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A note on Bertrand curves and constant slope surfaces according to Darboux frame.

1. INTRODUCTION

In the theory of curves in Euclidean space, one of the important and interesting problems is the characterization of a regular curve. In the solution of the problem, the curvature [kappa] and the torsion [tau] of a regular curve have an effective role. For example: if [kappa] = [tau] = 0, then the curve is a geodesic. If [kappa] [not equal to] 0 (constant) and [tau] = 0, then the curve is a circle with radius 1/[kappa]. So we can determine the shape and size of a regular curve by using its the curvature and the torsion.

In general, a curve is called a general helix or a constant slope curve if the tangent lines make a constant angle with a fixed direction. Assume that [tau] [not equal to] 0, a necessary and sufficient condition that a curve be a general helix is that the ratio [kappa]/[tau] is constant. If both [kappa] and [tau] are non-zero constants, it is called a circular helix.

There are a lot of interesting applications of helices (e.g., DNA double and collagen triple helix, helical staircases, helical structures in fractal geometry and so on). All these make authors say that the helix is one of the most fascinated curves in science and nature.

Another approach to solution of the problem is to consider the relation between the Frenet vectors of the curves. For instance, Bertrand curves:

Bertrand curves discovered by J. Bertrand in 1850 are one of the important and interesting topics of classical special curve theory. A Bertrand curve is defined as a special curve which shares its principal normals with another special curve, called Bertrand mate or Bertrand partner curve. The curve is a Bertrand curve if and only if there exist non-zero real numbers A, B such that A[kappa](s) + B[tau](s) = 1 for any s [member of] I. Hence a circular helix is a Bertrand curve (see [7]).

Bertrand mates represent particular examples of offset curves [1] which are used in computer-aided design (CAD) and computer-aided manufacture (CAM). Furt-hermore, Bertrand curves may be regarded as 1-dimesional analogue of Weingarten surfaces.

Constant slope surfaces were defined in [5]. Such surfaces are those whose position vectors make a constant angle with the normals at each point on the surface. Munteanu [5] showed that they can be constructed by using an arbitrary curve on the sphere S2 or an equiangular spiral. Constant slope surfaces have nice shapes and they are interesting in terms of differential geometry. The study of these surfaces is similar to that of the logarithmic spirals and general helices. At least for their shapes, one can say that constant slope surfaces are one of the most fascinated surfaces in Euclidean 3-space.

In [4], Babaarslan and Yayli studied constant slope surfaces and Bertrand curves in Euclidean 3-space. They found parametrization of constant slope surfaces for the tangent, the principal normal, the binormal and the Darboux indicatrices of a space curve. Furthermore, they investigated Bertrand curves corresponding to parameter curves of constant slope surfaces. By using the Darboux frame for the curves lying on the surfaces, Bertrand D-curves were defined and given the characterizations of them in [6].

In this paper, we will give some different characterizations of Bertrand curves and constant slope surfaces with respect to the Darboux frame. Afterwards, we will express some interesting relations and give some examples of our main results.

2. BASIC NOTIONS AND SOME PROPERTIES

Let [??]: I [right arrow] [R.sup.3] be a curve with [??]'(t) [not equal to] 0, where [??]'(t) = d[??]/dt(t). We say that the curve [??] is parametrized by the arc-length if it satisfies [parallel][??]'(s)[parallel] = 1. Denote by {T, N, B}, the moving Frenet frame along the curve [??] in the space [R.sup.3]. For the unit speed curve f with the first and second curvatures, [kappa] and [tau] in the space [R.sup.3], the following Frenet-Serret formulae are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Here, the curvature functions are defined by [kappa] = [kappa](s) = [parallel]T'[parallel] and [tau] = [tau](s) = -<N', B>. For a general parameter t of a space curve [??], we can calculate the curvature and the torsion as follows:

[kappa](t) = [[parallel][??]'(t) x [??]"(t)[parallel]/[[parallel][??]'(t)[parallel].sup.3]]; [tau](t) = [det([??]'(t), [??]"(t), [??]'"(t))/[[parallel][??]'(t) x [??]"(t)[parallel].sup.2]

Let f: I [right arrow] [S.sup.2] be a unit speed spherical curve. We denote v as the arc-length parameter of f .We denote t(v) = f'(v)and we call t(v) a unit tangent vector of f at v. We now set a vector s(v) = f(v) x t(v), where f denotes the position vector of the curve. By definition of the curve f, we have an orthonormal frame {f(v), t(v), s(v)} along f. This frame is called the Sabban frame of f [3]. Then we have the following spherical Frenet-Serret formulae of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [[kappa].sub.g] (v) is the geodesic curvature of the curve f on [S.sup.2]

which is given by [[kappa].sub.g](v) = det (f(v), t(v), t'(v)).

