Slant Ruled Surfaces in the Euclidean 3-space E 3
Mehmet Önder
Celal Bayar University, Faculty of Science and Arts, Department of Mathematics, Muradiye
Campus, 45047 Muradiye, Manisa, Turkey. E-mail: [email protected]
Abstract
In this study, we define some new types of ruled surfaces called slant ruled surfaces when the
angle between the Frenet vectors of the surface and a fixed direction is constant. We give
some characterizations for a regular ruled surface to be a slant ruled surface in E 3 . Moreover,
we obtain some corollaries which give the relationships between a slant ruled surface and its
striction line.
MSC: 53A25, 14J26.
Key words: Ruled surface; Frenet frame; slant ruled surface.
1. Introduction
In the local differential geometry, the curves whose curvatures satisfy some special
conditions have an important role. The well-known of such curves are helices. A general
helix in Euclidean 3-space E 3 is defined by the property that the tangent makes a constant
angle with a fixed straight line (the axis of the general helix) [4]. Therefore, a general helix
can be equivalently defined as one whose tangent indicatrix is a planar curve. Certainly, the
helices in E n correspond with those whose unit tangent indicatrices are contained in
hyperplanes. A classical result for the helices stated by M.A. Lancret in 1802 and first proved
by B. de Saint Venant in 1845 (see [21] for details) is: A necessary and sufficient condition
that a curve to be a general helix is that the ratio of the first curvature to the second
curvature be constant i.e., κ / τ is constant along the curve, where κ and τ denote the first
and second curvatures of the curve, respectively. Helices have been studied not only in
Euclidean spaces but also in Lorentzian spaces by some mathematicians and different
characterizations of these curves have been obtained according to the properties of the spaces
[5,6,7,10,19].
Recently, Izumiya and Takeuchi have introduced the concept of slant helix by saying that
the normal lines of the curve make a constant angle with a fixed direction and they have
given a characterization of slant helix in Euclidean 3-space E 3 [8]. Later, Kula and Yaylı
have investigated spherical images, the tangent indicatrix and the binormal indicatrix of a
slant helix and they have obtained that the spherical images of a slant helix are spherical
helices [11]. Moreover, Kula and et al. have also studied slant helices in Euclidean 3-space
and given some other characterizations [12]. Monterde has shown that for a curve with
constant curvature and non-constant torsion the principal normal vector makes a constant
angle with a fixed constant direction, i.e, the curve is a slant helix [13]. Ali has studied the
position vectors of slant helices in Euclidean 3-space [3]. Then Ali and Turgut have given the
corresponding characterizations for the position vector of a timelike slant helix in Minkowski
3-space E13 [2]. Recently, Ali and Lopez have given some new characterizations of slant
helices in Minkowski 3-space E13 [1].
Analogue to the definition of slant helix, Önder and et al. have defined B2 -slant helix in
Euclidean 4-space E 4 by saying that the second binormal vector of a space curve makes a
constant angle with a fixed direction and they have given some characterizations of B2 -slant
1
helix in Euclidean 4-space E 4 [16]. They have also given the corresponding
characterizations for spacelike B2 -slant helix in Minkowski 4-space E14 [17].
Moreover, it is well know a ruled surface has an orthonormal base along its striction line
and called Frenet frame of the ruled surface. In this paper, by considering the Frenet vectors
of a ruled surface we give the definitions of some special ruled surfaces for which the Frenet
vectors make a constant angle with some fixed directions in the space and we called these
surfaces as slant ruled surfaces.
2. Ruled Surfaces in Euclidean 3-space E 3
Let I be an open interval in the real line , f = f (u ) be a regular curve in E 3 defined
on I and q = q (u ) be a unit direction vector of an oriented line in E 3 . Then the parametric
representation of a ruled surface N is given as follows
(1)
r (u , v ) = f (u ) + v q (u ) .
The curve f = f (u ) is called base curve or generating curve of the surface and various
positions of the generating lines q = q (u ) are called rulings. In particular, if the direction of
q is constant, then the ruled surface is said to be cylindrical, and non-cylindrical otherwise.
The distribution parameter of N is defined by
f , q , q
d= ,
(2)
q , q
df dq
where f =
, q =
. If f , q , q = 0 , then the normal vectors are collinear at all points of
du
du
same ruling and at nonsingular points of the surface N , the tangent planes are identical, i.e,
tangent plane contacts the surface along a ruling. Such a ruling is called a torsal ruling. If
f , q , q ≠ 0 , then the tangent planes of the surface N are distinct at all points of same ruling
which is called nontorsal [9].
Definition 2.1. ([9]) A ruled surface whose all rulings are torsal is called a developable ruled
surface. The remaining ruled surfaces are called skew ruled surfaces. From (2) it is clear that a
ruled surface is developable if and only if at all its points the distribution parameter is zero.
For the unit normal vector m of a ruled surface N we have
ru × rv
( f + vq ) × q
m= =
.
2
ru × rv
f + vq , f + vq − f , q
(3)
The unit normal of the surface along a ruling u = u1 approaches a limiting direction as v
infinitely decreases. This direction is called the asymptotic normal (central tangent) direction
and from (3) defined by
q × q
a = lim m(u1 , v) = .
v →±∞
q
The point at which the unit normal of N is perpendicular to a is called the striction point (or
central point) C and the set of striction points of all rulings is called striction curve of the
surface. The parametrization of the striction curve c = c (u ) on a ruled surface is given by
2
q
,f c (u ) = f (u ) + v0 q (u ) = f − q ,
(4)
q , q
q , f
where v0 = − is called strictional distance.
q , q
The vector h defined by h = a × q is called central normal vector which is the surface
normal along the striction curve. Then the orthonormal system C ; q , h , a is called Frenet
frame of the ruled surface N where C is the central point and q , h = a × q , a are unit vectors
of ruling, central normal and central tangent, respectively.
For the Frenet formulae of the ruled surface N with respect to the arc length s of
striction curve we have
 dq / ds   0
k1
0  q 
  
