On the dual Bishop Darboux rotation axis
of the dual space curve
Murat Kemal Karacan, Bahaddin Bukcu and Nural Yuksel
Abstract. In this paper, the Dual Bishop Darboux rotation axis for dual
space curve in the dual space D3 is decomposed in two simultaneous rotation. The axes of these simultaneous rotations are joined by a simple
mechanism.
M.S.C. 2000: 53A04, 53A17, 53A25.
Key words: Dual space curve, Dual Bishop frame, Dual parallel transport frame,
Dual Bishop Darboux vector, Dual Bishop Darboux rotation axis.
1
Preliminaries
The Bishop frame or parallel transport frame is an alternative approach to defining a
moving frame that is well defined even when the curve has vanishing second derivative.
We can parallel transport an orthonormal frame along a curve simply by parallel
transporting each component of the frame. The parallel transport frame is based on
the observation that, while T (s) for a given curve model is unique, we may choose
any convenient arbitrary basis (N1 (s), N2 (s)) for the remainder of the frame, as long
as it is in the normal plane perpendicular to T (s) at each point. If the derivatives of
(N1 (s), N2 (s)) depend only on T (s) and not on each other, then we can make N1 (s)
and N2 (s) vary smoothly throughout the path regardless of the curvature. Therefore,
we have the alternative frame equations
 0  


T
0
k1 k2
T
 N10  =  −k1 0 0   N1 
N20
−k2 0 0
N2
where
κ(s) =
µ ¶
q
dθ(s)
k2
, τ (s) = −
k12 + k22 , θ(s) = arctan
k1
ds
[6, 7, 2], so that k1 and k2 effectively correspond
to a Cartesian coordinate system for
R
the polar coordinates κ, θ with θ = − τ (s)ds. The orientation of the parallel transport frame includes the arbitrary choice of integration constant θ0 , which disappears
from τ (and hence from the Frenet frame) due to the differentiation.
Applied Sciences, Vol.10, 2008, pp.
115-120.
c Balkan Society of Geometers, Geometry Balkan Press 2008.
°
116
2
Murat Kemal Karacan, Bahaddin Bukcu and Nural Yuksel
Introduction
In the Euclidean 3-Space E 3 , lines combined with one of their two directions can be
represented by unit dual vectors over the the ring of dual numbers. The important
properties of real vector analysis are valid for the dual vectors. The oriented lines E 3
are in one to one correspondence with the points of the dual unit sphere D3 .
A dual point on D3 corresponds to a line in E 3 , two different points of D3 represents two skew lines in E 3 .A differentiable curve on D3 represents a ruled surface E 3 .
If ϕ and ϕ∗ are real numbers and ε2 = 0 the combination ϕ
b = ϕ +εϕ∗ is called a dual
number. The symbol ε designates the dual unit with the property ε2 = 0. In analogy
with the complex numbers W.K. Clifford defined the dual numbers and showed that
they form an algebra, not a field. The our dual numbers are εa∗ .
According to the definition, the pure dual numbers εa∗ are zero divisors. No εa∗
number has an inverse in this algebra. Later, E.Study introduced the dual angle
subtended by two nonparallel lines E 3 , and defined it as ϕ
b = ϕ +εϕ∗ in which ϕ and
∗
ϕ are, respectively, the projected angle and the shortest distance between the two
lines.
The set
D = {b
x = x + εx∗ |x, x∗ ∈ R }
of dual numbers is a commutative ring with respect to the operations
i) (x + εx∗ ) + (y + εy ∗ ) = (x + y) + ε(x∗ + y ∗ )
ii) (x + εx∗ ).(y + εy ∗ ) = (x + y) + ε(xy ∗ + yx∗ ).
The division
x
b
y
b
is possible and unambiguous if yb 6= 0 and it can be easily seen that
x
b
x + εx∗
x
x∗ y − xy ∗
=
+
+
ε
.
yb
y + εy ∗
y
y2
The set
D3 = D × D × D =




x
b|x
b = (x1 + εx∗1 , x2 + εx∗2 , x3 + εx∗3 )

