Special Smarandache Curves According to Bishop Frame in Euclidean Spacetime

2017·0.17 MB·

by  E. M. Solouma

International J.Math. Combin. Vol.1(2017), 01-09
Special Smarandache Curves According to Bishop
Frame in Euclidean Spacetime
E. M. Solouma
(DepartmentofMathematics,FacultyofScience,Beni-SuefUniversity,Egypt)
M. M. Wageeda
(MathematicsDepartment,FacultyofScience,AswanUniversity,Aswan,Egypt)
E-mail: [email protected],[email protected]
Abstract: In this paper, we introduce some special Smarandache curves according to
Bishop frame in Euclidean 3-space E3. Also, we study Frenet-Serret invariants of a special
case in E3. Finally, we give an example to illustrate these curves.
Key Words: Smarandache curve,Bishop frame, Euclidean spacetime.
AMS(2010): 53A04, 53A05.
§1. Introduction
Inthe theory ofcurvesinthe EuclideanandMinkowskispaces,one ofthe interesting problems
isthecharacterizationofaregularcurve. Inthesolutionoftheproblem,thecurvaturefunctions
κandτ ofaregularcurvehaveaneffectiverole. Itisknownthattheshapeandsizeofaregular
curvecanbedeterminedbyusingitscurvaturesκandτ ([7],[8]). Forinstance,Bertrandcurves
andMannheimcurvesarisefromthisrelationship. AnotherexampleistheSmarandachecurves.
They are the objects of Smarandache geometry, that is, a geometry which has at least one
Smarandachely denied axiom [1]. The axiom is said to be Smarandachely denied if it behaves
in at least two different ways within the same space. Smarandache geometries are connected
with the theory of relativity and the parallel universes.
By definition, if the positionvectorofa curveβ is composedbythe Frenetframe’s vectors
of another curve α, then the curve β is called a Smarandache curve [9]. Special Smarandache
curves in the Euclidean and Minkowski spaces are studied by some authors ([6], [10]). For
instance, the special Smarandache curves according to Darboux frame in E3 are characterized
in [5].
In this work, we study special Smarandache curves according to Bishop frame in the Eu-
clidean3-spaceE3. Wehopetheseresultswillbehelpfultomathematicianswhoarespecialized
on mathematical modeling.
1ReceivedAugust23,2016,Accepted February2,2017.
2 E.M.SoloumaandM.M.Wageeda
§2. Preliminaries
The Euclidean 3-space E3 provided with the standard flat metric given by
, =dx2+dx2+dx2,
h i 1 2 3
where(x ,x ,x )isarectangularcoordinatesystemofE3. Recallthat,thenormofanarbitrary
1 2 3
vector v E3 is given by v = v,v . A curve α is called an unit speed curve if velocity
∈ k k |h i|
vector α of satisfies α = 1. For vectors u,v E3 it is said to be orthogonal if and only if
′ k ′k p ∈
u,v =0. Let ̺=̺(s) be a regularcurve in E3. If the tangent vectorfield of this curve forms
h i
a constant angle with a constant vector field U, then this curve is called a general helix or an
inclined curve.
Denote by T,N,B the moving Frenet frame along the curve α in the space E3. For
{ }
an arbitrary curve α E3, with first and second curvature, κ and τ respectively, the Frenet
∈
formulas is given by ([7]).
T (s) 0 κ 0 T(s)
′
 N (s) = κ 0 τ  N(s) , (1)
′
−
 B (s)   0 τ 0  B(s) 
 ′   −  
    
where T,T = N,N = B,B = 1, T,N = T,B = N,B = 0. Then, we write Frenet
h i h i h i h i h i h i
invariants in this way: T(s) = α(s), κ(s) = T (s) , N(s) = T (s)/κ(s), B(s) = T(s) N(s)
′ ′ ′
k k ×
and τ(s)= N(s),B (s) .
′
−h i
The Bishop frame or parallel transport frame is an alternative approach to defining a
moving frame that is well defined even when the curve has vanishing second derivative. One
canexpressBishopofanorthonormalframe alonga curvesimply by paralleltransportingeach
component of the frame [2]. The tangent vector and any convenient arbitrary basis for the
remainder of the frame are used (for details, see [3]). The Bishop frame is expressed as ([2],
[4]).
T (s) 0 k (s) k (s) T(s)
′ 1 2
 N (s) = k (s) 0 0  N (s) . (2)
1′ − 1 1
 N (s)   k (s) 0 0  N (s) 
 2′   − 2  2 
    
