On the Involutes of the Spacelike Curve with a Timelike

On the Involutes of the Spacelike Curve with a Timelike

International Mathematical Forum, 4, 2009, no. 31, 1497 - 1509
On the Involutes of the Spacelike Curve with a
Timelike Binormal in Minkowski 3-Space
Mustafa Bilici
Amasya Üniversitesi, Merzifon Meslek Yüksekokulu
05300 Merzifon, Amasya, Turkey
[email protected]
Mustafa Çalışkan
Ondokuz Mayıs Üniversitesi, Fen Edebiyat Fakültesi
Matematik Bölümü, 55139 Kurupelit, Samsun, Turkey
[email protected]
Abstract
The involute of a given curve is a well-known concept in 3-dimensional Euclidean
space IR 3 (see [7],[8]).
According to reference [6], M 1 is a timelike curve then the involute curve M 2 is a
spacelike curve with a spacelike or timelike binormal. On the other hand, it has been
investigated the involute and evolute curves of the spacelike curve M 1 with a spacelike
binormal in Minkowski 3-space and it has been seen that the involute curve M 2 is
timelike, [1].
In this paper, we have defined the involute curves of the spacelike curve M 1 with a
timelike binormal in Minkowski 3-space IR13 . We have seen that the involute curve M 2
must be a spacelike curve with a spacelike or timelike binormal.
The relationship between the Frenet frames of the involüte-evolüte curve couple and
some new characterizations with relation to the curve couple have been found.
Mathematics Subject Classification(2000): 53A04, 53B30
Keywords: Minkowski space, involüte-evolüte curve couple, Frenet frames
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M. Bilici and M. Çalışkan
1. INTRODUCTION
Let Minkowski 3-space IR13 be the vector space IR 3 provide with the Lorentzian
inner product g given by g( X , X ) = − x12 + x22 + x32 , where X = ( x1 , x 2 , x 3 ) ∈ IR 3 .
A vector X = ( x1 , x2 , x3 ) ∈ IR 3 is said to be timelike if g ( X , X ) < 0 , spacelike if
g ( X , X ) > 0 and lightlike (or null) if g ( X , X ) = 0 . Similarly, an arbitrary curve
α = α (s ) in IR13 where s is a pseudo-arclenght parameter, can locally be timelike
spacelike or null (lightlike), if all of its velocity vectors α ′(s ) are respectively timelike,
spacelike or null, for every s ∈ I ⊂ IR . A lightlike vector X is said to be pozitive (resp.
negative) if and only if x1 > 0 (resp. x1 < 0 ) and a timelike vector X is said to be pozitive
(resp. negative) if and only if x1 > 0 (resp. x1 < 0 ). The norm of a vector X is defined by
X
IL
=
g( X , X )
The vectors X = ( x1 , x2 , x3 ) , Y = ( y1 , y2 , y3 ) ∈ IR13 are orthogonal if and only if
g ( X ,Y ) = 0 , [7].
Now let X and Y be two vectors in IR 13 , then the Lorentzian cross product is given
by
X × Y = ( x3 y2 − x2 y3 , x1 y3 − x3 y1 , x1 y2 − x2 y1 ) , [5].
(1)
We denote by {T (s ) , N (s ) , B(s )} the moving Frenet frame along the curve α . Then T,
N and B are the tangent, the principal normal and the binormal vector of the curve α ,
respectively. Depending on the causal character of the curve α , we have the following
Frenet formulae and instantaneous rotation vectors:
i ) Let α be a unit speed timelike space curve with curvature κ and torsion τ . Let
Frenet frames of α be {T , N , B}.
In this trihedron, T is timelike vector, N and B are spacelike vectors. For this vectors, we
can write
T × N = − B , N × B = T , B ×T = −N ,
where × is the Lorentzian cross product, [5] in space IR13 . In this situation, the Frenet
formulas are given by
T ′ = κ N , N ′ = κ T − τ B , B′ =τ N , [4].
The Frenet instantaneous rotation vector for the timelike curve is given by
W = τ T − κ B , [2].
