Printer Friendly

Representation formulae for Bertrand curves in the Minkowski 3-space.

[section]1. Introduction

Bertrand curves are one of the important and interesting topic of classical spatial curve theory [4, 6, 10]. A Bertand curve is defined as a spatial curve which shares its principal normals with another spatial curve (called Bertrand mate). Note that Bertrand mates are particular examples of offset curves used in computer-aided design CADG, (see [5]).

Bertrand curves are characterised as spatial curves whose curvature and torsion are in linear relation. Thus Bertrand curves may be regraded as one-dimensional analogue of Weingarten surfaces [9]. Application of Weingarten surfaces to CADG, (see [8]).

Bertrand curves and their geodesic imbedding in surfaces are recently rediscovered and studied in the context of modern soliton theory by Schief [7].

Straightforward modification of classical theory to spacelike or timelike curves in Minkowski 3-space is easily obtained, (see [1]). Null Bertrand curves in Minkowski 3-space are studied in [2]. Nonnull Bertrand curves in the n-dimensional Lorentzian space are examined in [3].

However in [1]-[2], representation formulae for Bertrand curves are not obtained.

In this paper, we study representation formulae for Bertrand curves in Minkowski 3-space.

[section]2. Bertrand curves and Representation Formulae in Minkowski 3-space

In this section, we collect classical results on Bertrand curves in Minkowski 3-space [E.sup.3.sub.1]. Let [E.sup.3.sub.1] be the Minkowski 3-space and [gamma] a regular non-null curve. Then 7 can be parametrised by the unit speed parameter s;

<[gamma]'(s),[gamma]'(s)> = [[epsilon].sub.i] = [+ or -] 1.

If [gamma](s) is spacelike (resp. timelike), s is called the arclength parameter (resp.proper time parameter ). Let us denote by T the tangent vector field of [gamma];

T(s) := [gamma]'(s).

Hereafter, in case [[epsilon].sub.1] = 1 (spacelike curve), we assume that the acceleration vector field T' is nonnull. Then there exist vector fields N and B along [gamma] such that

T' = [[epsilon].sub.2][kappa]N, N' = -[[epsilon].sub.1][kappa]T - [[epsilon].sub.3][tau]B, B' = [[epsilon].sub.2][tau]N. (1)

Here [[epsilon].sub.2] and [[epsilon].sub.3] are second and third causal characters of [gamma] defined by

[[epsilon].sub.2] = <N, N>, [[epsilon].sub.3] = <B, B>.

The vector field N and B are called the principal normal and binormal vector field of [gamma] respectively. The functions [kappa] and [tau] are called the curvature and torsion of [gamma] respectively.

If there exists a spatial curve [bar.[gamma]]([bar.s]) whose principal normal direction coincides with that of original curve, then [gamma] is said to be a Bertrand curve. The pair ([gamma],[[bar.[gamma]]) is said to be a Bertrand mate.

There are several possibilities for Bertrand mates denoted by {[[bar.[epsilon]].sub.i]}, the causal characters of the Bertrand mate [bar.[gamma]]. Then by definition, [[bar.[epsilon]].sub.2] = [[epsilon].sub.2].

1. [gamma] is spacelike with [[epsilon].sub.2] = 1. In this case there are two subcases.

(a) ([[bar.[epsilon]].sub.1], [[bar.[epsilon]].sub.2], [[bar.[epsilon]].sub.3]) = (+1, +1, -1) : In this case, the mate is also spacelike. Both the rectifying planes of [gamma] and [gamma] are timelike. Thus the tangent vector fields are related by

[bar.T] = [+ or -](cosh [theta]T + sinh [theta]B)

for some function [theta] = [theta](s).

(b) ([[bar.[epsilon]].sub.1], [[bar.[epsilon]].sub.2], [[bar.[epsilon]].sub.3]) = ( -1, +1, +1) : In this case, the mate is timelike. Both the rectifying planes of [gamma] and [bar.[gamma]] are timelike. Thus the tangent vector fields are related by

[bar.T] = [+ or -](sinh [theta]T + cosh [theta]B)

for some function [theta] = [theta](s).

2. [gamma] is spacelike with [[epsilon].sub.2] = -1. Then ([[bar.[epsilon]].sub.1], [[bar.[epsilon]].sub.2], [[bar.[epsilon]].sub.3]) = (+1, -1, +1). Both the rectifying planes are spacelike. Thus

[bar.T] = cos [theta]T + sin [theta]B

for some function [theta] = [theta](s).

3. [gamma] is timelike. In this case there are two subcases.