Now we can express this lemma:

Lemma 2.1. Let f: I [right arrow] [S.sup.2] be a unit speed spherical curve. Then

[??](v) = a[[integral].sup.v.sub.0] f(t)dt + a tan[xi] [[integral].sup.v.sub.0] f(t) x f'(t)dt

is a Bertrand curve, where a, [xi] are constant numbers. Moreover all Bertrand curves can be constructed by this method [4].

As a consequence of this lemma, we get the following corollary.

Corollary 2.1. The spherical curve f is a circle if and only if the corresponding Bertrand curve is a circular helix.

Proof. By using Lemma 2.1, we have [kappa]'(v) = -[[epsilon]/2a]([[kappa]'.sub.g](v)sin 2[xi]) and [tau]'(v) = [1-a] ([cos.sup.2] [xi][[kappa]'.sub.g](v)). The spherical curve f is a circle if and only if [[kappa].sub.g](v) = constant. This condition is equivalent to the condition that both [kappa](v) and [tau](v) are non-zero constants. The proof is completed.

Constant slope surfaces are parametrized as follows:

Theorem 2.1. Let [??]: S [right arrow] [R.sup.3] be an isometric immersion of a surface S in the Euclidean 3-space. Then S is a constant slope surface if and only if either it is an open part of the Euclidean 2-sphere centered in the origin, or it can be parametrized by

[??](u, v) = u sin [theta] (cos [xi]f(v) + sin [xi]f (v) x f'(v)), (2.1)

where [theta] is a constant (angle) different from 0, [xi] = [xi](u) = cot [theta] log u and f is a unit speed curve on the Euclidean 2-sphere [S.sup.2], [5].

Accordingly, the following theorem holds.

Theorem 2.2. Let [??] : S [right arrow] [R.sup.3] be an isometric immersion of a surface S in the Euclidean 3-space and [??](v) be a v-parameter curve ofthe constant slope surface [??](u, v).Then [[integral].sup.v.sub.0] [??](v)dv is a Bertrand curve, [4].

Let M = M(u,v) be an oriented surface in the space [R.sup.3] and we consider a unit speed curve [??](s) lying on M. Instead of the Frenet and the Sabban frame fields on [??], let us consider the Darboux frame fields T, g, n, where T is the unit tangent of [??], n is the surface normal restricted to [??] and g is a unit vector given by g = n x T. So we have an orthonormal frame {T(s),g(s),n(s)} along [??]. This frame is called the Darboux frame of [??]. Since the unit tangent T is common in both the Frenet frame and the Darboux frame, the vectors N, B, g and n lie on the same plane. Thus, the relations between these frames can be given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [gamma] is the angle between the vectors g and N. The derivative formulae of the Darboux frame are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [k.sub.n](s) = <[??]"(s), n(s)), [k.sub.g](s) = <[??]"(s), g(s)> and [[tau].sub.g](s) = -<n'(s), g(s)> are called the normal curvature, the geodesic curvature and the geodesic torsion of [??], respectively.

The relations for the normal curvature, the geodesic curvature and the geodesic torsion are given by

[k.sub.n] = [kappa] sin [gamma], [k.sub.g] = [kappa] cos [gamma], [[tau].sub.g] = [tau] - [gamma]'.

For the curve [??](s) lying on a surface M, the following conditions are provided:

--[??] (s) is an asymptotic curve if and only if [k.sub.n] = 0;

--[??] (s) is a geodesic curve if and only if [k.sub.g] =0;

--[??] (s) is a principal curve if and only if [[tau].sub.g] =0, [2].

Spherical images are well-known in classical differential geometry of curves. When we move all unit tangent vectors T of a curve [??], their end points generate a curve (T) on the unit sphere [S.sup.2].The curve (T) is called the Darboux spherical image of the curve [??]. If [??] = [??] (s) is a natural representation of [??], then(T) = T(s) is a representation of (T). Similarly, one considers the Darboux spherical images of the curve [??], (g) = g(s)and (n) = n(s).

3. SOME CHARACTERIZATIONS BERTRAND CURVES AND CONSTANT SLOPE SURFACES ACCORDING TO DARBOUX FRAME

In this section, by using the Darboux spherical images, we give some characte-rizations of Bertrand curves and constant slope surfaces.