(2)
ii ) Let α be a unit speed spacelike space curve with a spacelike binormal. In this
trihedron, we assume that T and B spacelike vectors and N timelike vector. In this
situation,
T × N = − B , N × B = −T , B × T = N ,
The Frenet formulas are given by
Involutes of the spacelike curve
1499
T′ =κ N ,
N ′ = κ T + τ B,
B′ = τ N , [4].
The Frenet instantaneous rotation vector for the spacelike curve is given by
W = −τ T + κ B , [2].
(3)
iii ) Let α be a unit speed spacelike space curve. In this trihedron, we assume that T and
N spacelike vektors and B timelike vector. For this trihedron we write
T × N = B , N × B = −T , B × T = − N ,
The Frenet formulas are given by
T ′ = κ N , N ′ = − κ T + τ B , B ′ =τ N , [4].
The Frenet instantaneous rotation vector for the α curve is given by
W = τ T − κ B , [2].
(4)
If the curve α is non-unit speed curve in space IR13 , then
α ′ × α ′′ IL
det (α ′,α ′′,α ′′′)
κ=
, τ =
.
3
α ′ IL
α ′ × α ′′ IL 2
(5)
Lemma 1. Let X and Y be nonzero Lorentz orthogonal vektors in IR13 . If X is timelike,
then Y is spacelike, [8].
Lemma 2. Let X, Y be positive (negative) timelike vectors in IR13 . Then
g ( X ,Y ) ≤ X Y with equality if and only if X and Y are linearly dependent, [8].
Lemma 3. i ) The Timelike Angle between Timelike Vectors
Let X and Y be positive (negative) timelike vectors in IR13 . By the Lemma 2, there is a
unique nonnegative real number ϕ ( X ,Y ) such that
g ( X ,Y ) = X Y cosh ϕ ( X ,Y )
The Lorentzian timelike angle between X and Y is defined to be ϕ (X , Y ) .
ii ) The Spacelike Angle between Spacelike Vectors
Let X and Y be spacelike vectors in IR13 that span a spacelike vector subspace. Then we
have g ( X ,Y ) ≤ X Y . Hece, there is a unique real number ϕ ( X ,Y ) between 0 and π
such that
g ( X ,Y ) = X Y cos ϕ ( X ,Y )
The Lorentzian spacelike angle between X and Y is defined to be ϕ ( X ,Y ) .
iii ) The Timelike Angle between Spacelike Vectors
Let X and Y be spacelike vectors in IR13 that span a timelike vector subspace. Then we
have g ( X ,Y ) > X Y . Hence, there is a unique positive real number ϕ ( X ,Y ) such
that
g ( X ,Y ) = X Y cosh ϕ ( X ,Y )
The Lorentzian timelike angle between X and Y is defined to be ϕ ( X ,Y ) .
iv ) The Angle between Spacelike and Timelike Vectors
Let X be a spacelike vector and Y a positive timelike vector in IR13 . Then there is a unique
nonnegative real number ϕ ( X ,Y ) such that
1500
M. Bilici and M. Çalışkan
g ( X ,Y ) = X Y sinh ϕ ( X ,Y ) .
The Lorentzian timelike angle between X and Y is defined to be ϕ ( X ,Y ) , [8].
2. THE INVOLUTES OF THE SPACELIKE CURVE WITH A TIMELIKE
BINORMAL IN IR13
Definition 1. Let M 1 , M 2 ⊂ IR13 be two curves which are given by (I ,α ) and (I , β )
coordinate neighbourhoods, resp. Let Frenet frames of M 1 and M 2 be {T , N , B}
and T * , N * , B* , resp. M 2 is called the involüte of M 1 ( M 1 is called the evolute of
M 2 ) if
{
}
g( T ,T * ) = 0 .
(6)
If M 1 is a unit speed spacelike curve with a timelike binormal then the involute
curve M 2 must be a spacelike curve with a spacelike or timelike binormal. In this
situation, the causal characteristic of the Frenet frames of the curves M 1 and M 2 must be
of the form
{T spacelike, N spacelike, B timelike}
and
T * spacelike, N * timelike, B* spacelike or T * spacelike, N * spacelike, B* timelike .
So we can write
g T * ,T * = +1 , g N * , N * = m1 = ε 0 , g B* , B* = m1 = ε 0 .
