An. Şt. Univ. Ovidius Constanţa
Vol. 17(2), 2009, 131–142
ON MANNHEIM PARTNER CURVE IN
DUAL SPACE
Siddika Özkaldi, Kazim İlarslan and Yusuf Yayli
Abstract
In this paper, we define Mannheim partner curves in three dimensional dual space D3 and we obtain the necessary and sufficient conditions for the Mannheim partner curves in dual space D3 .
1
Introduction
In the differential geometry of a regular curve in the Euclidean 3-space E3 , it
is well-known that one of the important problem is the characterization of a
regular curve. The curvature functions k1 (curvature κ) and k2 (torsion τ )
of a regular curve play an important role to determine the shape and size of
the curve ([2, 6]). For example: If k1 = k2 = 0, then the curve is a geodesic.
If k1 6= 0 (constant) and k2 = 0, then the curve is a circle with radius 1/k1 .
If k1 6= 0 (constant) and k2 6= 0 (constant), then the curve is a helix in the
space, etc.
Another way to classification and characterization of curves is the relationship between the Frenet vectors of the curves. For example (in 1845)
Saint Venant proposed the question whether upon the surface generated by
the principal normal of a curve, a second curve can exist which has for its
principal normal the principal normal of the given curve. This question was
answered by Bertrand in 1850; he showed that a necessary and sufficient condition for the existence of such a second curve is that a linear relationship
with constant coefficients exists between the first and second curvatures of
Key Words: Mannheim partner curve, dual space, dual space curve
Mathematics Subject Classification: 53A04, 53A25, 53A40
Received: March, 2009
Accepted: September, 2009
131
132
Siddika Özkaldi, Kazim İlarslan and Yusuf Yayli
the given original curve. The pairs of curves of this kind have been called
Conjugate Bertrand Curves, or more commonly Bertrand Curves ([2, 6, 8]) .
There are many works related with Bertrand curves in the Euclidean space
and Minkowski space. Another kind of associated curves are called Mannheim
curve and Mannheim partner curve. If there exists a corresponding relationship between the space curves α and β such that, at the corresponding points
of the curves, the principal normal lines of α coincides with the binormal lines
of β, then α is called a Mannheim curve, and β Mannheim partner curve of
α. Mannheim partner curves was studied by Liu and Wang ([7]) in Euclidean
3− space and in the Minkowski 3−space.
Dual numbers had been introduced by W.K. Clifford (1849 - 1879) as a
tool for his geometrical investigations. After him E. Study used dual numbers
and dual vectors in his research on line geometry and kinematics. He devoted
special attention to the representation of oriented lines by dual unit vectors
and defined the famous mapping: The set of oriented lines in an Euclidean
three-dimension space E3 is one to one correspondence with the points of a
dual space D3 of triples of dual numbers ([3]).
In this paper we study Mannheim partner curves in dual space D3 .
2
Preliminary
By a dual number x
b, we mean an ordered pair of the form (x, x∗ ) for all
∗
x, x ∈ R. Let the set R × R be denoted as D. Two inner operations and an
equality on D = { (x, x)| x, x∗ ∈ R} are defined as follows:
(i) ⊕ : D × D → D for x
b = (x, x∗ ), yb = (y, y ∗ ) defined as
x
b ⊕ yb = (x, x∗ ) ⊕ (y, y ∗ ) = (x + y, x∗ + y ∗ )
is called the addition in D.
(ii) ⊙ : D × D → D for x
b = (x, x∗ ), yb = (y, y ∗ ) defined as
x
b ⊙ yb = x
byb = (x, x∗ ) ⊙ (y, y ∗ ) = (xy, xy ∗ + x∗ y)
is called the multiplication in D.
(iii) If x = y, x∗ = y ∗ for x
b = (x, x∗ ), yb = (y, y ∗ ) ∈ D, x
b and yb are equal,
and it is indicated as x
b = yb.
If the operations of addition, multiplication and equality on D = R × R
with set of real numbers R are defined as above, the set D is called the dual
numbers system and the element (x, x∗ ) of D is called a dual number. In a
dual number x
b = (x, x∗ ) ∈ D, the real number x is called the real part of x
b and
the real number x is called the dual part of x
b. The dual number (1, 0) = 1 is
called unit element of multiplication operation in D or real unit in D. The dual
ON MANNHEIM PARTNER CURVE IN DUAL SPACE
133
number (0, 1) is to be denoted with ε in short, and the (0, 1) = ε is to be called
dual unit. In accordance with the definition of the operation of multiplication,
it can easily be seen that ε2 = 0. Also, the dual number x
b = (x, x∗ ) ∈ D can
∗
be written as x
b = x + εx (see[9, 5]).