Theorem 3.1. Let [??](s) be a unit speed curve lying on oriented surface M in the space [R.sup.3] and (T): I [right arrow] [S.sup.2] be the Darboux spherical image of the curve [??]. Then the constant slope surface can be parametrized by

[??]T (u, v) = u sin [theta](cos[xi]T(v) + sin[xi][[k.sub.g](v)n)(v) - [k.sub.n](v)g(v)/[kappa](v)]),

where v = [[integral].sup.s.sub.0][parallel]T'(s)[parallel]ds.

Proof. Since (T): I [right arrow] [S.sup.2] is a spherical curve, we can take (T) = f(v). Substituting this into the Eq. (2.1), we have [[??].sub.T](u, v) = u sin [theta] (cos[xi]T(v) + sin[xi]T(v) x T'(v)). From the derivative formulae of the Darboux frame, we get

T' = [dT/dv] = [dT/ds] x [ds/dv] = ([k.sub.g]g + [k.sub.n]n) x [ds/dv].

Taking the norm of both sides, it follows that [ds/dv] = [1/[k.sup.2.sub.g]] = [1/[kappa]]and then T' = [k.sub.g]g + [k.sub.n]n/[kappa]. By using the Darboux frame, we obtain T x T' = [[k.sub.g]n - [k.sub.n]g/[kappa]].

As a result, we have

[[??].sub.T](u, v) = u sin [theta] (cos[xi]T(v) + sin[xi][[k.sub.g](v)n(v) - [k.sub.n](v)g(v)/[kappa](v)].

This completes the proof.

Corollary 3.1. Let [??](s) be a geodesic curve. Then the constant slope surface [[??].sub.T](u, v) can be parametrized by

[[??].sub.T](u, v) = u sin [theta](cos[xi]T(v) - [xi]sin[xi]g(v)),

where [epsilon] = [+ or -]1.

Considering this corollary, we give the following remark.

Remark 3.1. Since [??](s) is a geodesic curve, we can say that g = B. Thus Corollary 3.1 coincides with Proposition 1 in [4].

We give an example to illustrate our obtained result.

Example 3.1. We consider the cylinder M, with the usual parametrization given by

M(u, v) = (cos u, sin u, v).

Then u-parameter curve [alpha], passing through P = M(0,1) is given by

[alpha](s) = (cos s, sin s, 1).

The Darboux frame of [alpha] can be obtained as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then the normal curvature, the geodesic curvature and the geodesic torsion of a are given by [k.sub.n](s) = -1, [k.sub.g](s) = 0 and [[tau].sub.g](s) = 0. So [alpha] is both a geodesic curve and a principal curve. By using Corollary 3.1, we can easily find the constant slope surface as

[[??].sub.T](u, v)= u sin [theta](cos[xi](-sin s, cos s, 0) + sin [xi](0, 0, 1)).

The binormal vector of [alpha] is equal to B =(0, 0, 1). Thus we can see that g = B.

We also have the following corollaries of Theorem 3.1.

Corollary 3.2. Let [??](s) be an asymtotic curve. Then the constant slope surface [[??].sub.T](u, v) can be parametrized by

[[??].sub.T](u, v) = u sin [theta](cos [xi]T(v) + [epsilon] sin [xi]n(v)),

where [epsilon] = [+ or -]1.

Corollary 3.3. Let [[??].sub.T](v) be a v-parameter curve of the constant slope surface [[??].sub.T](u,v). Then

[[integral].sup.v.sub.0][[??].sub.T](v)dv = u sin [theta](cos[xi][[integral].sup.v.sub.0]T(v)dv + sin [xi] [[integral].sup.v.sub.0][[k.sub.g](v)n(v) - [k.sub.n](v)g(v)/[kappa](v)]dv)

is a Bertrand curve.

Corollary 3.4. The spherical curve T(v) is a circle if and only if the corresponding Bertrand curve [[integral].sup.v.sub.0][[??].sub.T](v)dv is a circular helix.

In the following we will give analogue results in terms of the Darboux spherical images (g) and (n) of a curve [??].

Theorem 3.2. Let [??](s) be a unit speed curve lying on oriented surface M in the space [R.sup.3] and (g): I [right arrow] [S.sup.2] be the Darboux spherical image of the curve [??]. Then the constant slope surface can be parametrized by

[[??].sub.g](u, v) = sin [theta](cos[xi]g(v) + sin[xi][[k.sub.g](v)n(v) + [[tau].sub.g](v)T(v)/[square root of [k.sup.2.sub.g](v) + [[tau].sup.2.sub.g]]),

where v = [[integral].sup.s.sub.0][parallel]g'(s)[parallel]ds.