(a) ([[bar.[epsilon]].sub.1], [[bar.[epsilon]].sub.2], [[bar.[epsilon]].sub.3]) = (+1, +1, -1) : In this case, the mate is timelike. The tangent vector fields are related by

[bar.T] = [+ or -](sinh [theta]T + cosh [theta]B)

for some function [theta] = [theta](s).

(b) ([[bar.[epsilon]].sub.1] [[bar.[epsilon]].sub.2] [[bar.[epsilon]].sub.3]) = ( -1, +1, +1) : In this case, the mate is timelike. The tangent vector fields are related by

[bar.T] = [+ or -](cosh [theta]T + sinh [theta]B)

for some function [theta] = [theta](s).

One can see that the case 3 is reduced to case 1. Thus we may restrict our study to case 1 and case 2.

Now let consider case 1-(a):

Let ([gamma], [bar.[gamma]]) be a Bertrand mate, then

[bar.[gamma]]([bar.s]) = [gamma](s)+ u(s)N(s) (2)

for some function u(s) [not equal to] 0. Differentiating this, we get

[bar.T]([bar.s])d[bar.s]/ds = (1 - [[epsilon].sub.1]u(s)[kappa](s))T(s) + u'(s) N(s) + [[epsilon].sub.3]u(s)[tau](s) B(s). (3)

Since [bar.T] [perpendicular to] N,

([bar.T], N)[[bar.s].sub.s] = u' = 0.

Hence u is a nonzero constant. Denote by [theta] the angle between [gamma] and [[bar.[gamma]] :

[bar.T] = [epsilon](cosh [theta] T + sinh [theta] B), [epsilon] = [+ or -] 1 (4)

Then computing the inner product of (3) and (4), we have

1 - [[epsilon].sub.1]u[kappa]/cosh [theta] = [[epsilon].sub.3]u[tau]/ sinh [theta] = d[bar.s]/ds. (5)

Differentiating (4),

[[epsilon].sub.2][bar.[kappa]][[bar.s].sub.s][bar.N] = ([epsilon][[epsilon].sub.2][kappa] cosh [theta] + [epsilon][[epsilon].sub.2] [tau] sinh [theta])N + [epsilon][theta]'(sinh [theta]T + cosh [theta]B).

By the assumption,

[bar.N] = [+ or -]N.

Hence

[theta]' = 0, [bar.[kappa]][[bar.s].sub.s] = [epsilon]([kappa] cosh [theta] + [tau] sinh [theta]).

Thus [theta] is a constant. If sinh [theta] = 0, then from (5), [tau] = 0. In this case, [gamma] is a planar curve. Note that planar curves are Bertrand curves. In fact, planar curves together with their parallel curves are Bertrand mates.

Next, if sin [theta] [not equal to] 0, then (5) is written in the form:

a[kappa] + b[tau] = 1, (6)

for constants a and b.

Conversely, if a spatial curve [gamma] satisfies (6), then define [bar.[gamma]] by (2). Then

[bar.T] = [epsilon](cosh [theta]T + sinh [theta]B).

Differentiating this by s, we obtain

[bar.[kappa]][bar.N] [[bar.s].sub.s] = [epsilon]([kappa] cosh [theta] + [tau] sinh [theta]) N.

Hence [gamma] is a Bertrand curve.

Thus we obatin the following result:

Theorem 1. A spatial curve is a Bertand curve in Minkowski 3-space [E.sup.3.sub.1] if and only if its curvature and torsion satisfy a[kappa] + b[tau] = 1 for some constants a and b.

Theorem 2. Let ([gamma], [bar.[gamma]]) be a Bertrand mate in Minkowski 3-space [E.sup.3.sub.1]. Then [tau](s)[bar.[tau]]([bar.s]) is a constant.

Proof of Theorem 2. From (2)-(6),

[tau] = sinh [theta]/u d[bar.s]/ds, [bar.[tau]] = -sinh [bar.[theta]]/[bar.u] ds/d[bar.s], [bar.u] = [+ or -]u.

Hence

[tau][bar.[tau]] = sinh [theta] sinh [bar.[theta]]/u[bar.u] = constant.

Corollary 1. Let [gamma] be a Bertrand curve with a[kappa] + b[tau] = 1 and [bar.[gamma]] a Bertrand mate. Then the fundamental quantities of the Bertrand mate are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Similarly, if we consider case 1-(b) we have following :

Theorem 3. A spatial curve is a Bertand curve in Minkowski 3-space [E.sup.3.sub.1] if and only if its curvature and torsion satisfy a[kappa] + b[tau] = 1 for some constants a and b.