  
(5)
 dh / ds  =  − k1 0 k2   h  ,
 da / ds   0 − k2 0   a 


 
ds3
ds1
where k1 =
, k2 =
and s1 , s3 are the arc lengths of the spherical curves circumscribed
ds
ds
by the bound vectors q and a , respectively and ruled surfaces satisfying k1 ≠ 0 , k2 = 0 are
called conoids (For details [9]).
{
}
Theorem 2.1. ([15]). Let the striction curve c = c ( s ) of the ruled surface N be unit speed
i.e., s is arc length parameter of c ( s ) and let c ( s ) be the base curve of the surface. Then N
is developable if and only if the unit tangent of the striction curve is the same with the ruling
along the curve.
3. q -Slant Ruled Surfaces in Euclidean 3-space E 3
In this section, we introduce the definition and characterizations of q -slant ruled surfaces
in E 3 . First, we give the following definition.
Definition 3.1. Let N be a regular ruled surface in E 3 given by the parametrization
r ( s, v) = c ( s ) + v q ( s), q ( s) = 1 ,
(6)
where c ( s ) is striction curve of N and s is arc length parameter of c ( s ) . Let the Frenet
frame and invariants of N be q , h , a and k1 , k2 , respectively. Then N is called a q -slant
ruled surface if the ruling makes a constant angle with a fixed non-zero direction u in the
space, i.e.,
π
q , u = cos θ = constant ; θ ≠ .
(7)
2
{
}
Then we give the following characterizations for q -slant ruled surfaces. Whenever we
talk about N we will mean that the surface has the parametrization and Frenet elements as
assumed in Definition 3.1.
3
Theorem 3.1. Let N be a regular ruled surface in E 3 with non-zero curvatures. Then N is a
q -slant ruled surface if and only if the function
k1
= tan θ ,
(8)
k2
is constant, where θ is the angle between the ruling q and a fixed direction.
Proof: Let N be a q -slant ruled surface in E 3 . Then denoting by u the unit vector of fixed
direction and by θ the angle between q and u , N satisfies
q , u = cos θ = constant .
(9)
Differentiating (9) with respect to s gives h , u = 0 . Therefore, u lies on the plane spanned
by the vectors q and a , i.e.,
(10)
u = (cos θ )q + (sin θ )a .
By differentiating (10) with respect to s it follows
0 = (cos θ k1 − sin θ k2 )h
(11)
and then we have k1 / k2 = tan θ is constant.
Conversely, given a regular ruled surface N the equation (8) is satisfied. We define
(12)
u = (cos θ )q + (sin θ )a .
Differentiating (12) and using (8) it follows u ′ = 0 , i.e., u is a constant vector. On the other
hand q , u = cos θ = constant . Then N is a q -slant ruled surface.
Theorem 3.2. Let N be a regular ruled surface in E 3 with non-zero curvatures. Then N is a
q -slant ruled surface if and only if det(q′, q′′, q′′′) = 0 .
Proof: Let N be a regular ruled surface in E 3 . From the Frenet formulae in (5) we have
q′ = k1h ,
q′′ = − k12 q + k1′h + k1k2 a ,
q′′′ = (−3k1k1′)q + (k1′′− k13 − k1k22 )h + (2k1′k2 + k2′ k1 )a ,
and then
 k ′
det(q′, q′′, q′′′) = k13k22  1  .
(13)
 k2 
k
Let now N be a q -slant ruled surface in E 3 . By Theorem 3.1 we have 1 is constant.
k2
Then from (13) it follows that det(q′, q′′, q′′′) = 0 .
Conversely, if det(q′, q′′, q′′′) = 0 , since the curvatures are non-zero from (13) it is obtained
k
that 1 is constant and Theorem 3.1 gives that N is a q -slant ruled surface in E 3 .
k2
Theorem 3.3. Let N be a regular ruled surface in E 3 with non-zero curvatures. Then N is a
q -slant ruled surface if and only if det(a′, a′′, a′′′) = 0 .
Proof: From the Frenet formulae we have
4
a′ = −k2 h ,
a′′ = k1k2 q − k2′ h − k22 a ,
a′′′ = (k1′k2 + 2k1k2′ )q + (−k2′′ + k23 + k12 k2 )h − 3k2 k2′ a ,
and so,
 k ′
det(a′, a′′, a′′′) = k25  1  .
 k2 
(14)
Let now N be a q -slant ruled surface in E 3 . By Theorem 3.1 we have
k1
is constant.
k2
Then from (14) it follows that det(a′, a′′, a′′′) = 0 .
Conversely, if det(a′, a′′, a′′′) = 0 , since the curvature k2 is non-zero from (14) it is
k
obtained that 1 is constant and Theorem 3.1 gives that N is a q -slant ruled surface in E 3 .
k2
Theorem 3.4. Let N be a regular ruled surface in E 3 with non-zero curvatures. Then N is a
q -slant ruled surface if and only if
q′′′ = mq′ + 3k1′h′ ,
(15)
k ′′
where m = 1 − (k12 + k22 ) .
k1
Proof: Assume that N is a q -slant ruled surface. From (5) we get
q′′ = − k12 q + k1′h + k1k2 a ,
(16)
2
2
q′′′ = (−3k1k1′)q + (k1′′− k1k2 )h + (2k1′k2 + k1k2′ )a − k1 q′ .
(17)
k
Since 1 is constant, by differentiation we have
k2
k1k2′ = k2 k1′ ,
(18)
and from (5)
1 (19)
h = q′ .
k1
Substituting (18) and (19) in (17) gives

