Mathematica Aeterna, Vol. 1, 2011, no. 08, 599 - 610
ON SLANT HELICES AND GENERAL
HELICES IN EUCLIDEAN n -SPACE
Yusuf YAYLI
Department of Mathematics, Faculty of Science, University of Ankara, Tandoğan,
Ankara,
Evren ZIPLAR
Department of Mathematics, Faculty of Science, University of Ankara, Tandoğan,
Ankara,
Abstract
In this paper, in Euclidean n -space E n , we investigate the relation between
slant helices and spherical helices. Moreover, in E n , we show that a slant helix
and the tangent indicatrix of the slant helix have the same axis (or direction). Also,
we give the important relations between slant helices, spherical helices in E n
and geodesic curves on a helix hypersurface in E n .
Keywords: Slant helices, General helices, Spherical helices, Ttangent indicatrix,
Helix hypersurfaces
Mathematics Subject Classification 2000: 53A04, 53B25, 53C40, 53C50
1. Introduction
Slant helice is one of the most important topics of differential geometry.
Izumiya and Takeuchi have investigated the many properties of slant helices that
the normal lines make a constant angle with a fixed direction in Euclidean 3-space
[14]. Moreover, they proved that a space curve is a slant helix if and only if the
geodesic curvature of the principal normal of the curve is a constant function [14].
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Yusuf YAYLI and Evren ZIPLAR
Besides, Izumiya and Takeuchi described a slant helix if and only if the function
2

( )
2
2 32

(   )
is constant, where  is curvature and  is torsion of the curve, respectively[14].
Monterde obtains a geometric characterization of Salkowski curves whose the
normal vector maintains a constant angle with a fixed direction in space [9].
On the other hand, Kula and Yayli have investigated spherical images of
tangent indicatrix of a slant helix in Euclidean 3-space and they proved that the
spherical images are spherical helix in Euclidean 3-space [11].
General helice whose tangents make a constant angle with a fixed direction is
also considerable subject of differential geometry. Many geometers have studied
on this type of curves [7, 13, 12]. In 1845, de Saint Venant first proved that a
space curve is a general helix if and only if the ratio of curvature to torsion be
constant [6]. Moreover, Ali and López consider the generalization of the concept
of general helices in the Euclidean n -space E n [3]. Also, Yılmaz and Turgut
introduce a new version of Bishop frame and introduce new spherical images
[16].
In differential geometry of surfaces, a helix hypersurface in E n is defined by
the property that tangent planes make a constant angle with a fixed direction [2].
Di Scala and Ruiz-Hernández have introduced the concept of these surfaces in [2].
And, A.I.Nistor has also introduced certain constant angle surfaces constructed on
curves in E 3 [1].
Özkaldı and Yayli give some characterization for a curve lying on a surface
for which the unit normal makes a constant angle with a fixed direction [15].
One of the main purposes of this work is to observe the relations between slant
helices and spherical helices. Another purpose of this study is to give an important
relation between helix hypersurfaces and spherical helices.
2. Preliminaries
Definition 2.1 Let  :   R  E n be an arbitrary curve in E n . Recall that
the curve  is said to be of unit speed (or parametrized by the arc-length
function s ) if  (s), (s)  1 , where , is the standart scalar product in the
Euclidean space E n given by
n
X , Y   xi y i ,
i 1
for each X  ( x1 , x 2 ,..., xn ), Y  ( y1 , y 2 ,..., y n )  E n .
Let V1 ( s), V2 ( s),...,Vn ( s) be the moving frame along  , where the vectors Vi
ON SLANT HELICES AND GENERAL HELICES IN EUCLIDEAN n-SPACE
are mutually orthogonal vectors satisfying
 are given by
0 0
k1
 V1   0
 V    k
0
k2 0
 2   1
 V3   0  k 2 0 k 3
.
.
.
 .    ..
.
.
.
.
 ..   .
.
.
Vn1   0
0
0 0

 
0
0 0
 Vn   0
601
Vi ,Vi  1 . The Frenet equations for
...
0
...
0
...
0
...
0
.
.
.
...  k n1
0   V1 
0   V2 
0   V3 
.
.
.
.  . 
.  . 
k n1  Vn1 


