Spherical Images of Special Smarandache Curves in E3

2015·0.44 MB·

by  Vahide Bulut

International J.Math. Combin. Vol.3(2015), 43-54
Spherical Images of Special Smarandache Curves in E3
Vahide Bulut and Ali Caliskan
(DepartmentofMathematics,EgeUniversity,Izmir,35100,Turkey)
E-mail: [email protected],[email protected]
Abstract: In this study, we introduce the spherical images of some special Smarandache
curvesaccordingtoFrenetframeandDarbouxframeinE3. Besides,wegivesomedifferential
geometric properties of Smarandache curvesand their spherical images.
Key Words: Smarandache curves, S.Frenet frame, Darboux frame, Spherical image.
AMS(2010): 53A04.
§1. Introduction
Curves especially regularcurves are used in many fields such as CAGD, mechanics, kinematics
and differential geometry. Researchers are used various curves in these fields. Special Smaran-
dache curves are one of them. A regular curve in Minkowski spacetime, whose position vector
is composed by Frenet frame vectors on another regular curve, is called a Smarandache curve
([7]). Some authors have studied on special Smarandache curves ([1, 2, 7]).
In this paper, we give the spherical images of some special Smarandache curves according
toFrenetframeandDarbouxframeinE3. Also,wegivesomerelationsbetweenthe arclength
parameters of Smarandache curves and their spherical images.
§2. Preliminaries
Let α(s) be an unit speed curve that satisfies α′(s) =1 in E3. S.Frenet frame of this curve
|| ||
in E3 parameterized by arc length parameter s is,
α′(s)=T, T′(s) =N(s), T(s) N(s)=B(s),
′
||T (s)|| ×
where T(s) is the unit tangent vector,N(s) is the unit principal normalvectorand B(s) is the
1ReceivedFebruary9,2015, AcceptedAugust10,2015.
44 VahideBulutandAliCaliskan
unit binormal vector of the curve α(s). The derivative formulas of S.Frenet are,
′
T 0 κ τ T
′
 N = κ 0 τ  N , (1)
−
′
 B   0 τ 0  B 
   −  
    
′ ′
where κ = κ(s) = T (s) and τ = τ(s) = B (s) are the curvature and the torsion of the
|| || || ||
curve α(s) at s, respectively [4].
Let S be a regular surface and a curve α(s) be on the surface S. Since the curve α(s) is
also a space curve, the curve α(s) has S.Frenet frame as mentioned above. On the other hand,
since the curve α(s) lies on the surface S, there exists another frame which is called Darboux
frame T,g,n of the curve α(s). T is the unit tangent vector of the curve α(s), n is the unit
{ }
normal of the surface S and g is a unit vector given by g = n T.The derivative formulas of
×
Darboux frame are
′
T 0 κ κ T
g n
′
 g = κ 0 τ  g , (2)
g g
−
′
 n   κ τ 0  n 
   − n − g  
    
where, κ is the geodesic curvature, κ is the normal curvature and τ is the geodesic torsion
g n g
of the curve α(s). The Darboux vector and the unit Darboux vector of this curve are given,
respectively as follows
d=τ T +κ g+κ n
g n g
d τ T +κ g+κ n
g n g
c= = . (3)
d
τ2+κ2 +κ2
|| || g n g
q
(1) α(s) is a geodesic curve if and only if κ =0.
g
(2) α(s) is an asymptotic line if and only if κ =0.
n
(3) α(s) is a principal line if and only if τ =0 ([6]).
g
The sphere in E3 with the radius r >0 and the center in the origin is defined by [3]
S2 = x=(x , x , x ) E3 : x,x =r2 .
1 2 3
∈ h i
(cid:8) (cid:9)
Letthevectorsofthemovingframeofacurveα(s)withnon-vanishingcurvaturearegiven.
Assume that these vectors undergo a parallel displacement and become bound at the origin O
of the Cartesian coordinate system in space. Then the terminal points of these vectors T(s),
N(s) and B(s) lie on the unit sphere S which are called the tangent indicatrix, the principal
normal indicatrix and the binormal indicatrix, respectively of the curve α(s).
