On the Dual Darboux Rotation Axis of the Timelike Dual Space Curve

Ahmet Yücesan, A. Ceylan Çöken and Nihat Ayyildiz

Abstract In this paper, the Dual Darboux rotation axis for timelike dual space curve in the semi-dual space D_{1}^{3} is decomposed in two simultaneous rotation. The axis of these simultaneous rotations are joined by a simple mechanism. This study is original corresponding in the semi-dual space D_{1}^{3} of the article entitled On the Dual Darboux Rotation Axis of the Dual Space Curve [6].

Mathematics Subject Classification: 53C50 Key words: dual Darboux vector, Darboux rotation axis, timelike dual curve, Semidual space.

1 Introduction

Theory of space curves of Riemannian manifold M is fully developed and its local and global geometry is well-known. In case M is proper semi-Riemannian, then, there are three categories of curves, namely, spacelike, timelike and null, depending on their causal character.

In the Euclidean 3−dimensional space R^{3}, lines combined with one of their two directions can be represented by unit dual vectors over the ring of dual numbers. The important properties of real vector analysis are valid for the dual vectors. The oriented lines R^{3} are in one to-one correspondence with the points of dual unit sphere D^{3} [13]. Similarly, Oriented timelike and spacelike lines in R_{1}^{3} may be represented by timelike and spacelike unit vectors with three components in the semi-dual space D_{1}^{3}, respectively.

If φ ve φ^{∗} are real numbers and ξ^{2} = 0 the combination

φ̂ = φ + ξφ^{∗} (1)

is called a dual number. The symbol ξ designates the dual unit with the property ξ^{2} = 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 pure dual numbers are ξa^{∗}.

According to the definition pure dual numbers ξa^{∗} are zero divisors. No number ξa^{∗} has an inverse in the algebra. But the other laws of the algebra of dual numbers (a + ib, i^{2} = −1). Later, E. Study introduced the dual angle subtended by two nonparallel lines in R^{3} and defined it as φ̂ = φ + ξφ^{∗} in which φ and φ^{∗} are, respectively, the projected angle and the shortest distance between the two lines [8].

∗ Balkan Journal of Geometry and Its Applications, Vol.7, No.2, 2002, pp. 137-142.

\begin{matrix}c \\ ◯ \end{matrix} Balkan Society of Geometers, Geometry Balkan Press 2002.

2 Preliminaries

Let L^{2} be the vector space R^{2} provided with Lorentzian inner product

< x, y >= x_{1}y_{1} − x_{2}y_{2} , x = (x_{1}, x_{2}), y = (y_{1}, y_{2}). (2)

We denote by G the proper Lorentz group SO^{+}(1, 1) consisting of all matrices of the form

A(u) = [ \begin{matrix}chu & shu \\ shu & chu \end{matrix} ] , u ∈ R (3)

where we are writing ch and sh for hyperbolic functions cosh and sinh [3].

In L^{2}, the norm ∥x∥ is defined to be \sqrt{|< x, x >|}. We also consider the time orientation as follows. Let e = (0, 1). A timelike vector x = (x_{1}, x_{2}) is future-pointing (resp., past-pointing) if < x, e >< 0 (resp., < x, e >> 0). So a vector x = (x_{1}, x_{2}) is timelike and future-pointing if and only if x_{1}^{2} − x_{2}^{2} < 0 and x_{2} > 0; in other words, if and only if |x_{1}| < x_{2}. We note that the group G is in fact the group of all linear transformations of L^{2} which preserve inner product, orientation and time-orientation [3].

The set D = {x̂ = x + ξx^{∗}|x, x^{∗} ∈ R} (4)

of dual numbers is a commutative ring with respect to the operations i) (x + ξx^{∗}) + (y + ξy^{∗}) = (x + y) + ξ (x^{∗} + y^{∗}) ii) (x + ξx^{∗}) · (y + ξy^{∗}) = xy + ξ (xy^{∗} + yx^{∗}). The division \frac{x̂}{ŷ} is possible and unambiguous if y ≠ 0 and it easily see that

\frac{x̂}{ŷ} = \frac{x + ξx^{∗}}{y + ξy^{∗}} = \frac{x}{y} + ξ \frac{x^{∗}y − xy^{∗}}{y^{2}} . (5)

The set D^{3} = D × D × D = {x̂|x̂ = (x_{1} + ξx_{1}^{∗}, x_{2} + ξx_{2}^{∗}, x_{3} + ξx_{3}^{∗}) (6)

= (x_{1}, x_{2}, x_{3}) + ξ (x_{1}^{∗}, x_{2}^{∗}, x_{3}^{∗}) (7)

= x + ξx^{∗}, x ∈ R^{3}, x^{∗} ∈ R^{3}}. (8)

is a module over the ring D.