(7)
Definition 2.( Unit Vectors C of Direction W for Nonnull Curves):
i ) For the curve M 1 with a timelike tangent, θ being a Lorentzian timelike angle
between the spacelike binormal unit vector -B and the Frenet instantaneous rotation vector
W,
a) If κ > τ , then W is a spacelike vector. In this situation, from the Lemma 3. iii) we can
{
} {
(
)
(
)
}
(
)
write
⎧⎪κ = W cosh θ
⎨
⎪⎩τ = W sinh θ
,
W
2
= g (W ,W ) = κ 2 − τ 2
(8)
and
W
(9)
= sinh θ T − cosh θ B ,
W
where C is unit vector of direction W.
b) If κ < τ , then W is a timelike vector. In this situation, from the Lemma 3.iv) we can
C=
write
⎧⎪κ = W sinh θ
⎨
⎪⎩τ = W cosh θ
and
,
W
C = cosh θ T − sinh θ B .
2
(
= − g (W ,W ) = − κ 2 − τ 2
)
(10)
(11)
Involutes of the spacelike curve
1501
ii ) For the curve M 1 with a timelike principal normal, θ being an angle between the B
and the W, if B and W spacelike vectors that span a spacelike vector subspace then by the
Lemma 3. ii) we can write
⎧⎪κ = W cos θ
2
,
W = g (W ,W ) = κ 2 + τ 2 .
(12)
⎨
⎪⎩τ = W sin θ
and
C = − sinθ T + cosθ B
(13)
iii ) For the curve M 1 with a timelike binormal, θ being a Lorentzian timelike angle
between the -B and the W,
a) If τ > κ , then W is a spacelike vector. From the Lemma 3.iv) , we can write
⎧⎪κ = W sinh θ
⎨
⎪⎩τ = W cosh θ
,
g (W ,W ) = W
2
(
= τ 2 −κ2
)
(14)
and
C = coshθT − sinh θB
(15)
b) If τ < κ , then W is a timelike vector. In this situation, from the Lemma 3.i) we have
⎧⎪κ = W cosh θ
⎨
⎪⎩τ = W sinh θ
and
,
g (W ,W ) = − W
2
(
= − τ 2 −κ2
)
C = sinh θT − coshθB .
(16)
(17)
Theorem 1. Let (M 2 , M 1 ) be the involüte-evolüte curve couple which are given by
( I ,α ) and ( I , β ) coordinate neighbourhoods, respectively. The distance between the
points α (s ) ∈ M 1 and β (s ) ∈ M 2 of the (M 2 , M 1 ) is given by
d IL (α (s ), β (s )) = c − s , c = cons tan t ∀s ∈ I .
Proof: If M 2 is the involüte of M 1 , we have
β (s ) = α (s ) + λT (s ) , λ = c − s , α ′ = T .
Let us derivative both side with respect to s:
dβ dα dλ
dT
=
+
.
T +λ
ds
ds ds
ds
dT
For being
= T ′ = κN ,
ds
dβ ⎛
dλ ⎞
= ⎜1 +
⎟T + λκN
ds ⎝
ds ⎠
where s and s* are arc parameter of M 1 and M 2 , respectively. Thus we have
ds* ⎛
dλ ⎞
= ⎜1 +
⎟T + λκN .
ds ⎝
ds ⎠
Making inner product with T this equation’s both side, we have
T*
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M. Bilici and M. Çalışkan
dλ ⎞
ds*
⎛
g T ,T * = ⎜ 1 +
⎟ g (T ,T ) + λκg (T , N )
ds ⎠
ds
⎝
From the definition of the involüte-evolüte curve couple, g T ,T * = 0 . If M 1 is a
spacelike then we can write g (T ,T ) = +1 and g (T , N ) =0. Thus we obtain
dλ
1+
= 0 ⇒ λ = c − s , c = cons tan t
ds
From the definition of the distance in the Lorentzian space, we easily find
d IL (α (s ), β (s )) = β (s ) − α (s ) ,
(
)
(
)
= λT = λ T ,
=λ
= c−s .
Theorem 2. Let (M 2 , M 1 ) be the involüte-evolüte curve couple which are given by
( I ,α ) and ( I , β ) coordinate neighbourhoods, respectively. Let Frenet frames of M 1
and M 2 in the points α (s ) ∈ M 1 and β (s ) ∈ M 2 be {T , N , B} and T * , N * , B* ,
respectively. For the curvature and torsion of curve M 2 , we have
ε0 κ 2 −τ 2
κτ ′ − κ ′τ
*2
(18)
, τ* =
κ =
.