The set of D = {b
x = x + εx∗ | x, x∗ ∈ R} of dual numbers is a commutative
ring according to the operations
(i)(x + εx∗ ) + (y + εy ∗ ) = (x + y) + ε(x∗ + y ∗ ),
(ii)(x + εx∗ )(y + εy ∗ ) = xy + ε(xy ∗ + y ∗ x).
The dual number x
b = x + εx∗ divided by the dual number yb = y + εy ∗
provided y 6= 0 can be defined as
The set of
D3
x
b
x + εx∗
x
x∗ y − xy ∗
=
=
+
ε
.
yb
y + εy ∗
y
y2
= D×D×D
−
→ −
→
= {x
b| x
b = (x1 + εx∗1 , x2 + εx∗2 , x3 + εx∗3 )}
−
→ −
→
= {x
b| x
b = (x1 , x2 , x3 ) + ε (x∗1 , x∗2 , x∗3 )}
−
→ → −
→
−
→ −
→ −
= {x
b| x
b =→
x + εx∗ , −
x , x∗ ∈ R3 }
−
→ −
→ →
−
→ −
−
→
is a module on the ring D. For any x
b =→
x + εx∗ , yb = −
y + εy ∗ ∈ D3 , the
−
→
−
→
scalar or inner product and the vector product of x
b and yb are defined by,
respectively,
−
→
−
→ →
−
→ −
→
→
→
→
h x
b , yb i = h −
x,−
yi+ε h −
x , y ∗ i + h x∗ , −
yi ,
−
→ −
→
x
b Λ yb = (b
x2 yb3 − x
b3 yb2 , x
b3 yb1 − x
b1 yb3 , x
b1 yb2 − x
b2 yb1 ) ,
−
→
where x
bi = xi + εx∗i , ybi = yi + εyi∗ ∈ D, 1 ≤ i ≤ 3. If x 6= 0, the norm x
b
−
→ −
−
→
of x
b =→
x + εx∗ is defined by
−
→
q −
−
→
→ −
→
h−
x , x∗ i
→
→
b = h x
b, x
b i = k−
xk+ε −
.
x
k→
xk
−
→
−
→ −
−
→
A dual vector x
b with norm 1 is called a dual unit vector. Let x
b =→
x + εx∗
∈ D3 . The set
134
Siddika Özkaldi, Kazim İlarslan and Yusuf Yayli
−
→ −
→
−
→ −
→
−
→
S2 = { x
b =→
x + εx∗ | x
b = (1, 0); −
x , x∗ ∈ R3 }
b in D3 .
is called the dual unit sphere with the center O
∗
If every xi (t) and xi (t), 1 ≤ i ≤ 3 real valued functions, are differentiable,
the dual space curve
x
b :
t →
=
I ⊂ R → D3
−
→
x
b (t) = (x1 (t) + εx∗1 (t), x2 (t) + εx∗2 (t), x3 (t) + εx∗3 (t))
−
→
−
→
x (t) + εx∗ (t)
−
→
→
in D3 is differentiable. We call the real part −
x (t) the indicatrix of x
b (t). The
−
→
dual arc length of the curve x
b (t) from t1 to t is defined as
Zt D E
Zt
Zt →
→ −
→ −
−
−
→
t , x∗ ´ = s + εs∗ ,
b (t)´ dt = k x (t)´k dt + ε
sb = x
t1
t1
t1
(2.1)
−−→
where b
t is a unit tangent vector of x(t). From now on we will take the arc
−−→
length s of x(t) as the parameter instead of t.
Now we will obtain equations relatively to the derivatives of dual Frenet
vectors throughout the curve in D3 . Let
x
b
:
s →
I → D3
−−→ −−→
−−−→
x
b(s) = x(s) + εx∗ (s)
be a dual curve with the arc length
−
→
dx
b
=
db
s
parameter s of the indicatrix. Then,
−
→
→
dx
b ds −
= b
t
ds db
s
−−→
is called the dual unit tangent vector of x
b(s). With the aid of equation(2.1),
we have
Zs D E
→
−
→ −
t , x∗ ´ ds
sb = s + ε
s1
b
ds
ds
= 1+ε∆, where the prime denotes differentiation with respect
D−
−
→
→ E
→ −
to the arc length s of indicatrix and ∆ = t , x∗ ´ . Since b
t has constant
length 1, its differentiation with respect to sb, which is given by
−
→
−
→
−
→
−
→
db
t
db
t ds
d2 x
b
=
=
=κ
bn
b,
2
db
s
ds db
s
db
s
and from this
ON MANNHEIM PARTNER CURVE IN DUAL SPACE
135
b
→
−
measures the way the curve is turning in D3 . The norm of the vector ddsbt is
−−→
called curvature function of x
b(s). We impose the restriction that the function
→
−
−
→
b
κ
b : I → D is never pure dual. Then, the dual unit vector n
b = κb1 ddsbt is called
−
→
−−→
−−→
the principal normal of x
b(s). The dual vector bb is called the binormal of x
b(s).