= (x1 + x2 + x3 ) + ε (x∗1 + x∗2 + x∗3 ) ,

= x + εx∗ , x ∈ R3 , x∗ ∈ R3
is a module over the ring D. It is clear that any dual vector x
b in D3 , consists of
∗
3
any two real vectors x and x in R , which are expressed in the natural right handed
orthonormal frame in the 3-dimensional Euclidean space E 3 . We call the elements of
D3 , dual vectors. If x 6= 0, then the norm kb
xk of x
b is defined by
kb
xk =
p
hb
x, x
bi = kxk + ε
xx∗
.
kxk
Let x
b : I → D3 , s → x
b(s) = x(s)+εx∗ (s) be a C 4 curve with the arc-length parameter
s of the indicatrix. Then
db
x ds
db
x
=
= Tb
db
s
ds db
s
On the dual Bishop Darboux rotation axis
117
is called the unit tangent vector of x
b(s) [5]. Since Tb has constant length 1, its differentiation with respect to sb, which is given by
dTb
dTb ds
d2 x
b
c1 + kb2 N
c2 , c
=
=
= kb1 N
k1 = k1 + εk1∗ , c
k2 = k2 + εk2∗
db
s
ds db
s
db
s
measures the way the curve is turning in D3 . We impose the restriction that the dual
c1 ,
natural curvatures c
k1 , c
k2 : I → D are never pure dual. We call the vectors c
T, N
c
N2 dual Bishop trihedron of x
b(s). Writing
dTb
db
s
c1
dN
db
s
c
dN2
db
s
=
c1 + a13 N
c2
a11 Tb + a12 N
=
c1 + a23 N
c2
a21 Tb + a22 N
=
c1 + a33 N
c2
a31 Tb + a32 N
and using the properties of inner product and differentiations of the inner products
c
c1 and N
c2 , we may express the dual Bishop formulas of the dual Bishop trihedron
T, N
in matrix form:

 


Tb
0
kb1 kb2
Tb0
 c 
 b0   b
 N 1  =  −k 1 0 0   N
1 .
b0
b
c
N
−
k
0
0
N
2
2
2
3
The dual Bishop Darboux rotation axis of the
dual space curve
n
o
c1 , N
c2 be the dual Bishop frame of the differentiable dual space curve in
Let c
T,N
the dual space D. Then the dual Bishop frame equations are
Tb0
c1
N
0
c2
N
0
=
k1 N1 + k2 N2 + ε (k1∗ N1 + k1 N1∗ + k2∗ N2 + k2 N2∗ )
=
−k1 T − ε (k1∗ T + k1 T ∗ )
=
−k2 T − ε (k2∗ T + k2 T ∗ )
where c
k1 = k1 + εk1∗ and c
k2 = k2 + εk2∗ are nowhere pure dual natural curvatures
and
q
d = κ + εκ∗ = k 2 + k 2 + 2ε (k1 k ∗ + k2 k ∗ ),
κ(t)
1
2
1
2
à !
µ
¶
∗
∗
b
b = θ + εθ∗ = arctan k2 = arctan k2 + ε (k1 k2 − k1 k2 ) ,
θ(t)
k1
k12
kb1
b
dθ(t)
.
dt
These equations form a dual Bishop rotation motion with dual Bishop Darboux vector,
τb(t) =
c1 + kb1 N
c2 = −k2 N1 + k1 N2 + ε (k1∗ N2 + k1 N2∗ − k2∗ N1 − k2 N1∗ ) .
$
b = $ + ε$∗ = −kb2 N
118
Murat Kemal Karacan, Bahaddin Bukcu and Nural Yuksel
Also momentum dual rotation vector is expressed as follows:
Tb0
0
c1
N
c2
N
=
$
b ∧ Tb
c1
$
b ∧N
=
c2 .
$
b ∧N
=
0
The dual Bishop Darboux rotation of the dual Bishop frame can be separated into
c1 rotates with an angular speed kb1 about the
two rotation motions. The vector N
c
vector N2 , that is
³
´
³
´
0
c1 = kb1 N
c2 ∧ N
c1 = kb1 N
c2 ∧ N
c1 = −kb1 Tb
N
c2 rotates with a angular speed −kb2 about the vector N
c1 , that is
and the vector N
³
´
³
´
0
c2 = −kb2 N
c1 ∧ N
c2 = −kb2 N
c1 ∧ N
c2 = −kb2 T.
b
N
The separation of the rotation motion of the momentum dual Bishop Darboux axis
b
into two rotation motions can be indicated as follows: The vector k$
$k
b rotates with a
angular speed
0
0
b1 kb2 − kb1 kb2
k
c=
W
2
2
kb1 + kb2
about the tangent vector Tb, also
¶0 ³
µ
´
b
$
b
c .Tb ∧ $
= W
k$k
b
k$k
b
and the tangent vector Tb rotates with a angular speed k$k
b about
Darboux axis, also
Tb0 = $
b ∧ Tb.
$
b
k$k
b .
Dual Bishop
¿From now on we shall do a further study of momentum dual Bishop Darboux axis.
b
We obtain the unit vector E:
b
−k2 N1 + k1 N2 + ε (k1∗ N2 + k1 N2∗ − k2∗ N1 − k2 N1∗ )
b= $
p
.
E
=
k$k
b
k12 + k22 + 2ε (k1 k1∗ + k2 k2∗ )
¿From the dual Bishop Darboux vector,
0
0
$
b0
0
0
=
c1 − kb2 N
c1 + kb1 N
c2 + kb1 N
c2
−kb2 N
³
´
³
´
0
0
c1 − kb2 −kb1 Tb + kb1 N
c2 + kb1 −kb2 Tb
−kb2 N
=
c1 + kb1 N
c2 .
−kb2 N
=
0
0
b
Differentiation of E,
µ
E0 =
$
b
k$k
b
¶0
0
0
=
µ 0
¶
0
kb1 kb2 − kb1 kb2
$
b k$k
b −$
b k$k
b
2
( k$k)
b
=
2
2
kb1 + kb2
³
´
c1 + kb2 N
c2
kb1 N
p
k12 + k22 + 2ε (k1 k1∗ + k2 k2∗ )
On the dual Bishop Darboux rotation axis
is found. From this,
119
³
´
b0 = W
c E
b ∧ Tb
E
is written. According to the dual Bishop frame,
³
´
b ∧ Tb
Tb0 = k$k
b
E
and
³
b ∧ Tb
E
´0
b 0 ∧ Tb + E
b ∧ Tb0
= E
³
´
³
´
c .Tb ∧ E
b ∧ Tb + k$k
b E
b ∧ Tb
= W
b .E
hD
E
D
E i
hD
E
D
E i
c Tb, Tb E
b − Tb, E
b Tb + k$k
b Tb E
b − E,
b E
b Tb
= W
b
E,
c .E
b − k$k
= W
b Tb
are obtained. These three equations are in the form of dual Bishop frames that is

