Applied Mathematical Sciences, Vol. 7, 2013, no. 75, 3739 - 3748
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2013.35257
Characterizations of Slant Helices
According to Quaternionic Frame
Hüseyin KOCAYİĞİT and Beyza Betül PEKACAR
Celal Bayar University, Faculty of Arts and Sciences, Department of Mathematics
45047 Muradiye Campus, Muradiye, Manisa, Turkey
[email protected], [email protected]
Copyright © 2013 Hüseyin KOCAYİĞİT and Beyza Betül PEKACAR. This is an open access
article distributed under the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract. In this study, we studied some integral characterizations of slant helices
according to quaternionic frame in 3- and 4-dimensional Euclidean spaces E 3
and E 4 . Furthermore, we obtain some necessary and sufficient conditions for a
space curve to be a slant helix according to quaternionic frame.
Mathematics Subject Classification : 20G20, 14H50, 14H45
Keywords : Quaternionic curve, quaternionic frame, slant helix.
1. Introduction
The quaternions were first defined by Hamilton in 1843. They are actually
multi-dimensional complex numbers and imaginary part of a quaternion is an
imaginary vector based on three imaginary orthogonal axes.
Some curves are special in differential geometry. They satisfy some
relationships between their curvatures and torsions and have an important role.
One of these curves is general helix which is defined by the property that the
tangent of the curve makes a constant angle with a fixed straight line called the
axis of the general helix. Furthermore, recently new special curves have been
defined and studied by Izumiya and Takeuchi. They have defined slant helix
which is a special curve whose principal normal vector makes a constant angle
with a fixed direction [7]. Kula and et al. studied some characterizations of slant
helices in the Euclidean 3- space [9]. After that, Önder and et al. have regarded
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Hüseyin KOCAYİĞİT and Beyza Betül PEKACAR
the notion of slant helix in E 4 and defined these new curves as B2 -slant helix
[10].
Baharathi and Nagaraj defined the quaternionic curves in E 3 and E 4 and
studied the differential geometry of space curves and introduced Frenet frames
and formulae by using quaternions [1]. Following, quaternionic inclined curves
have been defined and studied by Karadağ and Sivridağ [8]. In [2], Çöken and
Tuna have studied these curves in the semi-Euclidean space E24 . Quaternionic
rectifying curves have been studied by Güngör and Tosun [5]. Gök and et al. have
considered the definition given by Önder and et al. for spatial quaternionic curves
and have defined quaternionic B2 -slant helix. They have given new
characterizations for these curves in E 4 and E24 [3, 4].
In this study, we define quaternionic slant helices and obtain some necessary
and sufficient conditions for a space curve to be a slant helix according to
quaternionic frame in the Euclidean spaces E 3 and E 4 .
2. Preliminaries
In [6,11] and [1], a more complete elementary treatment of quaternions
quaternionic curves can be found respectively.
A real quaternion q is defined by
q = a1e1 + a2 e2 + a3e3 + a4 e4
where ai , (1 ≤ i ≤ 4) are ordinary numbers, and ei , (1 ≤ i ≤ 4), e4 = +1
quaternionic units which satisfy the non-commutative multiplication rules
⎧⎪ei × ei = −e4 , (1 ≤ i ≤ 3)
⎨
⎪⎩ei × e j = −e j × ei = ek , (1 ≤ i, j , k ≤ 3)
and
(1)
are
(2)
where (ijk ) is an even permutation of (123) in the Euclidean space. The algebra
of the quaternions is denoted by Q and its natural basis is given by
{e1 , e2 , e3 , e4 } . We can express a real quaternion in (1) by the form
q = sq + vq ,
where
(3)
sq = a4 is scalar part and vq = a1e1 + a2 e2 + a3e3 is vector part of q . So
the conjugate of q = sq + vq is defined by
q = sq − vq .
(4)
Using these basic products we can write the symmetric real-valued,
non-degenerate, bilinear form as follows:
1
h : Q × Q → IR, (q, p) → h (q, p) = (q × p + p × q )
(5)
2
which is called the quaternion inner product [6]. The norm of q is
Characterizations of slant helices
3741
2
q = h(q, q ) = q × q = q × q = a12 + a22 + a32 + a42 .
(6)
If q = 1 , then q is called unit quaternion. Then, inverse of the quaternion q
is given by
q −1 =
q
.
q
(7)
Let q = sq + vq = a1e1 + a2e2 + a3e3 + a4 e4 and p = s p + v p = b1e1 + b2e2 + b3e3 + b4 e4
be two quaternions in Q . Then the quaternion product of q and p is given by
q × p = sq s p − vq , v p + sq v p + s p vq + vq ∧ v p
where
(8)
and ∧ denote the inner product and vector product in Euclidean
,
3
3-space E , respectively.