Thelinearelementsds ,ds andds oftheseindicatricesorsphericalimagescanbeeasily
T N B
obtainedbymeansof(1). SinceT(s),N(s)andB(s)arethevectorfunctionsrepresentingthese
SphericalImagesofSpecialSmarandacheCurvesinE3 45
curves we find
ds2 =κ2ds2,
T
ds2 = κ2+τ2 ds2, (4)
N
ds2 =τ(cid:0)2ds2. (cid:1)
B
Curvatureandtorsionappearhereasquotientsoflinearelements;choosingtheorientation
of the spherical image by the orientation of the curve α(s) we have from (4)
ds ds
T B
κ= , τ = . (5)
ds | | ds
Moreover,from (5) we obtain the Equation of Lancret ([5])
ds2 =ds2 +ds2. (6)
N T B
§3. Special Smarandache Curves According to S.Frenet Frame In E3
3.1 TN- Smarandache Curves
Let α(s) be a unit speed regular curve in E3 and T,N,B be its moving S.Frenet frame. A
{ }
Smarandache TN curve is defined by ([1])
1
β(s∗)= (T +N). (7)
√2
Let moving S. Frenet frame of this curve be T∗,N∗,B∗ .
{ }
3.1.1 Spherical Image of the Unit Vector T∗
β
We can find the relation between the arc length parameters ds∗ and ds as follows
ds∗ 2κ2+τ2
= . (8)
ds 2
r
From the equations (5) and (8) we have
√2κ2+τ2
ds∗ = ds . (9)
T
κ√2
From the equation (5) we obtain the spherical image of the unit vector T∗ as
β
ds∗ √2 δ2+µ2+η2
T =κ∗ = , (10)
ds∗ p√2κ2+τ2 2
(cid:0) (cid:1)
where
√2 δ2+µ2+η2
κ∗ = . (11)
2
p√2κ2+τ2
(cid:0) (cid:1)
46 VahideBulutandAliCaliskan
Here ([1]),
δ = κ2 2κ2+τ2 +τ τκ′ κτ′ ,
− −
µ= hκ2(cid:0)2κ2+3τ(cid:1)2 +τ(cid:16) τ3 τκ′(cid:17)+i κτ′ ,
η =κ−hτ 2(cid:0)κ2+τ2 (cid:1)2 τκ(cid:16)′ −κτ′ . (cid:17)i
− −
Then, from the equations (9) ahnd(cid:0)(10) (cid:1) (cid:16) (cid:17)i
δ2+µ2+η2
ds∗ = ds (12)
T κp(2κ2+τ2)3/2 T
is obtained.
3.1.2 Spherical Image of the Unit Vector N∗
β
If we use the equation (6) we have
ds∗
N = (κ∗)2+(τ∗)2. (13)
ds∗
q
Besides, from the equations (6), (8) and (13)
√2κ2+τ2
ds∗ = (κ∗)2+(τ∗)2 ds (14)
N √2√κ2+τ2 N
q
is obtained, where
√2 κ2+τ2 κ′ (κσ+τω)+κ κτ +τ′ (φ ω)+ κ2+κ′ (κσ τφ)
τ∗ = − − − (15)
h(cid:16) [τ(2κ2+(cid:17)τ2)+κ′τ κτ′(cid:16)]2+(τκ′(cid:17) κτ′)2+(2(cid:16)κ3+κτ(cid:17)2)2 i
− −
and
ω =κ3+κ τ2 3κ′ κ′′,
− −
φ= κ3 κ(cid:16) τ2+3κ(cid:17)′ 3ττ′ +κ′′,
σ =−κ2τ− τ(cid:16)3+2τκ′ +(cid:17)−κτ′ +τ′′.
− −

3.1.3 Spherical Image of the Unit Vector B∗
β
From the equations (5) and (15) we have
ds∗
B =τ∗. (16)
ds∗
On the other hand, the following formula is found from the equations (5), (8) and (16).
√2κ2+τ2
ds∗ =τ∗ ds (17)
B τ√2 B
Example 1 Let the curve α(s)= 4sint, 2 cost, 3sint is given. TN-Smarandachecurve
5 − 5
(cid:0) (cid:1)
SphericalImagesofSpecialSmarandacheCurvesinE3 47
of this curve is found as
4 1 3
β(s∗)= (cost sint), (sint+cost), (cost sint) .