0   Vn 
Recall that the functions k i (s ) are called the i -th curvatures of  [8].
Definition 2.2 A unit speed curve  :   R  E n is called general helix if
its tangent vector V1* makes a constant angle with a fixed direction U [3].
Theorem 2.1 Let  :   E n be a unit speed curve and a general helix in E n .
Then the unit direction of the helix:
n
U  cos 1 [V1*   Gi*Vi* ] .
i 3
V1* ( s ),U  cos1
Here
G1*  1, G *2  0, Gi* 
V
*
1
1 * *
[ ki  2Gi  2  (Gi*1 )] , 3  i  n .
ki*1
and
On
the
other
hand,

,V2* ,..., Vn* is the Frenet frame of  and k i*1 , k i* 2 are curvatures of  ,
where 3  i  n [3].
Definition 2.3 A unit speed curve  :   R  E n is called slant helix if its
unit principal normal V 2 makes a constant angle with a fixed direction L [4].
Theorem 2.2 Let  :   E n be a unit speed curve in E n . Define the
functions
k
1
G1   k1 (s ) ds , G2  1 , G3  1 G1 , Gi 
[ k i  2 Gi 2  Gi1 ] ,
k2
k i 1
where 4  i  n . Then  is a slant helix if and only if the function
n
G
2
i
C
i 1
is constant and non-zero. Moreover, the constant C  sec 2  2 , being  2 the
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Yusuf YAYLI and Evren ZIPLAR
angle that makes V 2 with the fixed direction L that determines  [4].
Theorem 2.3 Let  :   E n be a unit speed curve and a slant helix in E n .
Then the unit direction of the helix:
n
L  cos  2 [  GiVi ] .
i 1
Here, V2 ( s ), L  cos  2 and
G1   k1 ( s )ds , G2  1 , G3 
On the other hand, V1 ,V2 ,...,Vn 
k1
1
G1 , Gi 
[ k i 2 Gi 2  Gi1 ] , 4  i  n .
k2
k i 1
is the Frenet frame of  [4].
Definition 2.4 Let  :   E n be a unit speed curve in E n . Harmonic
curvatures of  is defined by
H i :   R  IR , i  0,1,2,..., n  2 ,


0,
k
Hi   1 ,
k 2

1
,
V1[ H i 1 ]  H i 2 k i 
k i 1

i0
i 1
i  2,3,..., n  2
[5].
Theorem 2.4 Let  :   E n be a unit speed curve and a general helix in E n .
Then the unit direction of the helix:
  cos1 [V1*  H 1V3*  ...  H n 2Vn* ] .
Here, V1* ,   cos1 . On the other hand, V1* ,V2* ,..., Vn*  is the Frenet frame of
 and H 1 , H 2 ,..., H n2  are the harmonic curvatures of  [5].
Remark 2.1 If the Frenet frame of the tangent indicatrix  of a space curve
 is V1*  T ,V2*  N , V3*  B, then
ON SLANT HELICES AND GENERAL HELICES IN EUCLIDEAN n-SPACE
603
V1*  T  n
V2*  N 
V3*  B 
1
2
  2
1
 2  2
(  t   b)
( t   b)
 2  2
    
is the torsion of
is the curvature of  ,   

 ( 2   2 )
 , where V1  t ,V2  n,V3  b is the Frenet frame of  and  is the
curvature of  ,  is the torsion of  [10].
and
 
3. SLANT HELICES AND SPHERICAL HELICES
In the following Theorem, we give the relation between slant helices in E n
and spherical helices on the unit hypersphere in Sn -1  E n .
Theorem 3.1 Let  :   R  E n be a unit speed curve ( parametrized by
arclength function s ) in E n and let  :   Sn1  E n be the tangent indicatrix
of the curve  , where S n1 is the unit hypersphere in E n . Then the curve  is
a slant helix with direction L in E n if and only if the curve  is a general
helix (spherical helix) with direction L on S n1  E n . In other words,  and
 have the same direction L .
Proof: The tangent indicatrix of the curve  is defined by
d
  (s ) .
ds
We assume that the arclength parameter of  is s  . Then we can write
d d ds
.