−
→
−
→ −
−−→
→ b
The dual vectors b
t,n
b , b are called the dual Frenet trihedron of x
b(s) at
the point x
b(s). The equalities relative to derivatives of dual Frenet vectors
→
−
→ −
→ −
b
t,n
b , bb throughout the dual space curve are written in the matrix form
 → 
 −
→  
 −
b
b
t
t
0
κ
b
0
 −
d 
→ 
→ 


 −


−b
κ
0
τ
b
(2.2)
=
n
b ,
n
b
 −
db
s −
→ 
→ 
0
−b
τ
0
bb
bb
where κ
b = κ + εκ∗ is nowhere pure dual curvature and τb = τ + ετ ∗ is nowhere
pure dual torsion. The formulae (2.2) are called the Frenet formulae of dual
curve in D3 (see [5] ).
3
Mannheim partner curves in D3
In this section, we define Mannheim partner curves in dual space D3 and we
give some characterization for Mannheim partner curves in the same space.
Definition 1. Let D3 be the dual space with the standard inner product h, i .
If there exists a corresponding relationship between the dual space curves α
b
and βb such that, at the corresponding points of the dual curves, the principal
b then α
normal lines of α
b coincides with the binormal lines of β,
b is called
b
a dual Mannheim curve, and β a dual Mannheim partner curve of α
b. The
b is said to be a dual Mannheim pair.
pair {b
α, β}
Let α
b : x
b(b
s) be a dual Mannheim curve in D3 parameterized by its arc
length sb and βb : x
b1 (b
s1 ) the dual Mannheim partner
curve of with an arc
−
→ −
→
−
→
length parameter sb1 . Denote by b
s) the Frenet frame field
t (b
s), n
b (b
s), bb (b
along α
b:x
b(b
s).
In the following theorems, we give a necessary and sufficient condition for
a dual space curve to be a Mannheim curve.
Theorem 1. A dual space curve in D3 is a dual Mannheim curve if and only
b κ2 + τb2 ), where λ
b
if its curvature κ
b and torsion τb satisfy the formula κ
b = λ(b
is never pure dual constant.
136
Siddika Özkaldi, Kazim İlarslan and Yusuf Yayli
Proof. Let α
b : x
b(b
s) be a dual Mannheim curve in D3 with the arc length
parameter sb and βb : x
b1 (b
s1 ) the dual Mannheim partner curve of with an arc
length parameter sb1 . Then by the definition we can assume that
→
b s)−
x
b1 (b
s) = x
b(b
s) + λ(b
n
b (b
s)
(3.1)
b s). By taking the derivative of (3.1) with
for some never pure dual constant λ(b
respect to sb and applying the Frenet formulas we have
−
→
b−
→ dλ
→ b −
db
x1 (b
s) bκ b
n
b + λb
τ bb.
= 1 − λb
t +
db
s
db
s
−
→
−
→
Since b
t1 is coincident with bb1 in direction, we get
b s)
dλ(b
= 0.
db
s
b is a never pure dual constant. Thus we have
This means that λ
−
−
→
→
db
x1 (b
s) bτ bb .
bκ b
t + λb
= 1 − λb
db
s
On the other hand, we have
−
→ db
s
x1 db
b
=
t1 =
db
s db
s1
bκ
1 − λb
−
−
→ db
→
s
b
b
.
t + λb
τ bb
db
s1
By taking the derivative of this equation with respect to sb1 and applying the
Frenet formulas we obtain
−
→
s
db
t1 db
db
s db
s1
=
→
→ db
→ τ−
κ−
s
bb
bκ2 − λb
bτ 2 −
b db
b db
b
t + κ
b − λb
n
b +λ
−λ
db
s
db
s
db
s1
−
−
→ 2
→
bκ b
bτ bb d sb.
+ 1 − λb
t + λb
db
s21
From this equation we get
This completes the proof.
bκ2 − λb
bτ 2
κ
b − λb
db
s
= 0,
db
s1
b κ2 + τb2 ).
