# 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

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

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[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