Here, we shall callthe set T,N ,N as Bishoptrihedra andk (s) and k (s) as Bishopcurva-
1 2 1 2
{ }
tures. The relation matrix may be expressed as
T(s) 1 0 0 T(s)
 N (s) = 0 cosϑ(s) sinϑ(s)  N(s) , (3)
1
−
 N (s)   0 sinϑ(s) cosϑ(s)  B(s) 
 2    
    
SpecialSmarandacheCurvesAccordingtoBishopFrameinEuclideanSpacetime 3
where
k
2
ϑ(s)=arctan , k =0
1
k 6
1
 dϑ(s)(cid:16) (cid:17)
(4)
τ(s)=− ds
κ(s)= k2(s)+k2(s)
1 2
Here, Bishop curvatures are defined by p
k (s)=κ(s)cosϑ(s),
1
(5)

k (s)=κ(s)sinϑ(s).
 2
Let α=α(s) be a regular non-null curve parametrized by arc-length in Euclidean 3-space
E3 withits Bishopframe T,N ,N . ThenTN , TN ,N N andTN N -Smarandachecurve
1 2 1 2 1 2 1 2
{ }
of α are defined, respectively as follows ([9]):
1
= (℘(s))= T(s)+N (s) ,
1
B B √2
(cid:16) (cid:17)
1
= (℘(s))= T(s)+N (s) ,
2
B B √2
(cid:16) (cid:17)
1
= (℘(s))= N (s)+N (s) ,
1 2
B B √2
(cid:16) (cid:17)
1
= (℘(s))= T(s)+N (s)+N (s) .
1 2
B B √3
(cid:16) (cid:17)
§3. Special Smarandache Curves According to Bishop Frame in E3
Definition 3.1 A regular curve in Euclidean space-time, whose position vector is composed by
Frenet frame vectors on another regular curve, is called a Smarandache curve.
Inthelightoftheabovedefinition,weadaptittoregularcurvesaccordingtoBishopframe
in the Euclidean 3-space E3 as follows.
Definition 3.2 Let α=α(s) be a unit speed regular curve in E3 and T,N ,N be its moving
1 2
{ }
Bishop frame. TN -Smarandache curves are defined by
1
1
= (℘(s))= (T(s)+N (s)). (6)
1
B B √2
Let us investigate Frenet invariants of TN -Smarandache curves according to α = α(s).
1
By differentiating Eqn.(6) with respected to s and using Eqn.(2), we get
d d℘ 1
′ = B = k1T +k1N1+k2N2 , (7)
B d℘ ds √2 −
(cid:16) (cid:17)
4 E.M.SoloumaandM.M.Wageeda
and hence
k T +k N +k N
1 1 1 2 2
T = − , (8)
B 2k2+k2
1 2
where p
d℘ 2k2+k2
= 1 2. (9)
ds 2
r
In order to determine the first curvature and the principal normal of the curve , we
B
formalize
dT d℘ d℘ ζ T +ζ N +ζ N
TB′ = d℘B ds =T˙Bds = 1 2k22+1k2 323 2, (10)
1 2
where (cid:0) (cid:1)
ζ = k (2k k +k k ) (2k2+k2)(k +k2+k2) ,
1 1 1 1′ 2 2′ − 1 2 1′ 1 2
ζζ32 ==(cid:2)(cid:2)((22kk1122++kk2222))((kk21′′ −−kk112k)2−)−k1k(22k(21kk11′k+1′ +k2kk22′k)(cid:3)2′,) . (cid:3) (11)
Then, we have (cid:2) (cid:3)
√2
T˙ = ζ T +ζ N +ζ N . (12)
B 2k2+k2 2 1 2 1 3 2
1 2 (cid:16) (cid:17)
So, the first curvature and th(cid:0)e princip(cid:1)al normal vector field are respectively given by
√2 ζ2+ζ2+ζ2
κ = T˙ = 1 2 3. (13)
B k Bk p2k12+k22 2
(cid:0) (cid:1)
and
ζ T +ζ N +ζ N
1 2 1 3 2
N = . (14)
B ζ2+ζ2+ζ2
1 2 3
On other hand, we express p
T N N
1 2
1
T N = (cid:12) k k k (cid:12), (15)
B× B pq (cid:12) − 1 1 2 (cid:12)
(cid:12) (cid:12)
(cid:12) ζ ζ ζ (cid:12)
(cid:12) 1 2 3 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
where p= 2k2+k2 and q = ζ2+ζ2+ζ2(cid:12). So, the binorma(cid:12)l vector is
1 2 1 2 3
p p
1