κ
b = λ(b
ON MANNHEIM PARTNER CURVE IN DUAL SPACE
137
Theorem 2. Let α
b:x
b(b
s) be a dual Mannheim curve in D3 with the arc length
parameter sb. Then βb : x
b1 (b
s1 ) is the dual Mannheim partner curve of if and
only if the curvature κ
b1 and the torsion τb1 of βb satisfy the following equation
κ
b1
db
τ1
=
1+µ
b2 τb12
db
s1
µ
b
for some never pure dual constant µ
b.
Proof. Suppose that α
b:x
b(b
s) is a dual Mannheim curve. Then by the definition
we can assume that
−
→
x
b(b
s1 ) = x
b1 (b
s1 ) + µ
b(b
s1 )bb1 (b
s1 ) ,
(3.2)
for some function µ
b(b
s1 ). By taking the derivative of (3.2) with respect to sb1
and applying the Frenet formulas, we have
−
→
−
→ db
−
→
−
→
s
b
t
= b
t 1 + bb − µ
bτb1 n
b 1.
db
s1
(3.3)
−
→
−
→
Since bb is coincident with n
b in direction, we get
db
µ
= 0.
db
s1
This means that µ
b is never a pure dual constant. Thus we have
−
→ db
−
→
−
→
s
b
t
= b
t 1−µ
bτb1 n
b 1.
db
s1
On the other hand, we have
−
→ −
→
−
→
b
b
t = b
t 1 cos θb + n
b 1 sin θ,
(3.4)
(3.5)
−
→
−
→
where θb is the dual angle between b
t and b
t 1 at the corresponding points of α
b
b By taking the derivative of this equation with respect to sb1 , we obtain
and β.
!
!
−
→
b
−
→
−
→
−
→ db
dθb
d
θ
s
=− κ
b1 +
sin θb b
t1+ κ
b1 +
cos θb n
b 1 + τb1 sin θb bb 1 .
κ
bn
b
db
s1
db
s1
db
s1
−
→
−
→
From this equation and the fact that the direction of n
b is coincident with bb 1 ,
we get
 b
 κ
b1 + ddsbθ1 sin θb = 0
b
 κ
b1 + ddsbθ1 cos θb = 0.
138
Siddika Özkaldi, Kazim İlarslan and Yusuf Yayli
Therefore we have
dθb
= −b
κ1 .
(3.6)
db
s1
−
→
−
→
From (3.4), (3.5) and notice that b
t 1 is orthogonal to bb 1 , we find that
Then we have
1
µ
bτb1
db
s
=
=−
.
db
s1
cos θb
sin θb
b
µ
bτb1 = − tan θ.
By taking the derivative of this equation and applying (3.6), we get
that is
µ
b
db
τ1
=κ
b1 1 + µ
b2 τb12 ,
db
s1
db
τ1
κ
b1
=
1+µ
b2 τb12 .
db
s1
µ
b
Conversely, if the curvature κ
b1 and torsion τb1 of the dual curve βb satisfy
κ
b1
db
τ1
1+µ
b2 τb12
=
db
s1
µ
b
for some never pure dual constant µ
b(b
s), then we define a dual curve by
−
→
x
b(b
s1 ) = x
b1 (b
s1 ) + µ
bbb1 (b
s1 )
(3.7)
and we will prove that α
b is a Mannheim curve and βb is the partner curve of
α
b. By taking the derivative of (3.7) with respect to sb1 twice, we get
−
→ db
−
→
−
→
s
b
t
= b
t 1−µ
bτb1 n
b 1,
db
s1
2
−
→
−
→
−
→ d2 sb
→
−
→ db
db
τ1 −
s
b
b
b1 − µ
b
bκ
b1 τb1 t 1 + κ
n
b1−µ
bτb12 bb 1 ,
+ t 2 =µ
κ
bn
b
db
s1
db
s1
db
s1
(3.8)
(3.9)
respectively. Taking the cross product of (3.8) with (3.9) and noticing that
we have
κ
b1 − µ
b
db
τ1
+µ
b2 κ
b1 τb12 = 0,
db
s1
3
−
→ db
−
→
−
→
s
b
κ
bb
t 1+µ
bτb12 n
b 1.
=µ
b2 τb13 b
db
s1
(3.10)
ON MANNHEIM PARTNER CURVE IN DUAL SPACE
139
By taking the cross product of (3.10) with (3.8), we obtain also
−
→
κ
bn
b
db
s
db
s1
4
→
−
b2 τb12 bb 1 .