2 2
(c − s ) κ
c − sκ τ 2 − κ 2
{
(
}
)
Proof: If M 2 is the involüte of M 1 , we have
β (s ) = α (s ) + λT (s ) , λ = c − s , α ′ = T .
Let us derivative both side with respect to s:
*
dβ dβ ds*
* ds
=
=T
= (c − s )κN ,
ds ds* ds
ds
where s and s* are arc parameter of M 1 and M 2 , respectively. We can find
ds*
= (c − s )κ
ds
(19)
(20)
thus we have
T* = N .
Taking the derivative of the last equation
dT * ds* dN
=
= κT − τB ,
ds* ds
ds
ds
κ * N * = (− κT + τB ) * ,
ds
(
) {
(21)
}
2
⎛ ds ⎞
κ g N , N = κ g (T ,T ) + τ g (B , B ) ⎜ * ⎟ .
⎝ ds ⎠
From the equations (6),(7) and (20), we have
2
1
κ* ε0 = κ 2 −τ 2
,
(c − s )2 κ 2
*2
*
(
*
)
2
2
Involutes of the spacelike curve
κ
*2
1503
ε 0 (κ 2 − τ 2 )
=
.
(c − s )2 κ 2
From the equation (19) we can write
β ′ = (c − s )κN
and
β ′′ = −(c − s )κ 2T + ( (c − s )κ ′ − κ )N + (c − s )κτB .
If we calculate vector β ′ × β ′′ , then we get
(22)
β ′ × β ′′ = −(c − s )2 κ 2 τ T + (c − s )2 κ 3 B ,
(23)
β ′ × β ′′ IL = c − s κ τ − κ .
(24)
2
4
4
2
2
If we take the derivative of equation (22) we have
β ′′′ = (2κ 2 − 3λκκ ′)T + (− λκ 3 − 2κ ′ + λκ ′′ + λκτ 2 )N + (2λκ ′τ − 2κτ + λκτ ′)B
From the Lemma1. we find
3
3
det (β ′, β ′′, β ′′′) = (c − s ) κ 4τ ′ − (c − s ) κ 3κ ′τ .
(25)
det (β ′ , β ′′ , β ′′′)
For being τ * =
, from the equations (24) and (25), we get
2
β ′ × β ′′ IL
3
(
c − s ) κ 3 (κτ ′ − κ ′τ )
,
τ =
4
4 2
2
*
c − s κ τ −κ
τ* =
κτ ′ − κ ′τ
.
c − sκ τ 2 −κ 2
Theorem 3. Let (M 2 , M 1 ) be the involüte-evolüte curve couple. The Frenet vectors of
the curve couple (M 2 , M 1 ) as follow.
(1) If W is a spacelike vector( κ < τ ), then
⎡T * ⎤ ⎡ 0
1
0 ⎤⎡T ⎤
⎢ *⎥ ⎢
⎥⎢ ⎥
⎢ N ⎥ = ⎢ sinh θ 0 − cosh θ ⎥ ⎢ N ⎥ .
⎢ B* ⎥ ⎢⎣− cosh θ 0
sinh θ ⎥⎦ ⎢⎣ B ⎥⎦
⎣ ⎦
(2) If W is a timelike vector( κ > τ ), then
(26)
⎡T * ⎤ ⎡ 0
1
0 ⎤⎡T ⎤
⎢ *⎥ ⎢
⎥⎢ ⎥
(27)
⎢ N ⎥ = ⎢− coshθ 0 sinhθ ⎥ ⎢ N ⎥ .
*
⎢ B ⎥ ⎢⎣ − sinhθ 0 cosh hθ ⎥⎦ ⎢⎣ B ⎥⎦
⎣ ⎦
Proof: (1) If M 2 is the involüte of M 1 , we have
β (s ) = α (s ) + λT (s ) , λ = c − s , α ′ = T (c =constant).
From the equation (21) we have
T* = N .