 k ′′
q′′′ =  1 − k12 − k22  q′ − (3k1k1′)q + (3k2 k1′)a .
(20)
 k1

Using the second equation of (5), (15) is obtained from (20).
Conversely, let us assume that (15) holds. Differentiating (19) we obtain
 k′   1  h′ = −  12  q′ +   q′′ ,
(21)
 k1 
 k1 
and so,
 k ′ ′  k′   1  h′′ = −  12  q′ − 2  12  q′′ +   q′′′ .
(22)
 k1 
 k1 
 k1 
Substituting (15) in (22) it follows
5


 k′   k ′ ′ m  k′  (23)
h′′ = −2  12  q′′ −  12  +  q′ + 3  1  h′ .
 k1  k1 
 k1 
 k1 


Now, writing (16) in (23) and using (5) we have
2


′
 k k′  
′


′

k
m
k
1
1
(24)
h′′ = −  2  +  q′ − k ′q − 2   h +  2 1  a .
 k1  k1 
 k1 
 k1 


On the other hand, from (5) it is obtained
(25)
h′′ = −k1q′ − k1′q − k22 h + k2′ a .
Substituting (25) in (24) we have
k2′ k1′
(26)
= .
k2 k1
k
Integrating (26) we get that 1 is constant and by Theorem 3.1, N is a q -slant ruled surface.
k2
Theorem 3.5. Let N be a developable ruled surface in E 3 . Then N is a q -slant ruled
surface if and only if the striction line c ( s ) is a general helix in E 3 .
Proof: Since N is a developable ruled surface in E 3 , from Theorem 2.1 we have
c′( s) = t ( s) = q ( s ) . Then from Definition 3.1, it is clear that N is a q -slant ruled surface if
and only if the striction line c ( s ) is a general helix in E 3 .
4. h -Slant Ruled Surfaces in Euclidean 3-space E 3
In this section, we introduce the definition and characterizations of h -slant ruled surfaces
in E 3 . First, we give the following definition.
Definition 4.1. Let N be a regular ruled surface in E 3 given by the parametrization
r ( s, v) = c ( s) + v q ( s), q ( s ) = 1 ,
(27)
where c ( s ) is striction curve of N and s is arc length parameter of c ( s ) . Let the Frenet
frame and invariants of N be q , h , a and k1 , k2 , respectively. Then N is called a h -slant
ruled surface if the central normal vector h makes a constant angle with a fixed non-zero
direction u in the space, i.e.,
π
h , u = cos θ = constant ; θ ≠ .
(28)
2
{
}
Then, under the assumptions given in Definition 4.1, we can give the following theorems
characterizing h -slant ruled surfaces.
Theorem 4.1. Let N be a regular ruled surface in E 3 . Then N is a h -slant ruled surface if
and only if the function
 k2 ′
k12
(29)

3 
2
2 2  k1 
( k1 + k2 )
is constant.
6
Proof: Assume that N is a h -slant ruled surface in E 3 . Let u be a fixed constant vector
such that h , u = cos θ = c = constant . Then for the vector u we have
u = b1 ( s )q ( s ) + ch ( s ) + b2 ( s )a ( s) ,
(30)
where b1 = b1 ( s ) and b2 = b2 ( s ) are smooth functions of arc length parameter s . Since u is
constant, differentiation of (30) gives
b1′ − ck1 = 0,

(31)
b1k1 − b2 k2 = 0,
b′ + ck = 0.
 2
2
From the second equation of system (31) we have
k
(32)
b1 = b2 2 .
k1
Moreover,
u , u = b12 + c 2 + b22 = constant .
(33)
Substituting (32) in (33) gives
  k 2 
2
b2  1 +  2   = n 2 = constant .
(34)
  k1  