B = k ζ k ζ T + k ζ +k ζ N +k ζ +ζ N . (16)
1 3 2 2 1 3 2 1 1 1 1 2 2
B pq −
n(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) o
In order to calculate the torsionof the curve , we differentiate Eqn.(7) with respected to
B
s, we have
1
= k +k2+k k + T + k k2ζ N + k k k N . (17)
B′′ √2 − 1′ 1 1 2 1′ − 1 1 1 2′ − 1 2 2
n (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) o
and thus
ν T +ν N +ν N
1 2 1 2 2
= , (18)
′′′
B √2
SpecialSmarandacheCurvesAccordingtoBishopFrameinEuclideanSpacetime 5
where
ν = k′′+k′(3k +k )+k′(k +k ) k (k2+k2) ,
1 − 1 1 1 2 2 1 2 − 1 1 2
νν2 ==kk1′′(cid:2)′′−kk1k(k′12+k3k(k1′2++k21kk2′)+, k k ). (cid:3) (19)
3 2 − 1 2− 2 1 1 1 2
The torsion isthen given by:
√2 (k2 k′)(k ν +k ν )+k (k′ k k )(ν +ν )+(k2+k′ +k k )(k ν k ν )
τB = (cid:2) 1− 1(k11k2′3−k1′2k21)2+ 1k1k22′−+k12(2k1′ +1k1k22) 2+1k12(2k112+k11k22)21 3− 2 2 (cid:3). (20)
(cid:2) (cid:3)
Corollary 3.1 Let α=α(s) be a curve lying fully in E3 with the moving frame T,N,B . If α
{ }
iscontainedinaplane, thentheBishop curvaturesbecomesconstantandtheTN -Smarandache
1
curve is also contained in a plane and its curvature satisfying the following equation
2 k2(k2+1)+(k2+k2)2
1 2 1 2
κ = .
B q (cid:2) 2k12+k22 (cid:3)
.
Definition 3.3 Let α=α(s) be a unit speed regular curve in E3 and T,N ,N be its moving
1 2
{ }
Bishop frame. TN -Smarandache curves are defined by
2
1
= (℘(s))= T(s)+N (s) . (21)
2
B B √2
(cid:16) (cid:17)
Remark 3.1 The Frenet invariantsof TN -Smarandachecurves canbe easily obtainedby the
2
apparatus of the regular curve α=α(s).
Definition 3.4 Let α=α(s) be a unit speed regular curve in E3 and T,N ,N be its moving
1 2
{ }
Bishop frame. N N -Smarandache curves are defined by
1 2
1
= (℘(s))= N (s)+N (s) . (22)
1 2
B B √2
(cid:16) (cid:17)
Remark 3.2 The Frenet invariants of N N -Smarandache curves can be easily obtained by
1 2
the apparatus of the regular curve α=α(s).
Definition 3.5 Let α=α(s) be a unit speed regular curve in E3 and T,N ,N be its moving
1 2
{ }
Bishop frame. TN N -Smarandache curves are defined by
1 2
1
= (℘(s))= T(s)+N (s)+N (s) . (23)
1 2
B B √3
(cid:16) (cid:17)
Remark 3.3 The Frenet invariants of TN N -Smarandache curves can be easily obtained by
1 2
the apparatus of the regular curve α=α(s).
Example 3.1 Let α(s)= 1 coss, sins,s be a curve parametrized by arc length. Then
√2 − −
it is easy to show that T(s)= 1 sins, coss,1 , κ = 1 = 0, tanh = 1 = 0 and ϑ(s) =
(cid:0)√2 − (cid:1) √2 6 −√2 6
1 s+c, c = constant. Here, we can take c = 0. From Eqn.(4), we get k (s) = 1 cos s ,
√2 (cid:0) (cid:1) 1 √2 √2
(cid:0) (cid:1)
6 E.M.SoloumaandM.M.Wageeda
k (s) = 1 sin s . From Eqn.(1), we get N (s) = k (s)T(s)ds, N (s) = k (s)T(s)ds,
2 √2 √2 1 1 2 2
Z Z
then we have (cid:0) (cid:1)
√2 √2
N (s) = cos (1+√2)s cos (1 √2)s ,
1
4(1+√2) − 4(1 √2) −
(cid:18) (cid:16) (cid:17) − (cid:16) (cid:17)
√2 √2 √2 s
sin (1+√2)s sin (1 √2)s , sin