1
(β ′ × β ′′) , from the equations (23) and (24) , we obtain
For being B* =
β ′ × β ′′ IL
1504
M. Bilici and M. Çalışkan
B* = −
τ
τ −κ
2
2
T+
κ
τ 2 −κ2
B,
substituting (14) into the last equation, we obtain
B* = − coshθT + sinhθB .
(28)
*
*
*
Since N = − B × T , then we have
N * = sinh θT − cosh θB .
(29)
If the equations (21), (28), (29) are written by matrix form, then the theorem is proved.
Note: In this situation, the causal characteristics of the Frenet frames of the curves
M 1 and M 2 are
{T spacelike, N spacelike, B timelike}
and
T * spacelike, N * spacelike, B* timelike .
(2) The proof is analogous to the proof of the statement (1). In this case, the causal
characteristics of the Frenet frames of the curves M 1 and M 2 must be of the form.
{T spacelike, N spacelike, B timelike}
and
T * spacelike, N * timelike, B* spacelike .
Theorem 4. Let (M 2 , M 1 ) be be the involüte-evolüte curve couple. If W and W * are the
Frenet instantaneous rotation vectors of M 1 and M 2 respectively, then
1
(θ ′ N − W ) , ( for κ > τ )
(i)
W* =
c − sκ
and
1
(θ ′ N + W ) , ( for κ < τ )
(ii)
W* =
c − sκ
(
)
{
}
{
}
Proof: (i) For the Frenet instantaneous rotation vector of the curve M 2 , from the equation
(3) we can write
W * = −τ *T * + κ * B*
(30)
Using the equations (18) , (26) in the equation (30), we have
τ 2 −κ2
κτ ′ − κ ′τ
W =−
N+
( − cosh θT + sinh θB )
c − sκ
c − sκ τ 2 −κ 2
*
′
⎛κ ⎞ 2
⎜ ⎟τ
W
⎝τ ⎠
*
(− cosh θT + sinh θB ) .
W =
N+
2
2
c − sκ
c − sκ τ −κ
Using the equation (14) in the last equation , then we obtain
1
{θ ′N − (τ T − κ B )}
W* =
c − sκ
and then , we get
Involutes of the spacelike curve
1505
1
(θ ′ N − W ) .
c − sκ
W* =
(ii) The proof is analogous to the proof of the statement (i).
Theorem 5. Let (M 2 , M 1 ) be a first type involüte-evolüte curve couple. If C and C *
are unit vectors of direction of W and W * ,respectively, then we have
(i)
C =−
(ii)
C =
θ′
*
θ ′2 + κ 2 − τ 2
θ′
*
θ′ + κ −τ
2
2
2
N+
κ 2 −τ 2
θ ′2 + κ 2 − τ 2
τ 2 −κ2
N+
θ′ + κ −τ
2
2
2
( for κ > τ )
C,
C,
( for κ < τ )
Proof: (i) Let θ * be the angle between B * and W * . Then we can write
C * = − sinhθ * T * + coshθ * B* .
For the curvatures and torsions of the curve M 2 , we have
⎧κ * = W * cosh θ *
⎪
⎨ *
*
*
⎪⎩τ = W sinh θ
Using the equation (18) and Theorem 4. into the equation (32) , we get
θ′
sinh θ * =
,
2
2
2
′
θ + κ −τ
cosh θ =
*
κ 2 −τ 2
θ ′2 + κ 2 − τ 2
.
(31)
(32)
(33)
(34)
Substituting by the equations (33) , (34) into the equation (31), the theorem is proved.
(ii) The proof is analogous to the proof of the statement (i).
Corollary 1. Let ( M 2 ,M 1 ) be the first type involüte-evolüte curve couple. If M 1 evolute
curve is helix, then
i) The vectors W * and B * of the involüte curve M 2 are linearly dependent .
ii) C = C *
Proof: i) If M 1 evolute curve is helix , then we have
τ
= tanh θ = constant.
κ
and then we have
θ ′= 0 .
Substituting by the equation (35) into the equations (33) , (34)
sinh θ * = 0
cosh θ * = 1
are found. Thus we have
(35)
1506
M. Bilici and M. Çalışkan
θ* =0.
ii ) Substituting by the equation (35) into the Theorem 5. , we have
C =C*
Theorem 6. Let (M 2 , M 1 ) be the first type involüte-evolüte curve couple. Let curvatures
and torsions of the curve couple (M 2 , M 1 ) be κ ,τ ,κ * ,τ * ( κ ≠ τ , κ ≠ 0 ) . If M 1 is a
cylindrical helix , then M 2 is a plane curve.