If n = 0 , then b2 = 0 and from (31) we have b1 = 0, c = 0 . This means that u = 0 which is a
contradiction. Thus, n ≠ 0 . Then from (34) it is obtained that
n
.
(35)
b2 = ±
2
 k2 
1+  
 k1 
Considering the third equation of system (31), from (35) we have





d 
n
(36)
±
 = −ck2 .
2
ds 
 k2  
 1+   
 k1  

This can be written as
 k2 ′ c
k12
 = = d = constant ,
3 
n
2
2 2  k1 
( k1 + k2 )
which is desired.
Conversely, assume that the function in (29) is constant, i.e.,
 k2 ′
k12
 = d = constant .
3 
2
2 2  k1 
k
+
k
( 1 2)
We define
u=
k2
k12 + k22
q + dh +
k1
k12 + k22
a.
(37)
7
Differentiating (37) with respect to s and using (29) we have u ′ = 0 , i.e., u is a constant
vector. On the other hand h , u = constant . Thus, N is a h -slant ruled surface in E 3 .
Theorem 4.2. Let N be a regular ruled surface in E 3 with first curvature k1 ≡ 1 . Then N is
a h -slant ruled surface if and only if the second curvature is given by
s
.
(38)
k2 ( s ) = ±
tan 2 θ − s 2
Proof: Let N be a h -slant ruled surface with k1 ≡ 1 . Then for a fixed constant unit vector u
we have
(39)
h , u = cos θ = constant .
Differentiating (39) with respect to s gives
−q + k2 a , u = 0 ,
and from (40) we have
q , u = k2 a , u .
If we put a , u = x , we can write
u = k2 xq + cos θ h + xa .
Since u is unit, from (42) we have
sin θ
x=±
.
1 + k22
Then, the vector u is given as follows
k sin θ sin θ u=± 2
q + cos θ h ±
a.
2
1 + k2
1 + k22
Differentiating (40) with respect to s , it follows
−(1 + k22 )h + k2′ a , u = 0 .
Writing a , u = x and (39) in (45) we have
(1 + k22 ) cos θ
.
k2′
From (43) and (46) we obtain the following differential equation,
k2′
± tan θ
+1 = 0 .
3/ 2
(1 + k22 )
x=
(40)
(41)
(42)
(43)
(44)
(45)
(46)
(47)
By integration from (47) we get
k2
± tan θ
+s+c = 0,
(48)
1 + k22
where c is integration constant. The integration constant can be subsumed thanks to a
parameter change s → s − c . Then (48) can be written as
k2
± tan θ
= −s
(49)
1 + k22
s
which gives us k2 ( s) = ±
.
2
tan θ − s 2
8
s
Conversely, assume that k2 ( s) = ±
x=∓
sin θ
1+ k
2
2
tan 2 θ − s 2
sin θ
=∓
s
1+
holds and let us put
= ∓ cos θ tan 2 θ − s 2 ,
2
(50)
tan 2 θ − s 2
where we are assuming that when k2 has the positive (negative) sign, then x gets the
negative (positive) sign and θ is constant. Thus, k2 x = − s cos θ . Let now consider the vector
u defined by
u = cos θ sq + h ± tan 2 θ − s 2 a
(51)
We will prove that u is constant and makes a constant angle θ with h . By differentiating
(51) and using Frenet formulae we have u ′ = 0 , i.e., the direction of u is constant and
h , u = cos θ = constant . Then N is a h -slant ruled surface.
(
))
(
On the other hand, if the striction line c ( s ) is a geodesic on N , then the principal normal
vector n of c ( s ) and the central normal vector h of N coincide. Then, we have the