Let x̂ = x + ξx^{∗}, ŷ = y + ξy^{∗}. The Lorentzian inner product of x̂ and ŷ is defined by

< x̂, ŷ >=< x, y > +ξ(< x, y^{∗} > + < x^{∗}, y >).

We call the dual space D^{3} together with this Lorentzian inner product as dual Lorentzian space and show by D_{1}^{3}. It is clear that any dual vector x̂ in D_{1}^{3}, consists of any two real vector x and x^{∗} in R_{1}^{3} which are expressed in the natural right handed orthonormal frame in the 3−dimensional semi-Euclidean space R_{1}^{3}. We call the elements of D_{1}^{3} the dual vectors [12]. If x ≠ 0 the norm ∥x̂∥ of x̂ is defined by ∥x̂∥ = (|< x̂ , x̂ >|)^{\frac{1}{2}} .

Let x̂ be dual vector. x̂ is said to be spacelike, timelike and lightlike (null) if the vector x is spacelike, timelike and lightlike (null), respectively.

For any x = (x_{1}, x_{2}, x_{3}), y = (y_{1}, y_{2}, y_{3}) ∈ R_{1}^{3}, the pseudo vector product of x and y is defined by

x × y = (x_{2}y_{3} − x_{3}y_{2}, x_{1}y_{3} − x_{3}y_{1}, x_{2}y_{1} − x_{1}y_{2}), (9)

where e_{1} × e_{2} = −e_{3}, e_{2} × e_{3} = e_{1} and e_{3} × e_{1} = −e_{2} [1].

Definition 1. Let x̂, ŷ ∈ D_{1}^{3}. We define the Lorentzian vectoral product of x̂ and ŷ by

x̂ × ŷ = (x̂_{2}ŷ_{3} − x̂_{3}ŷ_{2}, x̂_{1}ŷ_{3} − x̂_{3}ŷ_{1}, x̂_{2}ŷ_{1} − x̂_{1}ŷ_{2}),

where x̂ = (x_{1}, x_{2}, x_{3}), ŷ = (y_{1}, y_{2}, y_{3}) and ê_{1} × ê_{2} = −ê_{3}, ê_{2} × ê_{3} = ê_{1} and ê_{3} × ê_{1} = −ê_{2}.

Definition 2. Let n ≥ 2 and 0 ≤ ν ≤ n. Then

i) The pseudosphere of radius r > 0 in R_{υ}^{n+1} is the hyperquadric

S_{ν}^{n}(r) = q^{−1}(r^{2}) = {p ∈ R_{υ}^{n+1}| < p, p >= r^{2}},

with dimension n and indeks ν. ii) The pseudohyperbolic space of radius r > 0 in R_{υ+1}^{n+1} is the hyperquadric

H_{ν}^{n}(r) = q^{−1}(−r^{2}) = {p ∈ R_{υ+1}^{n+1}| < p, p >= −r^{2}},

with dimension n and indeks ν [11]. Definition 3. Let x̂ = x + ξx^{∗} ∈ D_{1}^{3}.

i) The set

S_{1}^{2}= \{x̂ = x + ξx^{∗}| \left∥{ \atop x̂}\right∥ = (1, 0) ; x, x^{∗} ∈ R_{1}^{3} and the vector x is spacelike \}

is called the dual Lorentzian unit sphere in D_{1}^{3}. ii) The set

H_{0}^{2}= \{x̂ = x + ξx^{∗}| \left∥{ \atop x̂}\right∥ = (1, 0); x, x^{∗} ∈ R_{1}^{3} and the vector x is timelike \}

is called the dual hyperbolic unit sphere in D_{1}^{3} [14].

