξ˙α) P (s)( ξ˙α+ξα (ξα)2)]ξα
3 2 1 − 1 − 1 − 2 2 − 1 2
+[P (s)(ξαξα ξ˙α) P (s)((ξα)2 ξ˙α)]ξα ,
2 2 1 − 1 − 1 2 − 2 3}
SmarandacheCurvesofaSpacelikeCurveAccordingtotheBishopFrameofType-2 37
where
P (s)=2ξ˙αξα+ξαξ˙α ξ¨α ξαξ˙α+ξαξα+(ξα)3,
1 2 1 2 1 − 1 − 1 1 1 2 1
P (s)=3ξ˙αξα ξ¨α (ξα)2 (ξα)2ξα,
2 2 2 − 2 − 2 − 1 2
P (s)=(ξα)2ξα (ξα)3+ξαξ˙α ξ¨α+ξ˙α 3ξ˙αξα.
3 1 2 − 2 2 2 − 2 2 − 1 1
3.4 Ω Ω B Smarandache Curves
1 2
−
Definition 3.4 Let α = α(s) be a unit speed regular curve in E3 and Ωα,Ωα,B be its
1 { 1 2 α}
moving Bishop frame. ΩαΩαB Smarandache curves are defined by
1 2 −
1
β(s∗)= √3(Ωα1 +Ωα2 +Bα). (3.18)
Now we can investigate Bishop invariants of ΩαΩαB Smarandache curves of the curve
1 2 −
α=α(s). Differentiating (3.18) with respect to s, we have
dβ ds 1
β˙ = . ∗ = ( ξαΩα ξαΩα+(ξα εα)B ), (3.19)
ds ds √3 − 1 1 − 2 2 1 − 2 α
and
ds 1
T . ∗ = ( ξαΩα ξαΩα+(ξα ξα)B ),
β ds √3 − 1 1 − 2 2 1 − 2 α
where
ds (ξα ξα)2+(ξα)2 (ξα)2
∗ = 1 − 2 2 − 1 . (3.20)
ds 3
r
The tangent vector of the curve β is found as follows
1
Tβ = (ξα ξα)2+(ξα)2 (ξα)2(−ξ1αΩα1 −ξ2αΩα2 +(ξ1α−ξ2α)Bα). (3.21)
1 − 2 2 − 1
p
Differentiating (3.21) with respect to s, we find
dT ds
β. ∗ =[ Q(s)ξ˙α Q(s)(ξα)2+Q(s)ξαξα Q(s)ξα]Ωα
ds ds − 1 − 1 1 2 − ′ 1 1
∗
+[ Q(s)ξ˙α Q(s)ξαξα+Q(s)(ξα)2 Q(s)ξα]Ωα (3.22)
− 2 − 1 2 2 − ′ 2 2
+[Q(s)(ξα ξα)-Q(s)(ξα)2+Q(s)(ξα ξα)]B ,
1 − 2 ′ 1 ′ 1 − 2 α
where
1
Q(s)= .
(ξα ξα)2+(ξα)2 (ξα)2
1 − 2 2 − 1
Substituting (3.20) into (3.22) bpy using (3.23) gives
√3
T = (M (s)Ωα+M (s)Ωα+M (s)B ),
β′ K(s) 1 1 2 2 3 α
38 YasinU¨nlu¨tu¨rkandSu¨haYılmaz
where
R (s)= Q(s)ξ˙α Q(s)(ξα)2+Q(s)ξαξα Q(s)ξα,
1 − 1 − 1 1 2 − ′ 1
R (s)= Q(s)ξ˙α Q(s)ξαξα+Q(s)(ξα)2 Q(s)ξα, (3.23)
2 − 2 − 1 2 2 − ′ 2
R (s)=Q(s)(ξα ξα) Q(s)(ξα)2+Q(s)(ξα ξα).
3 1 − 2 ′− 1 ′ 1 − 2
Thenthefirstcurvatureandtheprincipalnormalvectorfieldofβare,respectively,obtained
as follows
√3
κ = T = R2(s)+R2(s)+R2(s),
β β′ K(s) − 1 2 3
(cid:13) (cid:13) p
and (cid:13) (cid:13)
(cid:13) (cid:13)
1
B = − [(M (ξα ξα)+M ξα)Ωα
β K(s) R2(s)+R2(s)+R2(s) 2 1 − 2 3 2 1 (3.24)
− 1 2 3
+(M (ξα ξα)+M ξα)Ωα+(ξαM (s) ξαM (s))B ].
1 p1 − 2 3 2 2 2 1 − 1 2 α
Differentiating (3.19) with respect to s in order to calculate the torsionof the curve β, we
obtain
1
β¨= [( ξ˙α (ξα)2+ξαξα)Ωα
√3 − 1 − 1 1 2 1
+( ξ˙α+(ξα)2 ξαξα+(ξα)2)Ωα+(ξ˙α ξ˙α (ξα)2)B ],
− 2 2 − 1 2 2 2 1 − 2 − 1 α
and similarly
... 1
β = [( ξ¨α 2ξαξ˙α+ξ˙αξα+ξαξ˙α)Ωα
√3 − 1 − 1 1 1 2 1 2 1
+( ξ¨α+4ξαξ˙α ξ˙αξα 2ξαξα)Ωα
− 2 2 2 − 1 2 − 1 2 2
+(ξαξ˙α+ξα(ξα)2 (ξα)2 (ξα)3+(ξα)2ξα)B ].
2 2 1 2 − 1 − 1 1 2 α
The torsion of the curve β is
1 K2(s)
τ = [Q (s)( ξ˙α+2(ξα)2 ξαξα)
β 9 M2(s)+M2(s)+M2(s){ 3 − 2 2 − 1 2
− 1 2 3
Q (s)(ξα ξ˙α ξαξα)]ξα+[Q (s)( ξ˙α (ξα)2+ξαξα)
− 2 1 − 2 − 1 2 1 3 − 1 − 1 1 2
Q (s)( ξ˙α+2(ξα)2 ξαξα)]ξα [Q (s)( ξα (ξα)2+ξαξα)
− 1 − 2 2 − 1 2 2 − 2 − 1 − 1 1 2
Q (s)( ξα+2(ξα)2 ξαξα)](ξα ξα) ,
− 1 − 2 2 − 1 2 1 − 2 }
where
Q (s)= ξ˙α (ξα)2+ξαξα,
1 − 1 − 1 1 2
Q (s)= ξ˙αξα+2ξαξ˙α ξ¨α,
2 − 1 2 2 2 − 2
Q (s)=ξα(ξα)2 (ξα)2 (ξα)3+(ξα)2ξα+ξαξ˙α.
3 1 2 − 1 − 1 1 2 2 2
3.5 Example
Example 3.1 Next, let us consider the following unit speed curve w =w(s) in E3 as follows
1
w(s)=(s,√2ln(sech(s)),√2arctan(sinh(s))). (3.25)