5√2 − √2 5√2 −
(cid:20) (cid:21)
The spherical images of T∗, N∗ and B∗ for the curve β(s∗) are shown in Figures 1, 2 and 3,
respectively.
Figure 1 Spherical image of T∗
Figure 2 Spherical image of N∗
Figure 3 Spherical image of B∗
3.2 NB- Smarandache Curves
Let α(s) be a unit speed regular curve in E3 and T,N,B be its moving S. Frenet frame.
{ }
Smarandache NB curve is defined by ([1])
1
β(s∗)= (N +B). (18)
√2
48 VahideBulutandAliCaliskan
3.2.1 Spherical Image of the Unit Vector T∗
β
From the equations (5) and (8) we have
√2κ2+τ2
ds∗ = ds . (19)
T
κ√2
From the equation (5), we obtain the spherical image of the T∗ as
β
ds∗ √2 γ2+γ2+γ2
T =κ∗ = 1 2 3, (20)
ds∗ (2κ2+τ2)2
p
where
γ =κτ 2κ+τ′ +τ2 τ κ′ ,
1
−
γ = (cid:16)κ2 2κ2+(cid:17)3τ2+(cid:16)2τ′ +(cid:17)τ τ3 2κκ′ ,
γ2 =−2κh2 τ(cid:16)′ τ2 τ τ3+(cid:17)2κκ′(cid:16). − (cid:17)i
3
− −
Then, the following formula is o(cid:16)btained (cid:17)from t(cid:16)he equation(cid:17)s (9) and (20).
γ2+γ2+γ2
ds∗ = 1 2 3ds (21)
T κp(2κ2+τ2)3/2 T
3.2.2 Spherical Image of the Unit Vector N∗
β
The spherical image of N∗ can be found by using the equation (6) as
β
ds∗
N = (κ∗)2+(τ∗)2, (22)
ds∗
q
where
√2(κϕ +τϕ ) 2τ2 κ2
τ∗ = 3 1 − (23)
(2τ3 2κ2)2+(κτ′ τκ′)2(cid:0)+( κ3+(cid:1)κτ′ κ′τ)2
− − − −
and
ϕ = τ3 3ττ′ +κ2τ +τ′′,
3
− −
ϕ = κ3+κ τ2+2τ′ +τκ′ κ′′.
 1 − −
(cid:16) (cid:17)
Besides, from the equations (6), (8) and (22)
√2κ2+τ2
ds∗ = (κ∗)2+(τ∗)2 ds (24)
N √2√κ2+τ2 N
q
is obtained.
3.2.3 Spherical Image of the Unit Vector B∗
β
SphericalImagesofSpecialSmarandacheCurvesinE3 49
From the equations (5) and (23) we have
ds∗
B =τ∗. (25)
ds∗
On the other hand, from the equations (5), (8) and (25)
√2(κϕ +τϕ ) 2τ2 κ2 √2κ2+τ2
ds∗ = 3 1 − ds (26)
B "(2τ3 2κ2)2+(κτ′ τκ′)2(cid:0)+( κ3+(cid:1)κτ′ κ′τ)2# τ√2 B
− − − −
is found.
Example 2 Letthe curve α(s)= 4sint, 2 cost, 3sint is given. NB -Smarandachecurve
5 − 5
of this curve is
(cid:0) (cid:1)
1 4 3 3 4
β(s∗)= sint , cost, sint+ .
√2 −5 − 5 −5 5
(cid:20) (cid:21)
ThesphericalimagesofT∗andN∗forthecurveβ(s∗)areshowninFigures4and5,respectively.
Figure 4 Spherical image of T∗
Figure 5 Spherical image of N∗
The spherical image of B∗ for the curve β(s∗) is a point similar to the Figure 3.
3.3 TB- Smarandache Curves
Let α(s) be a unit speed regular curve in E3 and T,N,B be its moving S.Frenet frame.