Theorem 4. Let ([gamma], [bar.[gamma]]) be a Bertrand mate in Minkowski 3-space [E.sup.3.sub.1]. Then [tau](s)[bar.[tau]]([bar.s]) is a constant.

Corollary 2. Let [gamma] be a Bertrand curve with a[kappa] + b[tau] = 1 and [bar.[gamma]] a Bertrand mate. Then the fundamental quantities of the Bertrand mate are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In case 2, We obtained the classical results in Euclidean space.

Lemma 1. Let e(t) be a unit vector field which is not parallel to a fixed plane. Take a nonzero constant a. Then

[alpha](t) := -[[epsilon].sub.2]a [integral] e(t) x [??](t)dt

is a spatial curve of constant torsion -[[epsilon].sub.2]/a and binormal vector field [+ or -] e(t).

Proof of Lemma 1. Direct computations show

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Here we used the following formula in Minkowski 3-space [E.sub.3.sub.1]:

(x x y) x (z x w) = det(x, z, w)y - det(y, z, w)x

By the assumption, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the binormal vector field of [alpha] is [B.sub.[alpha]] = [+ or -]e.

Next, since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence the torsion of [alpha] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Conversely, let [alpha](s) be a curve of constant torsion -[[epsilon].sub.2]/a. Here s is the arclength parameter. Then put e(s) = B(s). Then the Frenet-Serret formula implies

e x [??] = [[epsilon].sub.2][tau] B x N = -[[epsilon].sub.2]/ a [alpha]'.

Hence [alpha](s) = -[[epsilon].sub.2]a [[integral].sup.s] e x eds.

Lemma 2. If a spatial curve a is of constant nonzero torsion [[tau].sub.[alpha]], then the curve

[beta](s) = -1/[[tau].sub.[alpha]] N(s) - [[epsilon].sub.3] [integral] B(s)ds

has constant curvature [absolute value of [[tau].sub.[alpha]]].

Proof of Lemma 2. We use the subscript [*.sub.[alpha]] for expressing geometric objects of [alpha]. By the Frenet-Serret formula for a, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence the unit tangent vectot field of [beta] is [T.sub.[beta]] = [epsilon]T, [epsilon] = sgn([tau]). Hence the arclength parameter [s.sub.[beta]] of [beta] is

[s.sub.[beta]] = [[integral].sup.s] [[epsilon].sub.1] [[kappa].sub.[alpha]]/[absolute value of [[tau].sub.[alpha]]] ds.

Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = constant.

Lemma 3. If a spatial curve a is of constant nonzero torsion [[tau].sub.[alpha]], then the curve

[beta](s) = a[alpha](s) + b(-1/[[tau].sub.[alpha]]N(s) - [[epsilon].sub.3] [integral] B(s)ds)

is a Bertand curve.

Proof of Lemma 3. Direct computations show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

From these

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Put [epsilon] = sgn{[[epsilon].sub.1] [[epsilon].sub.2] [[epsilon].sub.3] (a + (b[[epsilon].sub.1][[kappa].sub.[alpha]]/[[tau].sub.[alpha]])}. Then [epsilon]b[[kappa].sub.[beta]] + a[[tau].sub.[beta]] = -[[epsilon].sub.3][[tau].sub.[alpha]] = constant.

From these Lemma, one can deduce the following:

Theorem 5. (Representation formula) Let u([sigma]) be a curve in the [H.sup.2] parametrised by arclength. Then define three spatial curves [alpha], [beta] and [gamma] by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then [alpha] is a constant curvature curve, [beta] is a constant torsion curve and [gamma] is a Bertrand curve. Conversely, every Bertrand curve can be represented in this form.

Proof of Theorem 5. Here we give a detailed proof.

Let u = u([sigma]) be a timelike curve in [H.sup.2] parametrised by the arclength [sigma]. Then = {[xi] = u', [eta] = u x u', u} is a positive orthonormal frame field along u. Hence,

u" = u + [lambda][eta], u x u" = -[lambda][xi].

for some function [lambda]. From the definition of [gamma], we get

[gamma]' = a(u + tanh [theta][eta]), [gamma]" = a(1 - [lambda] tanh [theta])[xi], [gamma]'" = a(1 - [lambda] tanh [theta])(u + [lambda][eta])

The arclength parameter s of [gamma] is determined by

([gamma]', [gamma]') = [a.sup.2]/[cosh.sup.2] [theta] [(ds/d[sigma]).sup.2].