following corollary.
Corollary 4.1. Let the striction line c ( s ) be a geodesic on N . Then N is a h -slant ruled
surface if and only if the striction line is a slant helix in E 3 .
If the ruled surface N is developable, then by Theorem 2.1, the Frenet frame t , n , b of
the striction line c ( s ) coincides with the frame q , h , a and we can give the following
{
{
}
}
corollary.
Corollary 4.2. Let N be a developable surface. Then N is a h -slant ruled surface if and
only if the striction line is a slant helix in E 3 .
5. a -Slant Ruled Surfaces in Euclidean 3-space E 3
In this section we introduce the definition of a -slant ruled surfaces in E 3 .
Definition 5.1. Let N be a regular ruled surface in E 3 given by the parametrization
r ( s, v) = c ( s ) + v q ( s ), q ( s) = 1 ,
where c ( s ) is striction curve of N and s is arc length parameter of c ( s ) . Let the Frenet
frame and invariants of N be q , h , a and k1 , k2 , respectively. Then N is called a a -slant
ruled surface if the central tangent vector a makes a constant angle with a fixed non-zero
direction u in the space, i.e.,
π
a , u = cos θ = constant ; θ ≠ .
2
{
}
9
From (10) it is celar that a ruled surface N is a -slant ruled surface if and only if it is a q slant ruled surface. So, all the theorems given in Section 3 also characterize the a -slant ruled
surfaces.
After these definitions and characterizations of slant ruled surfaces we have the
followings:
Let N1 and N 2 be two regular ruled surfaces in E 3 with Frenet frames q1 , h1 , a1 and
q2 , h2 , a2 , respectively. If N1 and N 2 have common central normals i.e., h1 = h2 at the
{
{
}
}
corresponding points of their striction lines, then N1 and N 2 are called Bertrand offsets [20].
Similarly, if a1 = h2 at the corresponding points of their striction lines, then the surface N 2 is
called a Mannheim offset of N1 and the ruled surfaces N1 and N 2 are called Mannheim
offsets [14]. Considering these definitions we come to the following corollaries:
Corollary 5.1. Let N1 be a h -slant ruled surface. Then the Bertrand offsets of N1 form a
family of h -slant ruled surfaces.
Corollary 5.2. Let N1 and N 2 form a Mannheim offset. Then N1 is a q -slant (or a -slant)
ruled surface if and only if N 2 is a h -slant ruled surface.
6. Examples
Example 6.1. Let consider the ruled surface N given by the parametrization
1
1
1
1
1 
1
r ( s, v) =  (1 + s)3/ 2 + v (1 + s )1/ 2 , (1 − s)3/ 2 − v (1 − s )1/ 2 ,
s+v
.
2
3
2
2
2
3
(Fig. 6.1). It is easily seen that det(q′, q′′, q′′′) = 0 . Then Theorem 3.2 gives that N is a q slant ruled surface in E 3 . Since the q -slant ruled surfaces are also a -slant, N is also a a slant ruled surface.
Fig. 6.1.
10
Example 6.2. Let consider the ruled surface
r ( s, v) = (r1 , r2 , r3 ) where
r1 ( s, v) =
N
given by the parametrization
25
9
18
 50