And then if q + q = 0 , q is called a spatial quaternion and if q − q = 0 , q
is called a temporal quaternion [1].
Theorem 2.1. The three-dimensional Euclidean space E 3 is identified with the
space of spatial quaternions {q ∈ Q : q + q = 0} in an obvious manner. Let
I = [ 0,1] be an interval in real line IR and let
3
α : I ⊂ IR → Q, s → α ( s ) = ∑ α i ( s )ei
(9)
i =1
be an arc-lengthed curve with nonzero curvatures
{k , r }
and
r
r
r
{t ( s), n1 ( s), n2 ( s)}
denotes the Frenet frame of the curve α ( s) . Then the Frenet formulae of the
quaternionic curve α ( s) are given by
r
⎡ t′ ⎤ ⎡ 0
⎢ r′ ⎥ ⎢
⎢ nr1 ⎥ = ⎢ − k
⎢⎣ n2′ ⎥⎦ ⎢⎣ 0
r
0⎤ ⎡ t ⎤
⎢r ⎥
0 r ⎥⎥ ⎢ n1 ⎥
r
− r 0 ⎥⎦ ⎢⎣ n2 ⎥⎦
k
(10)
where k ( s ) is principal curvature and r ( s ) is torsion of α ( s ) [1].
Theorem 2.2. The four-dimensional Euclidean space E 4 is identified with the
space of unit quaternions. Let I = [ 0,1] be an interval in real line R and let
4
γ : I ⊂ IR → Q, s → γ ( s ) = ∑ α i ( s )ei ,
i =1
e4 = +1
(11)
be a smooth curve in E 4 with nonzero curvatures { K , k , r − K } and
r
r
r
r
T ( s ), N1 ( s ), N 2 ( s ), N 3 ( s ) denotes the Frenet frame of the curve γ ( s) . Then the
{
}
Frenet formulae of the quaternionic curve γ ( s ) are given by
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Hüseyin KOCAYİĞİT and Beyza Betül PEKACAR
⎡T′ ⎤ ⎡ 0
⎢ N ⎥ ⎢− K
⎢ 1⎥ = ⎢
⎢ N 2′ ⎥ ⎢ 0
⎢ ⎥ ⎢
⎣ N 3′ ⎦ ⎣ 0
0 ⎤⎡ T ⎤
0 ⎥⎥ ⎢⎢ N1 ⎥⎥
(12)
−k
r − K ⎥ ⎢ N2 ⎥
0
⎥⎢ ⎥
0 −(r − K )
0 ⎦ ⎣ N3 ⎦
where K ( s ) is principal curvature, k ( s ) is torsion and (r − K )( s ) is
bitorsion of γ ( s ) [1].
K
0
0
k
3. Characterizations of Slant Helices in E 3
In this section, we give the definition and characterizations of slant helices in E 3
according to quaternionic frame. First, we give the following definition:
Definition 3.1. Let α : I → E 3 be a regular spatial quaternionic curve with the
r r r
r
quaternionic frame {t , n1 , n2 } . If the unit vector n1 ( s ) makes a constant angle θ
r
r r
with some fixed unit vectors u , that is h(n1 , u ) = cos θ = const. for all s ∈ I ,
then α is called a spatial quaternionic slant helix according to quaternionic
frame.
Then we can give the followings:
Theorem 3.1. Let α = α ( s ) be a unit speed spatial quaternionic curve in E 3
with non-zero curvatures. Then α is a quaternionic slant helix if and only if the
function
k2
⎛ r ⎞′
(13)
⎟
3 ⎜
2
2 2 ⎝k ⎠
(r + k )
is constant everywhere.
r
Proof. We assume that α is a quaternionic slant helix and u be the constant
r
r
vector field such that the function h(n1 ( s ), u ) = c is constant. Then for the
smooth functions a1 = a1 ( s ) and a2 = a2 ( s ) we have the following form of the
r
vector field u
r
r
r
r
u = a1 ( s )t ( s) + cn1 ( s ) + a2 ( s )n2 ( s ), s ∈ I .
(14)
By differentiating (14) it follows
⎧ a1′ − ck = 0,
⎪
(15)
⎨a1k − a2 r = 0,
⎪ cr + a′ = 0.
2
⎩
Characterizations of slant helices
3743
From the second equation in (15) we have
⎛r⎞
a1 = a2 ⎜ ⎟ .