Proof. (g): I [right arrow] [S.sup.2] is a spherical curve, so we can take (g) = f(v). Substituting this into the Eq. (2.1), we obtain

[[??].sub.g](u, v) = u sin [theta] (cos[xi]g(v) + sin[xi]g(v) x g'(v)).

From the derivative formulae of the Darboux frame, we have

g' = [dg/dv] = [dg/ds] x [ds/dv] = (-[k.sub.g]T + [[tau].sub.g]n) x [ds/dv].

Taking the norm of both sides, we get

ds/dv = 1/square root of [k.sup.2.sub.g] + [[tau].sup.2.sub.g]

and then

g' = -[k.sub.g]T + [[tau].sub.g]n/[square root of [k.sup.2.sub.g] + [[tau].sup.2.sub.g]].

By using the Darboux frame, we obtain

g x g' = -[k.sub.g]n + [[tau].sub.g]T/[[k.sup.2.sub.g] + [[tau].sup.2.sub.g]].

As a result, we have

[[??].sub.g](u, v) = u sin [theta] (cos[xi]g(v) + sin[xi][[k.sub.g](v)n(v) + [[tau].sub.g](v)T(v)/[square root of [k.sup.2.sub.g](v) + [[tau].sup.2.sub.g]]).

Therefore, theorem is proved.

We have the following corollaries of Theorem 3.2.

Corollary 3.5. Let [??](s) be a geodesic curve. Then the constant slope surface [[??].sub.g](u, v) can be parametrized by

[[??].sub.g](u, v) = u sin [theta](cos [xi]g(v) + [epsilon]sin [xi]T(v)),

where [epsilon] = [+ or -]1.

Corollary 3.6. Let [??](s) be a principal curve. Then the constant slope surface [[??].sub.g](u, v) can be parametrized by

[[??].sub.g](u, v) = u sin [??](cos [xi]g(v) + [epsilon] sin [xi]n(v)),

where [epsilon] = [+ or -]1.

Corollary 3.7. Let [[??].sub.g](v) be a v-parameter curve of the constant slope surface [[??].sub.g](u, v). Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a Bertrand curve.

Corollary 3.8. The spherical curve g(v) is a circle if and only if the corresponding Bertrand curve [[integral].sup.v.sub.0] [[??].sub.g](v)dv is a circular helix.

Moreover, we have the following parametrizations of Bertrand curves and cons-tant slope surfaces.

Theorem 3.3. Let [??](s) be a unit speed curve lying on oriented surface M in the space [R.sup.3] and (n): I [right arrow] [S.sup.2] be the Darboux spherical image of the curve [??]. Then the constant slope surface can be parametrized by

[[??].sub.n](u, v) = u sin [theta] (cos[xi]n(v) + sin[xi][-[k.sub.n](v)g(v) + [[tau].sub.g](v)T(v)/[square root of [k.sup.2.sub.n](v) + [[tau].sup.2.sub.g](v)],

where v = [[integral].sup.s.sub.0][parallel]n'(s)[parallel]ds.

Proof. Since (n): I [right arrow] [S.sup.2] is a spherical curve, we can take (n) = f(v). Substituting this into the Eq. (2.1), we have

[[??].sub.n](u, v) = u sin [theta] (cos[xi]n(v) + sin [xi]n(v) x n'(v)).

Using the derivative formulae of the Darboux frame, we get

n' = [dn/dv] = [dn/ds] x [ds/dv] = (-[k.sub.n]T - [[tau].sub.g]g x [ds/dv].

Taking the norm of both sides, we have

[ds/dv] = [1/[square root of [k.sup.2.sub.n] + [[tau].sup.2.sub.g]]

and then

n' = -[[k.sub.n]T + [[tau].sub.g]g/[square root of [k.sup.2.sub.n] + [[tau].sup.2.sub.g]].

By the Darboux frame, we obtain

n x n' = [-[k.sub.n]g + [[tau].sub.g]T/[square root of [k.sup.2.sub.n] + [[tau].sup.2.sub.g]].

So, we conclude

[[??].sub.n](u, v) = u sin [theta] (cos[xi]n(v) + sin[xi][-kn(v)g(v) + [[tau].sub.g](v)T(v)/[square root of [k.sup.2.sub.n](v) + [[tau].sup.2.sub.g](v)].

This completes the proof.

We have the following corollaries of Theorem 3.3.

Corollary 3.9. Let [??](s) be an asymptotic curve. Then the constant slope surface [[??].sub.n](u, v) can be parametrized by

[[??].sub.n](u, v) = u sin [theta] (cos[xi]n(v) + [epsilon] sin [xi]T(v)),

where [epsilon] = [+ or -]1.