Proof: If M 1 is a cylindrical helix, then we can write
κ
= constant,
τ
′
⎛κ ⎞
⎜ ⎟ =0
⎝τ ⎠
and we easily obtain
κ ′τ − κτ ′ = 0 .
(36)
On the other hand from the equation (18) , we can write
τ
=
κ*
*
κτ ′ − κ ′τ
( c − s )κ ( κ 2 − τ 2 )
κ 2 −τ 2
,
( c − s )2 κ 2
substituting by the equation (36) into the last equation, we get
τ* = 0.
Example 2.1: Let α 1 (θ ) = 2 sinh θ , 2 cosh θ ,θ be a unit speed time-like curve. If
α 1 is a time-like curve then the involute curve is a space-like. In this situation, involute
curve β 1 of the curve α 1 can be given below.
(
β 1 (θ ) =
(
)
)
2 sinh θ + (c − θ ) 2 cosh θ , 2 cosh θ + (c − θ ) 2 sinh θ , c , c ∈ IR ,
for specially c=2 and 0 ≤ θ ≤ π We can draw involute-evolute curve couple (β 1 ,α 1 )
with helping the programme of Mapple 11 as follow.
Involutes of the spacelike curve
1507
Figure 2.1
Example 2.2: The involute of the unit speed space-like curve
⎛ 3
3
θ⎞
coshθ ,
sinh θ , ⎟⎟ is a space-like or time-like. The involute of the
α 2 (θ ) = ⎜⎜
2
2⎠
⎝ 2
curve α 2 can be given below.
⎛ 3
⎞
[cosh θ + (c − θ ) sinh θ ], 3 [sinh θ + (c − θ ) cosh θ ], c ⎟⎟, c ∈ IR .
2
2⎠
⎝ 2
β 2 (θ ) = ⎜⎜
β 2 is a space-like curve. For specially c=2 and θ ∈ [0 ,π ] , once again we can draw
involute-evolute curve couple (β 2 ,α 2 ) with helping the programme of Mapple 11 as
follow
1508
M. Bilici and M. Çalışkan
Figure 2.1
REFERENCES
[1] B. Bükcü, M.K. Karacan, On the Involute and Evolute Curves of the Spacelike Curve
with a Spacelike Binormal in Minkowski 3-Space, Int. J. Math. Sciences, Vol. 2, no. 5
(2007), 221-232.
[2] B. O’neill, Semi Riemann Geometry, Academic Press, New York, London, 1983.
[3] H.H. Uğurlu, On The Geometry of Time-like Surfaces, Commun. Fac. Sci. Ank.
Series A1 V.46, (1997), 211-223
[4] J. G. Ratcliffe, Foundations of Hyperbolic Manifolds, Springer-Verlag New York,
Inc., New York, 1994.
[5] K. Akutagawa, S. Nishikawa, The Gauss Map and Space-like Surfaces with
Prescribed Mean Curvature in Minkowski 3-Space, Töhoko Math., J. 42 (1990), 67-82.
[6] M. Bilici, M. Çalışkan, On The Involutes of Timelike Curves in IR13 , IV.
International Geometry Symposium, Zonguldak Karaelmas Üniversity 17-21 July 2006.
[7] M. Çalışkan, M. Bilici, Some Characterizations for The Pair of Involute-Evolute
Curves in Euclidean Space E 3 , Bulletin of Pure and Applied Sciences. Vol. 21E (No:2),
(2002), 289-294.
Involutes of the spacelike curve
1509
[8] R.S. Millman, G.D. Parker, Elements of Differential Geometry, Prentice-Hall Inc.,
Englewood Cliffs, New Jersey, 1977.
[9] V.D.I. Woestijne, Minimal Surfaces of the 3-dimensional Minkowski space. Proc.
Congres “Géométrie différentielle et applications” Avignon (30 May 1988), Word
Scientific Publishing. Singapore. (1990), 344-369.
Received: December, 2008