⎝k⎠
r
Moreover, since the vector u is constant we have
r r
h ( u , u ) = a12 + c 2 + a22 = const.
(16)
(17)
From (16) and (17), we obtain
⎛ ⎛ r ⎞2 ⎞
a ⎜ ⎜ ⎟ + 1⎟ = m 2 = const.
⎜⎝ k ⎠
⎟
⎝
⎠
r r
If m = 0 then a2 = 0 and from (15) we have a1 = c = 0 . This means that u = 0
which is a contradiction. Thus it must be assumed that m ≠ 0 . Then we have,
m
a2 = m
.
2
( kr ) + 1
2
2
The third equation of (15) yields
⎡
d ⎢
m
ds ⎢
⎣
⎤
⎥ = −cr .
2
⎥
( kr ) + 1 ⎦
m
This can be written as
c
⎛ r ⎞′
⎜ ⎟ = m = const.
2
2
m
(r + k ) ⎝ k ⎠
k2
3
2
which gives the desired.
Conversely, assume that the condition (13) is satisfied. In order to simplify the
computations, we assume that the equation (13) is a constant, namely c . We
r
define a vector field u as follows
r r
r
k
r
r
u=
t + cn1 +
n2 .
(18)
r2 + k2
r2 + k2
r
du
r
= 0 , that is, u is a constant vector. On the
By differentiating (18) it follows
ds
r
r
other hand, h ( n1 ( s ), u ) = 0 and this means that α is a slant helix.
4. The Slant Helices According to Quaternionic Frame in E 4
Definition 4.1. A unit speed quaternionic curve α : I → E 4 is called a slant helix
r
if its unit principal normal N1 makes a constant angle with a fixed direction u .
Theorem 4.1. Let α : I → E 4 be a unit speed quaternionic curve in E 4 . Define
the functions
K
1
(19)
G0 = ∫ K ( s )ds, G1 = 1, G2 = G0 , G3 =
[ kG1 + G2′ ] .
k
r−K
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Hüseyin KOCAYİĞİT and Beyza Betül PEKACAR
Then α is a slant helix if and only if the function
3
∑G
i =0
2
1
=c
(20)
is constant and non-zero. Moreover, the constant c = sec2 θ , being θ the angle
r
between N1 and a fixed direction u .
Proof. Let α be a unit speed quaternionic curve in E 4 . Assume that α is a
r
r
quaternionic slant helix curve. Let u be the direction with which N1 makes a
r r
constant angle θ and, without lost of generality, we suppose that h(u , u ) = 1 .
Consider the differentiable functions ai , 0 ≤ i ≤ 3 ,
3
r
r
r
u = a0 ( s )T ( s) + ∑ ai ( s ) N i ( s ), s ∈ I
(21)
i =1
that is,
r r
r r
a0 = h(T , u ), ai = h( N i , u ), 1 ≤ i ≤ 3
(22)
r
for any s . Because the vector field u is constant, a differentiation in (21)
together (19) gives the following system of ordinary differential equation:
a0′ − Ka1 = 0,
⎧
⎪
a0 K − ka2 = 0,
⎪
(23)
⎨
′
a
k
a
(
r
K
)
a
0,
+
−
−
=
1
2
3
⎪
⎪⎩ a2 (r − K ) + a3′ = 0.
Let us define the functions Gi = Gi ( s ) as follows
ai ( s ) = Gi ( s )a1 , 0 ≤ i ≤ 3.
(24)
We point out that a1 ≠ 0 : on the contrary, (24) gives ai = 0 , for 0 ≤ i ≤ 3 and so,
r
u = 0 , which is a contradiction. Since, (23) lead to
⎧G0 = K ( s )ds ,
∫
⎪
⎪G1 = 1,
⎪
(25)
⎨G = K G ,
2
0
⎪
k
⎪
1
⎪G3 =
[ kG1 + G2′ ].
⎩
r−K
The last equation (23) leads to following condition:
G3′ + (r − K )G2 = 0.
(26)
We do the change of variables:
s
dt
(27)
t ( s ) = ∫ (r − K )(u )du ,
= (r − K )( s ).
ds
In particular, and from the last equation of (25), we have
⎛ k (t ) ⎞
G2′ (t ) = G3 (t ) − ⎜
⎟ G1 (t ).
⎝ (r − K )(t ) ⎠
Characterizations of slant helices
3745
As a consequence, if α is a slant helix, substituting the equation (26) to the last
equation, we express
k (t )
G3′′(t ) + G3 (t ) =
G1 (t ).