sin(18s ) −
sin(50 s ) + v  cos(18s) − cos(50s )  ,
612
1700
68
 68

25
9
18
 50

cos(18s ) +
cos(50 s ) + v  sin(18s) − sin(50 s)  ,
612
1700
68
 68

15
15
r3 ( s, v) =
sin(16 s ) + v cos(16 s).
272
17
(Fig. 6.2). After some computations, the curvatures of the surface are obtained as
510
510
k1 =
sin(16 s ), k2 =
cos(16 s ) .
17
17
Then we have
 k2 ′
k12
136
,
 =−
3 
255
2
2 2  k1 
( k1 + k2 )
r2 ( s, v) = −
and Theorem 4.1 gives that N is a h -slant ruled surface.
Fig. 6.2.
6. Conclusions
New types of ruled surfaces in Euclidean 3-space E 3 are defined and called slant ruled
surfaces. Some properties of these special surfaces are obtained and the relationships between
slant ruled surfaces and their striction lines are introduced. Of course, one can consider the
definitions and characterizations given in this study for the ruled surfaces of the other spaces
and obtain the corresponding characterizations.
11
References
[1] Ali, A.T., Lopez, R., Slant Helices in Minkowski Space E13 , J. Korean Math. Soc. 48(1)
(2011) 159-167.
[2] Ali, A.T., Turgut, M., Position vector of a time-like slant helix in Minkowski 3-space, J.
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