ds 
ds ds 
And, from the Frenet equations (see Definition 2.1),
d
ds
 k1 ( s )V2 (s )
.
ds 
ds 
Thus, from the last equation, by taking norms on both sides, we obtain
ds
1
d

 V2 (s ) .
( k1 ( s )  0) . Hence, we have
ds k1 ( s)
ds
Now, we assume that the curve  is a slant helix with direction L in E n . So,
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Yusuf YAYLI and Evren ZIPLAR
the scalar product between the unit principal normal vector field V 2 with L is
V2 , L  cos  .
d
d
, L  cos .
 V2 . Hence, we have 
ds 
ds 
It follows that the curve  is a general helix (spherical helix) with the direction
On the other hand, we know that
L on S n1  E n .
Conversely, the curve  is a general helix (spherical helix) with direction L on
d
, L  cos , where s  is the arclength
S n1  E n . Then we have 
ds 
d
parameter of  . On the other hand, we know that V2 
. Hence, we have
ds 
V2 , L  cos  . So, we deduce that the curve  is a slant helix with direction L
in E n . This completes the proof.
This above Theorem has the following corollary.
Corollary 3.1 Let  :   R  E n be a slant helix with unit speed (or
parametrized by arclength function s ) and let  :   Sn1  E n be the tangent
indicatrix of the curve  (parametrized by arclength function s  ) , where S n1
is the unit hypersphere in E n .We assume that the direction of  is
n
L  cos [  GiVi ] . Then the direction L can be expressed in the forms
i 1
n
L  cos [V1*   Gi*Vi * ] and L  cos [V1*  H 1V3*  ...  H n 2Vn* ] ,
i 3
where  is constant.
Here, for the curve  :
k1
1
G1 , Gi 
[ k i 2 Gi 2  Gi1 ] , 4  i  n
k2
k i 1
and V1 , V2 ,...,Vn  is the Frenet frame of  .
Moreover, for the curve  :
1
G1*  1 , G2*  0 , Gi*  * [ k i*2 Gi*2  (Gi*1 ) ] , 3  i  n
k i 1
G1   k ( s )ds , G2  1 , G3 
and V1* ,V2* ,..., Vn*  is the Frenet frame of  ,
harmonic curvatures of  .
H 1 , H 2 ,..., H n2 
are the
ON SLANT HELICES AND GENERAL HELICES IN EUCLIDEAN n-SPACE
605
Proof: From the Theorem 3.1, since  and  have the same direction L
d
 V1*  V2 , then V1* , L  V2 , L  cos . So, we deduce that
and
ds 
n
n
cos  [ GiVi ] = cos  [V1*   Gi*Vi * ] = cos [V1*  H 1V3*  ...  H n 2Vn* ] .
i 1
i 3
This completes the proof.
In the following example, it has been obtained a general helix on the unit
hypersphere S2  E 3 by using a slant helix in E 3 .
2
1
2
1
4
Example 3.1 Let  ( s)  ( sin 2 s  sin 8s,
cos 2s  cos 8s, sin 3s ) be
5
40
5
40
15
3
a slant helix with unit speed in E , where  3  s  2 3 . It is easily obtain the
curvatures as follows:
 ( s)  4 sin 3s and  (s )  4 cos 3s ,
where,  is the curvature of  and  is the torsion of  .
Now, we are going to find out the tangent indicatrix  of the curve  :
d
4
1
4
1
4
 (s ) 
 ( cos 2s  cos 8s, sin 2s  sin 8s, cos 3s ) .
ds
5
5
5
5
5
2
That is,  (s) is a general helix on the hypersphere S in E 3 . The slant helix
 is shown in the following Figure 1 and the tangent indicatrix  of the curve
 is shown in the following Figure 2, where  3  s  2 3 .
0
-0.1
-0.2
-0.2
0
0.2
0.4
0.35
0.3
0.25
0.2
Figure 1: The slant helix 
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Yusuf YAYLI and Evren ZIPLAR
-1-0.8
0.5
-0.6-0.4
0.25
0
-0.25
-0.5
0.5
0
-0.5
Figure 2: The tangent indicatrix  of the curve 
For Corollary 3.1, we can give the following example.
Example 3.2 In this example, we are going to find out the directions of the
curves  and  defined in Example 3.1.
The direction of the slant helix  :
3
L  cos  [  GiVi ]  cos [G1t  G2 n  G3 b] ,
i 1
where G1    (s )ds , G2  1 , G3 