(r − K )(t )
By the method of variation of parameters, the general solution of this equation is
obtained
⎛
⎡ k (t )
⎤ ⎞
G3 (t ) = ⎜ A − ∫ ⎢
G1 (t ) sin t ⎥ dt ⎟ cos t
⎣ (r − K )(t )
⎦ ⎠
⎝
(28)
⎛
⎡ k (t )
⎤ ⎞
+⎜B+∫⎢
G1 (t ) cos t ⎥ dt ⎟ sin t
r
−
K
t
(
)(
)
⎣
⎦ ⎠
⎝
where A and B are arbitrary constants. Then (28) takes the following form
G3 ( s ) = A − ∫ ⎡⎣ k ( s)G1 ( s) sin ∫ (r − K )( s )ds ⎤⎦ ds cos ∫ (r − K )( s )ds
(29)
⎡
⎤
+ B + ∫ ⎣ k ( s )G1 ( s ) cos ∫ (r − K )( s )ds ⎦ ds sin ∫ (r − K )( s )ds.
(
(
)
)
From (26), the function G2 is given by
(
)
− ( B + ∫ ⎡⎣ k ( s )G ( s) cos ∫ (r − K )( s)ds ⎤⎦ ds ) cos ∫ (r − K )( s )ds.
G2 ( s ) = A − ∫ ⎡⎣ k ( s )G1 ( s ) sin ∫ (r − K )( s )ds ⎤⎦ ds sin ∫ (r − K )( s )ds
(30)
1
By using equation (25) we have
G02 + G12 = const.
Using equations (29) and (30), we have
(31)
(
)
+ ( B + ∫ ⎡⎣ k ( s )G ( s ) cos ∫ (r − K )( s )ds ⎤⎦ ds ) .
G32 + G22 = A − ∫ ⎡⎣ k ( s )G1 ( s ) sin ∫ (r − K )( s )ds ⎤⎦ ds
2
2
(32)
1
It follows from (31) and (32) that
3
∑G
i =0
2
i
= c.
Moreover this constant c can be calculated as follows. From (24) together the
equations (25), we have
3
1 3
1
c = ∑ Gi2 = 2 ∑ ai2 = 2 = sec 2 θ
a1 i =0
a1
i =0
r
where we have used (20) and the fact that u is a unit vector field.
We do the converse of theorem. Assume that the condition (25) is satisfied
r
for a curve α . Let θ be so that c = sec2 θ . Define the unit vector u by
r⎤
r
⎡ r 3
u = cos θ ⎢G0T + ∑ Gi N i ⎥ .
i =1
⎣
⎦
3746
Hüseyin KOCAYİĞİT and Beyza Betül PEKACAR
r
du
r
By taking account (25), a differentiation of u gives that
= 0 , which it means
ds
r
that u is a constant vector field. On the other hand, the quaternion scalar product
r
r
between the unit tangent vector field N1 with u is
r
r
h( N1 ( s ), u ) = cos θ .
Thus α is a quaternionic slant helix in the space E 4 .
Theorem 4.2. There are no quaternionic slant helices with constant and non-zero
curvatures in the space E 4 .
Proof. Let us suppose a quaternionic slant helix with constant and non-zero
curvatures. Then the equations in (32) and (34) hold. Then, we have
3
⎛ ⎛ K ( s) ⎞2 ⎞
(k ( s )) 2
2
⎜
⎟
=
+
+
+
G
1
K
(
s
)
ds
1
∑
i
⎟ ⎟ ∫
⎜ ⎜
(r − K )( s )
i =0
⎝ ⎝ k ( s) ⎠ ⎠
(
3
and it is easy to say that
∑G
i =0
2
i
)
is nowhere constant. By the theorem (4.1) we
arrive that there does not exist a quaternionic slant helix with constant and
non-zero curvatures in the space E 4 .
Theorem 4.3. Let α : I → E 4 be a unit speed quaternionic curve in E 4 . Then
α is a quaternionic slant helix if and only if there exist a c 2 -function G3 ( s)
such that
dG3
1
(33)
G3 =
= − k ( s )G2 ( s )
[ kG1 + G2′ ] ,
(r − K )
ds
where
K
1
G0 = ∫ K ( s )ds, G1 = 1, G2 = G0 , G3 =
[ kG1 + G2′ ].
k
r−K
Proof. Let assume that α is a slant helix curve. By using Theorem 4.1 and by
differentiation the (constant) function given in (29), we obtain
G3 (G3′ + (r − K )G2 ) = 0.
r
This shows (23). Conversely, if (23) holds, we define a vector field u by
r⎤
r
⎡ r 3
u = cos θ ⎢G0T + ∑ Gi N i ⎥ .
i =1
⎣
⎦
r
du
r
By the quaternion equations,
= 0 , and so, u is constant. On the other hand,
ds
r
r
h( N1 ( s ), u ) = cos θ is constant, and this means that α is a quaternionic slant
helix.