= −b
µτb12 1 + µ
−
→
This means that the principal normal direction n
b of α
b:x
b(b
s) coincides with
−
→
b1 (b
s1 ). Hence α
b:x
b(b
s) is a dual Mannheim
the binormal direction bb 1 of βb : x
b
curve and β : x
b1 (b
s1 ) is its dual Mannheim partner curve.
Definition 2. A dual helix is a dual curve for which the tangent makes a
constant dual angle with a dual fixed line.
Proposition 1. Let α
b:x
b(b
s) be a dual Mannheim curve in D3 with the arc
length parameter sb and βb : x
b1 (b
s1 ) the dual Mannheim partner curve of with
an arc length parameter sb1 . If α
b:x
b(b
s) is a generalized dual helix, then βb :
x
b1 (b
s1 ) is a dual straight line.
→
−
→ −
→ −
Proof. Let b
t,n
b , bb be the tangent, principal normal and binormal vector
field of the curve α
b : x
b(b
s), respectively. From the properties of generalized
dual helices and the definition of dual Mannheim curves, we have
−
→
−
→
bb 1 .b
p= n
b .b
p=0
for some constant dual vector pb. Then it is easy to obtain that τb1 = κ
b1 ≡ 0.
Proposition 2. If a generalized dual helix is the Mannheim partner curve of
some curve α
b:x
b(b
s) in D3 , then the ratio of torsion and curvature of the curve
α
b:x
b(b
s) is
b
c2
1 −bc1 sb
τb
= ebc1 sb −
e
,
κ
b
2
2b
c2
for some nonzero constant b
c1 and for some never pure dual constant b
c2 , and
sb is the arc length parameter of α
b. In particular, if we put b
c1 = b
c2 = 1, we
have
τb
esb − e−sb
=
= sinh sb.
κ
b
2
→
−
→ −
→ −
Proof. Let b
t,n
b , bb be the tangent, principal normal and binormal vector
field of the curve α
b : x
b(b
s), respectively. From the properties of generalized
dual helices and the definition of dual Mannheim curves, we have
−
→
bb 1 .b
p = sin θb0 ,
140
Siddika Özkaldi, Kazim İlarslan and Yusuf Yayli
for some constant dual vector pb and some constant dual angle θb0 . From the
last equation we know that sin θb0 6= 0 and κbτb 6= dual constant. By taking the
derivative of this equation with respect to sb twice, we get
−
→
−
→
−b
κb
t .b
p + τb n
b .b
p = 0,
→
→
db
τ−
db
κ−
b
n
b .b
p= κ
b2 + τb2 sin θb0 .
t .b
p+
db
s
db
s
b κ2 + τb2 ), we obtain
By a direct calculation and using κ
b = λ(b
−
−
→
b
t .b
p=
τb
−
→
n
b .b
p=
Taking the derivative, we have

κ
b=
τ
b
bκ d( κb )
λb
ds
b
1
τ
b
b d( κb )
λ
ds
b
1

1 −
b
λ
τb =
b
λ
sin θb0 ,
sin θb0 .
τ
b
d κ
b
ds2
κ
b
τb
ds
b
τ
b
d2 ( κ
b)
ds
b2

( ) 

b
2  ,
τ
b

d( κ
b)
2
τ
b
d( κ
b)
ds
b
2 ,
respectively. From these equations, we find that
τ
b
d2 ( κ
b)
τb
2
ds
b
=
.
2
τ
b
b
κ
b
2 τ
d( κ
b)
τb d ( κ
b)
−
ds
b
κ
b dsb2
s), then we get the following differential equation
Let κbτb = yb(b
1 + yb2
2
d2 yb
db
y
= 0.
−
y
b
db
s2
db
s
Solving this equation, we obtain that
yb(b
s) = b
c0
ON MANNHEIM PARTNER CURVE IN DUAL SPACE
or
yb(b
s) =
141
1 −bc1 sb
b
c2 bc1 sb
e −
e
,
2
2b
c2
for some nonzero constant b
c0 ,b
c1 and for some never pure dual constant b
c2 .
Thus, the proposition is proved.
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University of Ankara
Faculty of Science
Department of Mathematics
Tandoğan, Ankara, Turkey
Email: sozkaldi@kku.edu.tr
142
University of Kırıkkale
Faculty of Science and Arts
Department of Mathematics
Kırıkkale, Turkey
Email: kilarslan@yahoo.com
University of Ankara
Faculty of Science
Department of Mathematics
Tandoğan, Ankara, Turkey
Email: yayli@science.ankara.edu.tr
Siddika Özkaldi, Kazim İlarslan and Yusuf Yayli