3 On the dual Darboux rotation axis of the timelike dual space curve

Let {t̂, n̂, b̂} be the Dual Frenet trihedron of the differentiable timelike dual space curve in the semi-dual space D_{1}^{3}. Then the Frenet equations are

\begin{aligned}t̂′ & = & κn + ξ(κ^{∗}n + κn^{∗}) \\ n̂′ & = & κt + τb + ξ(κ^{∗}t + κt^{∗} + τ^{∗}b + τb^{∗}) \\ b̂′ & = & −τn − ξ(τ^{∗}n + τn^{∗}), \end{aligned} (10)

where κ̂ = κ + ξκ^{∗} is nowhere pure dual curvature and τ̂ = τ + ξτ^{∗} is nowhere pure dual torsion.

These equations form a dual rotation motion with dual Darboux vector,

ˆ∂ = ∂ + ξ∂^{∗} = −τt − κb − ξ(τ^{∗}t + τt^{∗} + κb^{∗} + κ^{∗}b). (11)

Also momentum dual rotation vector is expressed as follows:

\begin{aligned}t̂′ & = & ˆ∂ × t̂ \\ n̂′ & = & ˆ∂ × n̂ \\ b̂′ & = & ˆ∂ × b̂. \end{aligned} (12)

Dual Darboux rotation of dual Frenet frame can be seperated into two rotation motions:

t̂ tangent vectors rotates with a κ̂ angular speed round b̂ binormal vector, that is

t̂′ = (−κ̂b̂) × t̂ (13)

and b̂′ binormal vector rotates with a τ̂ angular speed round t̂ tangent vector, that is

b̂′ = (−τ̂t̂) × b̂. (14)

The seperation of the rotation motion of the momentum dual Darboux axis into two rotation motions can be indicated as such:

\frac{∂}{\left∥{ \atop ˆ∂}\right∥} vector rotates with a ŵ = \frac{τ̂′ κ̂−τ̂κ̂′}{−τ̂^{2}+κ̂^{2}} angular speed round n̂ principal normal,

also

(\frac{ˆ∂}{∥ˆ∂∥})′ = (ŵn̂) × \frac{ˆ∂}{∥ˆ∂∥}, (15)

and n̂ principal normal vector rotates with a ∥ˆ∂∥ angular speed round \frac{ˆ∂}{\left∥{ \atop ˆ∂}\right∥} dual

Darboux axis, also

n̂′ = ˆ∂ × n̂. (16)

From now on we shall do a further study of momentum dual Darboux axis. We obtain the unit vector ê from dual Darboux vector,

ê = \frac{ˆ∂}{∥ˆ∂∥} = \frac{−τt − κb − ξ(τ^{∗}t + τt^{∗} + κb^{∗} + κ^{∗}b)}{\sqrt{|−τ^{2} + κ^{2} + 2 ξ(−ττ^{∗} + κκ^{∗})|}}. (17)

From ˆ∂′ = −τ̂′ t̂− κ̂′b̂, differentation of ê

ê′ = (\frac{ˆ∂}{∥ˆ∂∥})′ = \frac{∥ˆ∂∥ ˆ∂′ − ˆ∂∥ˆ∂∥′}{∥ˆ∂∥^{2}} = \frac{τ̂′ κ̂ − τ̂κ̂′}{−τ̂^{2} + κ̂^{2}} · \frac{−κ̂t̂− τ̂b̂}{\sqrt{|−τ^{2} + κ^{2} + 2 ξ(−ττ^{∗} + κκ^{∗})|}} (18)

is found. From this, ê′ = −ε(ê) ŵ (ê × n̂) (19) is written. According to the 2.Frenet formula,

n̂′ = ε(ê) ∥ˆ∂∥ (ê × n̂) (20)

and (ê × n̂)′ = −ε(ê) ε(ê × n̂) ∥ˆ∂∥ n̂ + ε(ê × n̂) ŵ ê (21)

are obtained. These three equations are in the form of the Frenet formulas that is