Moreover we have

[gamma]' x [gamma]" = [a.sup.2] (1 - [lambda] tanh [theta]) ([eta] + tanh [theta]u),

det([gamma]', [gamma]", [gamma]'") = -<[gamma]' x [gamma]", [gamma]'"> = [a.sup.3] [(1 - [lambda] tanh [theta]).sup.2] (tanh [theta] - [lambda]).

Using these,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence we have

a([kappa] - tanh [theta][tau]) = 1.

Thus [gamma] is a Bertrand curve.

Next, we compute the curvature of [alpha] and torsion of [beta]. Direct computation shows that

[alpha]' = au, [alpha]" = a[xi], [alpha]' x [alpha]" = [a.sup.2][eta],

[beta]' = a tanh [eta], [beta]" = -[alpha][lambda] tanh [xi], [beta]' x [beta]" = -[a.sup.2][lambda] [tanh.sup.2] [theta]u,

det([beta]', [beta]", [beta]'") = [a.sup.3] [[lambda].sup.2] [tanh.sup.3] [theta].

Hence

[[kappa].sub.[alpha]] = 1/a, [[tau].sub.[beta]] = 1/a tanh [theta].

Conversely. let [gamma](s) be a timelike Bertrand curve with relation:

a([kappa] - tanh [theta][tau]) = 1.

Denote by [sigma] the arclength parameter of the spherical curve:

u = cosh [theta]T - sinh [theta]B.

Then

[u.sub.[theta]] = cosh [theta]/a N.

Hence d[sigma]/ds = cosh [theta]/a. Thus

au[[sigma].sub.s] = cosh [theta] (cosh [theta]T - sinh [theta]B),

a tanh [theta]u x [u.sub.[sigma]][[sigma].sub.s] = - sinh [theta](cosh [theta]B - sinh [theta]T).

Henceforth,

a [integral] ud[sigma] - a tanh [theta] [integral] u x du/d[sigma] = [integral] T (s)ds = [gamma](s).

Acknowledgements

The authors would like to thank professor Jun-ichi Inoguchi for suggesting us the present study and his invaluable comments.

Conclusion

In this paper, we gave some characterizations for Bertrand curves and spatial curves in Minkowski 3-space We obtained representation formulae for Bertrand curves in [E.sup.3.sub.1]. We hope these results will be helpful to mathematicians who are specialized on mathematical modeling.

References

[1] H. Balgetir, M. Bektas and M. Ergut, Bertrand curves for nonnull curves in 3-dimensional Lorentzian space, Hadronic Journal, 27(2004), 229-236.

[2] H. Balgetir, M. Bektas and J. Inoguchi, Null Bertrand curves in Minkowski 3-space, Note di Mat., 23(2004/2005), No. 1, 7-13.

[3] N. Ekmekci and K. Ilarslan, On Bertrand curves and their characterization, Differential Geometry-Dynamical Systems, 3(2001), 17-24.

[4] L. P. Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces, Ginn and Company, Boston, 1909.

[5] A. W. Nutbourne and R. R. Martin, Differential Geometry Applied to the Design of Curves and Surfaces, ellis Horwood, Chichester, UK, 1988.

[6] T. Otsuki, Differential Geometry (in Japanese), Asakuara, Tokyo, 1961.

[7] W. K. Schief, On the integrability of Bertrand curves and Razzaboni surfaces, J. Geom. Phys., 45(2003), 130-150.

[8] B. van-Brunt and K. Grant, Potential application of Weingarten surfaces in CADG, Part I: Weingarten surfaces and surface shape investigation, Computer Aided Geometric Design, 13(1996), 569-582.

[9] B. van-Brunt, Congruent characteristics on linear Weingarten surfaces in Euclidean 3-space, New Zealand J. Math., 27(1998), 269-276.

[10] C. E. Weatherburn, Differential Geometry of Three Dimensions I, Cambridge Univ. Press, 1927.

Handan Balgetir Oztekin and Mehmet Bektas

Department of Mathematics, Firat University, 23119, Elazig, Tiirkiye

E-mail: handanoztekin@gmail.com mbektas@firat.edu.tr
COPYRIGHT 2010 American Research Press
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2010 Gale, Cengage Learning. All rights reserved.

Article Details
Printer friendly Cite/link Email Feedback
Author:Oztekin, Handan Balgetir; Bektas, Mehmet
Publication:Scientia Magna
Article Type:Report
Geographic Code:7TURK
Date:Jan 1, 2010
Words:2491
Previous Article:On quotient binary algebras.
Next Article:Smarandache's Cevians theorem (II).
Topics:

Terms of use | Privacy policy | Copyright © 2022 Farlex, Inc. | Feedback | For webmasters |