{ }
Smarandache TB curve is defined by ([1])
1
β(s∗)= (T +B). (27)
√2
50 VahideBulutandAliCaliskan
3.3.1 Spherical Image of the Unit Vector T∗
β
We can find the spherical image of T∗ from the equation (5) and obtain
β
ds∗ √2 σ2+σ2+σ2
T =κ∗ = 1 2 3, (28)
ds∗ (2κ2+τ2)2
p
where
σ = 2κ2+τ2 κτ κ2 ,
1
−
σσ2 ==(cid:0)(22κκ2++ττ)2(cid:16)(cid:1)κ(cid:0)′κττ−κττ2′(cid:17)(cid:1).,
3
−
From the equations (5) and(8) (cid:0) (cid:1)(cid:0) (cid:1)
√2κ2+τ2
ds∗ = ds (29)
T
κ√2
is obtained. Then, the formula following is acquired from equations (28) and (29).
σ2+σ2+σ2
ds∗ = 1 2 3ds (30)
T pκ(2κ2+τ2)3/2 T
3.3.2 Spherical Image of the Unit Vector N∗
β
If we use the equation (6) we have the spherical image of N∗ as
β
ds∗
N = (κ∗)2+(τ∗)2. (31)
ds∗
q
Besides, from the equations (6), (8) and (31)
√2κ2+τ2
ds∗ = (κ∗)2+(τ∗)2 ds (32)
N √2√κ2+τ2 N
q
is obtained, where
√2(τ κ)2(κΦ +τΦ )
τ∗ = − 3 1 , (33)
2 2
τ(κ τ)2 + κ(κ τ)2
− −
h i h i
′ ′ ′
Φ =2τ κ τ +τ (κ τ),
3
− −
Φ =κ′((cid:16)τ κ)+(cid:17)2κ τ′ κ′ .
 1 − −
(cid:16) (cid:17)
3.3.3 Spherical Image ofthe Unit Vector B∗
β
From the equations (5) and (33) we have
ds∗
B =τ∗. (34)
ds∗
SphericalImagesofSpecialSmarandacheCurvesinE3 51
On the other hand, the following formula is found from the equations (5), (8) and (34).
√2(τ κ)2(κΦ +τΦ ) √2κ2+τ2
ds∗ = − 3 1 ds (35)
B  τ(κ τ)2 2+ κ(κ τ)2 2 τ√2 B
 − − 
h i h i 
Example 3 Let the curve α(s)= 4sint, 2 cost, 3sint is given. TB -Smarandache curve
5 − 5
of this curve is
(cid:0) (cid:1)
1 4 3 3 4
β(s∗)= cost , sint, cost+ .
√2 5 − 5 5 5
(cid:20) (cid:21)
ThesphericalimagesofT∗andN∗forthecurveβ(s∗)areshowninFigures6and7,respectively.
Figure 6 Spherical image of T∗
Figure 7 Spherical image of N∗
The spherical image of B∗ for the curve β(s∗) is a point similar to the Figure 3.
3.4 TNB- Smarandache Curves
Let α(s) be a unit speed regular curve in E3 and T,N,B be its moving S.Frenet frame.
{ }
Smarandache TNB curve is defined by ([1])
1
β(s∗)= (T +N +B). (36)
√3
Remark 1 Thesphericalimagesofthecurveβ(s∗)canbefoundinasimilarwayaspresented
above.
52 VahideBulutandAliCaliskan
§4. Spherical Images of Darboux Frame T,g,n
{ }
LetS be anorientedsurfaceinE3. Letα(s) be aunit speedregularcurvesinE3 and T,g,n
{ }
be Darboux frame of this curve.
4.1 Spherical Image of The Unit Vector T
The differential geometric properties of the spherical image of the unit vector T are given as
dT dT ds
= .
ds ds ds
T T
dT ds
=(κ g+κ n).
g n
ds ds
T T
ds
T = κ2+κ2. (37)
ds g n
q
On the other hand, from (4) and (37)
κ= κ2+κ2. (38)
g n
q
can be written.
4.2 Spherical Image of The Unit Vector g
The differential geometric properties of the spherical image of the unit vector g are found as
dg dg ds
= .
ds ds ds
g g
dg ds
=( κ T +τ n).
g g
ds − ds
g g
The relation between the arc length parameters are given as follows
ds
g = κ2+τ2. (39)
ds g g
q
4.3 Spherical Image of The Unit Vector n
The differential geometric properties of the spherical image of the unit vector n are given as
dn dn ds
= .
ds ds ds
n n
dn ds
=( κ T τ g).
n g
ds − − ds
n n