\begin{aligned}n̂′ & = & ε(ê) ∥ˆ∂∥ (ê × n̂) \\ (ê × n̂)′ & = & −ε(ê) ε(ê × n̂) ∥ˆ∂∥ n̂ + ε(ê × n̂) ŵ ê \\ ê′ & = & −ε(ê) ŵ (ê × n̂), \end{aligned} (22)

where the first coefficient ∥ˆ∂∥ is nowhere pure dual and second coefficient

ŵ = \frac{τ̂′ κ̂ − τ̂κ̂′}{−τ̂^{2} + κ̂^{2}} = \frac{(\frac{τ̂}{κ̂})′}{1 − (\frac{τ̂}{κ̂})^{2}} (23)

is related only to \frac{τ̂}{κ̂} harmonic curvature. Thus, the vectors n̂, (ê× n̂), ê define a dual rotation motion together the dual rotation vector,

ˆ∂_{1} = ŵn̂ + ∥ˆ∂∥ ê = ŵn̂ + ˆ∂. (24)

Also momentum dual rotation vector is expressed as follows:

n̂′ = ˆ∂_{1} × n̂ (25)

(ê × n̂)′ = ˆ∂_{1} × (ê × n̂) (26)

(ê)′ = ˆ∂_{1} × ê. (27) This dual rotation motion of dual Darboux axis can be seperated into two dual rotation motions again. Here ˆ∂_{1} dual rotation vector is the addition of the dual rotation vectors of the dual rotation motions.

When continued in the similar way, the dual rotation motion of dual Darboux axis is done in a consecutive manner. In this way the series of dual Darboux vectors are obtained.

That is ˆ∂_{0} = ˆ∂, ˆ∂_{1}, ... . (28)

References

[1] Akutagawa, K. and Nishikawa, S., The Gauss Map And Spacelike Surfaces With Prescribed Mean Curvature In Minkowski 3-Space, Tohoku Math. J. 42, 67-82, 1990.

[2] Barthel, W., Zum Drehvorgang Der Darboux-Achse Einer Raumkurve, Journal of Geometry Vol. 49, 46-49, 1994.

[3] Birman, S. and Nomizu, K., Trigonometry in Lorentzian Geometry, Am. Math. Mon. 91, 543-549, 1984.

[4] Blaschke, W., Vorlesungen Über Differential Geometry I. Verlag von Julieus Springer in Berlin, pp. 89, 1930.

[5] Clifford, W. K., Preliminary Sketch of Biquaternions, Proceedings of London Math. Soc. 4, 361-395, 1873.

[6] Çöken, A. C. and Görgülü, A., On The Dual Darboux Rotation Axis Of The Dual Space Curve, Demonstratio Mathematica No. 1, Vol. 35 (to appear), 2002.

[7] Guggenheimer, W., Differential Geometry, McGraw-Hill, New York, 1963.

[8] Hacisalihog̃lu, H. H., Acceleration Axes in Spatial Kinamatics I., Communications, Série A: Mathématiques, Physique et Astronomie, Tome 20 A, pp. 1-15, Année 1971.

[9] Hartl, J., Zerlegung der Darboux-Drehung in zwei ebene Drehungen, Journal of Geometry Vol. 47, 32-38, 1993.

[10] Köse, Ö., An Expilicit Characterization of Dual Spherical Curves, Doğa Mat., 12, No. 3, 105-113, 1998.

[11] O”Neill, B., Semi-Riemannian Geometry with Applications to relativity, Academic press Inc, London, 1983.

[12] Özyılmaz, E., Some Results On Space-like Line Congruences and Their space-like Parameter Ruled Surfaces, Tr. J. of Mathematics, 23, 333-344, 1999.

[13] Study, E., Geometrie der Dynamen, Leipzig, 1903.

[14] Uğurlu, H. H.and Çalışkan A., The Study Mapping For Directed Space-like and Time-like Lines In Minkowski 3-Space R_{1}^{3}, Mathematical & Computational Applications, Vol.1, No. 2, p.p 142-148, 1996.

Department of Mathematics, Süleyman Demirel University, 32260 Isparta, TURKEY A. Yücesan, email: yucesan@fef.sdu.edu.tr; A.C. Çöken, email: ceylan@fef.sdu.edu.tr; N. Ayyildiz. email: ayyildiz@fef.sdu.edu.tr