G1 and

V1  t ,V2  n,V3  b
is the
Frenet frame of the slant helix  . From the example 3.1, we know that
 ( s)  4 sin 3s and  ( s)  4 cos 3s . So, it is easily obtain the G1 and G3 as
follows:
4
4
G1  cos 3s and G3 
sin 3s .
3
3
On the other hand, from theorem 2.2, we know that G12  G 22  G32  sec 2  .
3
Hence, we can deduce from the last equality that cos  .
5
Finally, the direciton of the slant helix  is found out as
4
3
4
L  ( cos 3s) t  n  ( sin 3s)b .
5
5
5
The direction of the tangent indicatrix  :
ON SLANT HELICES AND GENERAL HELICES IN EUCLIDEAN n-SPACE
607
3
U  cos  [V1*   Gi*Vi* ]  cos  [T  G3* B ] ,
i 3
where G3* 
1

[  G1*  (G2* )] (   is the curvature of  and   is the torsion of
 ) and V1*  T ,V2*  N ,V3*  B is the Frenet frame of  .
From theorem 2.1, we know that G1*  1 and G2*  0 . So, G3* 

. On the

 2  2
     
other hand, from remark 2.1,   
. Hence, we
and   

 ( 2   2 )
( 2   2 ) 3 2
*
obtain G3 
. Again, from remark 2.1, T  n
and
     
B
1
 2  2
( t   b) .
Consequently, ıf we also consider that  ( s)  4 sin 3s and  ( s)  4 cos 3s , then
the direciton of the helix  is found out as
4
3
4
U  ( cos 3s)t  n  ( sin 3s)b ,
5
5
5
3
where cos  .
5
Notice that the directions of the curves  and  equal.
4. HELIX HYPERSURFACES AND HELICES
In this section, we give the important relations between slant helices, spherical
helices in E n and geodesic curves on a helix hypersurface in E n .
Definition 4.1 Given a hypersurface M  E n and an unitary vector d  0
in E n , we say that M is a helix hypersurface with respect to the fixed direction
d if d ,  is constant function along M , where  is a normal vector field on
M [2].
Theorem 4.1 Let M be a helix hypersurface with the direction d in E n
and let  :   R  M be a unit speed geodesic curve on M . Then, the curve
 is a slant helix with the direction d in E n [17].
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Yusuf YAYLI and Evren ZIPLAR
Proof: Let  be a normal vector field on M . Since M is a helix
hypersurface with respect to d , d ,   constant . That is, the angle between
d and  is constant on every point of the surface M . And,  (s )   |  ( s )
along the curve  since  is a geodesic curve on M . Moreover, by using the
Frenet equation  ( s )  V1  k1V2 , we obtain  |  ( s )  k1V2 , where k1 is first
curvature of  . Thus, from the last equation, by taking norms on both sides, we
obtain   V2 or   -V2 . So, d ,V2 is constant along the curve  since
d ,   constant . In other words, the angle between d and V 2 is constant
along the curve  . Consequently, the curve  is a slant helix with the direction
d in E n .
This completes the proof.
Now, we can give a new Theorem by using the above Theorem.
Theorem 4.2 Let  :   R  M  E n be a geodesic curve on M with
unit speed (or parametrized by arclength function s ), where M is a helix
hypersurface with respect to a fixed direction d in E n .Then the tangent
indicatrix  (s) of the curve  (s) is a spherical helix on unit the hypersphere
S n1  E n .
Proof: We assume that  is a geodesic curve on M . Then from Theorem
4.1,  is a slant helix with the direction d in E n . On the other hand, from
Theorem 3.1, the tangent indicatrix of the curve  is a spherical helix with the
direction d on the hypersphere S n1  E n .
This completes the proof.
This above Theorem has the following corollary.
Corollary 4.1 Let M  E n be a helix hypersurface in E n . Then, the tangent
indicators of all geodesic curves on the surface M are spherical helices whose
axes coincide.
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