Characterizations of slant helices
3747
Theorem 4.4. Let α : I → E 4 be a unit speed quaternionic curve in E 4 . Then
α is a quaternionic slant helix if and only if the following condition is satisfied
(
)
(
)
s
G2 ( s ) = A − ∫ ⎡ kG1 sin ∫ (r − K )ds ⎤ ds sin ∫ (r − K )(u )du
⎣
⎦
s
(34)
− B + ∫ ⎡ kG1 cos ∫ (r − K )ds ⎤ ds cos ∫ (r − K )(u )du
⎣
⎦
for some constants A and B .
Proof. Suppose that α is a slant helix. Let define m( s ) and n( s ) by
s
φ ( s ) = ∫ (r − K )(u )du ,
⎧m( s ) = G3 ( s ) cos φ + G2 ( s ) sin φ + kG1 sin φ ds,
∫
⎪
(35)
⎨
⎪⎩n( s ) = G3 ( s ) sin φ − G2 ( s ) cos φ − ∫ kG1 cos φ ds.
If we differentiate equations (35) with respect to s and taking into account of
dm
dn
(34) and (35), we obtain
= 0 and
= 0 . Therefore, there exist constants
ds
ds
A and B such that m( s ) = A and n( s ) = B . By substituting into (35) and
solving the resulting equations for G2 ( s ) , we get
G2 ( s ) = ( A − ∫ kG1 sin φ ds) sin φ − ( B + ∫ kG1 cos φ ds ) cos φ .
Conversely, suppose that (34) holds. In order to apply Theorem 4.3, we define
G3 ( s ) by
G3 ( s ) = ( A − ∫ kG1 sin φ ds ) cos φ + ( B + ∫ kG1 cos φ ds ) sin φ
s
with φ ( s ) = ∫ (r − K )(u )du . A direct differentiation of (34) gives
G2′ = (r − K )G3 − kG1.
This shows the left condition in (25). Moreover, a straight forward computation
leads to G3′ ( s ) = −(r − K )G2 , which finishes the proof.
References
[1] Baharathi, K., Nagaraj, M., Quaternion valued function of a real Serret-Frenet
formulae, Indian J. Pure Appl. Math. 16 (1985), 741-756.
[2] Çöken, A.C., Tuna, A., On the quaternionic inclined curves in the
semi-Euclidean space E24 , Applied Math. and Comp., 155 (2004), 373-389.
3748
Hüseyin KOCAYİĞİT and Beyza Betül PEKACAR
[3] Gök, İ., Okuyucu, O.Z., Kahraman, F., Hacisalihoğlu, H.H., On the
Quaternionic B2 -slant helix in the Euclidean space E 4 , Adv. Appl. Clifford
Algebras, 21 (2011), 707–719.
[4] Gök, İ., Kahraman, F., Hacisalihoğlu, H.H., On the Quaternionic B2 -slant
helices in the semi – Euclidean Space E24 , Applied Math. and Comp., Vol.
218(2011), 6391-6400.
[5] Güngor, M.A., Tosun, M., Some characterizations of quaternionic rectifying
curves, Differential Geometry-Dynamical Systems , Vol. 13, 2011, pp.
89-100.
[6] Hacisalihoglu, H.H., Hareket Geometrisi ve Kuaterniyonlar Teorisi, Gazi
Üniversitesi, Fen-Edebiyat Fakültesi Yayınları Mat. No: 2, 1983.
[7] Izumiya, S., Takeuchi, N., New special curves and developable
surfaces,Turk.J. Math. Vol. 28 (2004), 153-163.
[8] Karadağ, M., Sivridağ, A.I., Kuaterniyonik Eğilim Cizgileri için
karakterizasyonlar. Erc. Ünv. Fen Bil. Derg. 13 1-2 (1997), 37-53.
[9] Kula, L., Ekmekçi, N., Yaylı, Y., İlarslan, K., Characterizations of Slant
Helices in Euclidean 3- space, Turk.J. Math. Vol. 34(2010), 261-273.
[10] Önder, M., Kazaz, M., Kocayiğit, H., Kılıç, B2 -slant helix in Euclidean
4-space E 4 , Int. J. Cont. Math. Sci. vol. 3, no.29 (2008), 1433-1440.
[11] Ward, J.P., Quaternions and Cayley Numbers, Kluwer Academic Publishers,
Boston/London, 1997.
Received: March 11, 2013