Corollary 3.10. Let f (s) be a principal curve. Then the constant slope surface [[??].sub.n](u, v) can be parametrized by

[[??].sub.n](u, v) = u sin [theta] (cos [xi]n(v) - [epsilon] sin[xi]g(v)), (3.1)

where [epsilon] = [+ or -]1.

Corollary 3.11. Let [[??].sub.n](v) be a v-parameter curve of the constant slope surface [[??].sub.n](u, v). Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a Bertrand curve.

Corollary 3.12. The spherical curve n(v) is a circle if and only if the corresponding Bertrand curve [[integral].sup.v.sub.0] [[??].sub.n](v)dv is a circular helix.

We now illustrate an example of constant slope surfaces and Bertrand curves for the Darboux spherical image (n) and draw their pictures using Mathematica computer program.

Example 3.2. Let us consider a spherical curve f defined by

f(s) = (cos [square root of 2]s, sin [square root of 2]s, l).

Then the Darboux frame of f can be calculated as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The normal curvature, the geodesic curvature and the geodesic torsion of f are expressed as

[k.sub.n](s) = -1, kg(s) = 1 and [[tau].sub.g](s) = 0.

Thus, f is a principal curve.

Now we compute the Darboux spherical image (n) of the curve f. Since v = [[integral].sup.s.sub.0][parallel]n'(s)[parallel]ds and n'(s) = (- sin [square root of 2]s, cos [square root of 2]s, 0), we have v = s. So the Darboux spherical image (n) of the curve f is equal to

n(v) = [[square root of 2]/2](cos [square root of 2]v, sin [square root of 2]v, 1).

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

Substituting n(v), g(v)and [epsilon] = -1 into Eq. (3.1), we obtain the following constant slope surface

[[??].sub.n](u, v) = [[square root of 2]/2]u sin [theta](cos [xi](cos [square root of 2]v, sin [square root of 2]v, 1) + sin[xi](- cos [square root of 2]v, - sin [square root of 2]v, 1)) .

From Corollary 3.11, for u = e,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a Bertrand curve.

Since the spherical curve n(v) is a circle, from Corollary 3.12, Bertrand curve [[integral].sup.v.sub.0][[??].sub.n](v)dv is a circular helix (see Figure 3).

We can draw the pictures of n(v), [[??].sub.n](u, v) and [[integral].sup.v.sub.0][[??].sub.n](v)dv for [theta] = [square root of [pi]]/6 in Figures 1-3, respectively.

[FIGURE 3 OMITTED]

Acknowledgment. The authors are very grateful to referee for his/her valuable suggestions which improved the first version of the paper.

REFERENCES

[1] A. W. Nutbourne and R. R. Martin: Differential Geometry Applied to Design of Curves and Surfaces, Ellis Horwood, Chichester, UK, 1988.

[2] B. O'Neill: Elementary Differential Geometry, Academic Press Inc., New York, 1966.

[3] J. Koenderink: Solid shape, MIT Press, Cambridge, MA, 1990.

[4] M. Babaarslan and Y. Yayli: The characterizations of constant slope surfaces and Bertrand curves, Int. J. Phys. Sci., 6(2011), No. 8, 1868-1875.

[5] M. I. Munteanu: From Golden Spirals to Constant Slope Surfaces, J. Math. Phys., 51(2010), No. 7, The Art. No. 073507, 1-9.

[6] M. Kazaz, H. H. Ugurlu, M. Onder and S. Oral: Bertrand Partner D-Curves in Euclidean 3-space E3, arXiv:1003.2044v3 [math.DG], 2010.

[7] S.Izumiya and N.Takeuchi: Generic properties of helices and Bertrand curves, J. Geom., 74(2011), 97-109.

Bozok University

Faculty of Arts and Sciences

Department of Mathematics

66100, Yozgat, Turkey

E-mail address: murat.babaarslan@bozok.edu.tr

Bozok University

Faculty of Arts and Sciences

Department of Mathematics

66100, Yozgat, Turkey

E-mail address: yali.tandogan@bozok.edu.tr

Ankara University

Faculty Science

Department of Mathematics

06100, Tandogan, Ankara, Turkey

E-mail address: yayli@science.ankara.edu.tr

Received: September 26, 2011. Revised: December 27, 2011.

2010 Mathematics Subject Classification: 53A04, 53A05.
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Author:Babaarslan, Murat; Tandogan, Yusuf Ali; Yayli, Yusuf
Publication:Journal of Advanced Mathematical Studies
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Date:Jan 1, 2012
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