Tb0


0
 ³ b b ´0  
b
 = − k$k
 E∧T
0
b0
E
k$k
b
0
c
W


b
0
³ T ´
c 
b ∧ Tb 
W
 E

0
b
E
where the first coefficient k$k
b is nowhere pure dual and and second coefficient
³ ´0
0
0
c
k
1
b
b
b
b
c
k
k
−
k
k
1
2
1
2
k
2
c=
W
=
³ ´2
2
2
c
b
b
k1 + k2
1 + kc1
k2
n ³
´ o
c
b ∧ Tb , E
b
is related only to natural harmonic curvature kc1 . Thus, the vectors Tb, E
k2
define a dual rotation motion together with the dual rotation vector,
c .Tb + k$k
b = −W
c .Tb + $
$
c1 = −W
b .E
b
As well, the momentum rotation vector is expressed as follows:
Tb0
´0
³
b ∧ Tb
E
=
=
$
c1 ∧ Tb
³
´
b ∧ Tb
$
c1 ∧ E
b0
E
=
b
$
c1 ∧ E.
Remark. This dual rotation motion of dual Bishop Darboux axis can be separated
into two dual rotation motions again. Here, the dual rotation vector $
c1 is the sum of
the dual rotation vectors of the dual rotation motions. When continued in the similar
way, the dual rotation motion of dual Bishop Darboux axis is done in a consecutive
manner. In this way the series of dual Bishop Darboux vectors are obtained. That is
$
c0 = $,
b $
c1 . . . , etc.
120
Murat Kemal Karacan, Bahaddin Bukcu and Nural Yuksel
References
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49, (1994), 46-49.
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Monthly, 82 (1975), 246-251.
[3] O. Bottema and B. Roth, Theoretical Kinematics, North Holland Publ. Com.,
1979.
[4] B. Bukcu and M. K. Karacan, Special Bishop motion and Bishop Darboux rotation axis of the space curve in Euclidean 3 space, submitted for publication.
[5] A. C. Coken and A. Görgülü, On the dual Darboux rotation axis of dual space
curve, Demonstratio Mathematica, 37 (2004), 197-202.
[6] A. J. Hanson and H. Ma, Parallel transport approach to curve framing, Indiana
University, Techreports-TR425, 11 (1995), 3-7.
[7] A. J. Hanson and H. Ma, Quaternion frame approach to streamline visualization,
IEEE Transactions On Visualization and Computer Graphics, 1 (1995), 164-174.
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Cranfield University, PhD Thesis, 2004.
[9] J. M. Selig, Curves of stationary acceleration in SE(3), IMA Journal of Mathematical Control and Information, 7 (2006), 17-18.
Authors’ addresses:
Murat Kemal Karacan
Usak University, Faculty of Science and Arts,
Department of Mathematics, Usak, Turkey.
E-mail: mkkaracan@yahoo.com , murat.karacan@usak.edu.tr
Bahaddin Bukcu
Gazi Osman Pasa University, Faculty of Science and Arts,
Department of Mathematics, Tokat, Turkey.
E-mail: mbukcu@gop.edu.tr, bbukcu@yahoo. com
Nural Yuksel
Erciyes University, Faculty of Science and Arts,
Department of Mathematics, Kayseri, Turkey.
E-mail: yukseln@erciyes.edu.tr