−4(1+√2) − 4(1 √2) − 2 √2
(cid:16) (cid:17) − (cid:16) (cid:17) (cid:16) (cid:17)(cid:19)
√2 √2
N (s) = sin (1+√2)s sin (1 √2)s ,
2
4(1+√2) − 4(1 √2) −
(cid:18) (cid:16) (cid:17) − (cid:16) (cid:17)
√2 √2 √2 s
cos (1+√2)s + cos (1 √2)s , cos .
4(1+√2) 4(1 √2) − 2 √2
(cid:16) (cid:17) − (cid:16) (cid:17) (cid:16) (cid:17)(cid:19)
0.5
0.0
-0.5
8
6
4
2
0
-0.5
0.0
0.5
Figure 1 The curve α=α(s).
In terms of definitions, we obtain special Smarandache curves, see Figures 2 - 5.
1.0
0.5
0.0
-0.5
1.0
0.5
0.0
0.0
0.5
1.0
Figure 2 TN -Smarandache curve.
1
SpecialSmarandacheCurvesAccordingtoBishopFrameinEuclideanSpacetime 7
0.0
-0.5
-1.0
1.0
0.5
0.0
0.0
0.5
1.0
Figure 3 TN -Smarandache curve.
2
-0.6 0.8
1.0
-0.8 1.2
1.4
-1.0
-1.2
0.5
0.0
-0.5
Figure 4 N N -Smarandache curve.
1 2
8 E.M.SoloumaandM.M.Wageeda
-0.20.0
-0.4 0.5
-0.6
1.0
-0.8
1.0
0.5
0.0
Figure 5 TN N -Smarandache curve.
1 2
§4. Conclusion
Consider a curve α= α(s) parametrized by arc-length in Euclidean 3-space E3 that the curve
α(s) is sufficiently smooth so that the Bishop frame adapted to it is defined. In this paper, we
study the problem of constructing Frenet-Serret invariants T ,N ,B ,κ ,τ from a given
{ B B B B B}
some special curve accordingto Bishopframe in Euclidean3-spaceE3 that posses this curve
B
as Smarandache curve. We list an example to illustrate the discussed curves. Finally, we hope
these results will be helpful to mathematicians who are specialized on mathematical modeling.
References
[1] C.Ashbacher,Smarandachegeometries,SmarandacheNotionsJournal,Vol. 8(1–3)(1997)
, 212–215.
[2] L.R. Bishop, There is more than one way to frame a curve, Amer. Math. Monthly, 82 (3)
(1975), 246–251.
[3] B. Bukcu, M. K. Karacan,Paralleltransportframe of the spacelike curve with a spacelike
binormal in Minkowski 3-space, Sel¸cuk J. Appl. Math., 11 (1) (2010), 15–25.
[4] B. Bukcu, M. K. Karacan, Bishop frame of the spacelike curve with a spacelike binormal
in Minkowski 3-space, Sel¸cuk J. Appl. Math., 11 (1) (2010), 15–25.
[5] O¨. Bekta¸s, S. Yce, Special Smarandache curves according to Darboux frame in Euclidean
3- Space, Romanian Journal of Mathematics and Computer Science, 3 (2013), 48–59.
[6] M. C¸etin, Y. Tunc¸er, M. K. Karacan, Smarandache curves according to Bishop frame in
Euclidean 3- space, General Mathematics Notes, 20 (2) (2014), 50–66.
[7] M.P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, Englewood
SpecialSmarandacheCurvesAccordingtoBishopFrameinEuclideanSpacetime 9
Cliffs, NJ, 1976.
[8] B. O.Neill, Elementary Differential Geometry, Academic Press Inc. New York, 1966.
[9] M.Turgut,S.Yılmaz,SmarandachecurvesinMinkowskispace-time,InternationalJournal
of Mathematical Combinatorics, 3 (2008), 51–55.
[10] K. Ta¸sk¨opru¨ and M. Tosun, Smarandache curves according to Sabban frame on, Boletim
da Sociedade Paraneanse de Matematica, 32 (1) (2014), 51–59.