On Darboux Helices in Euclidean 3-Space - Global Journalshelices in terms of and thus prove that the class of Darboux helices coincide with the class of slant helices. In special,

© 2012. Evren Ziplar, Ali Senol & Yusuf Yayli. This is a research/review paper, distributed under the terms of the Creative Commons Attribution-Noncommercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/3.0/), permitting all non commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Global Journal of Science Frontier Research Mathematics and Decision Sciences Volume 12 Issue 13 Version 1.0 Year 2012 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA) Online ISSN: 2249-4626 & Print ISSN: 0975-5896

On Darboux Helices in Euclidean 3-Space

By Evren Ziplar, Ali Senol

& Yusuf Yayli

Ankara University, Turkey

Abstract -

In this paper, we introduce a Darboux helix to be a curve in 3-space whose Darboux vector makes a constant

angle with a fixed straight line. We completely characterize Darboux helices in terms of and thus prove that the class of Darboux helices coincide with the class of slant helices. In special, if we take = constant, the curves are curve of constant precession.

Keywords and phrases : helices, slant helices, curves of constant precession, darboux vector.

GJSFR-F Classification

: MSC 2000: 53C040, 53A05

On DarbouxHelicesinEuclidean3-Space

Strictly as per the compliance and regulations of :

� & �

t �+ 22

On Darboux Helices in

Euclidean 3-Space

Evren Ziplar

α, Ali Senol

σ

&

Yusuf Yayli

ρ

Abstract

-

In this paper, we introduce a Darboux helix to be a curve in

3-space whose Darboux vector makes a constant

Author α

:

Ankara University, Faculty of Science, Department of Mathematics, 06100 Ankara,

Turkey. E-mail : [email protected]

Author σ

: Cankiri Karatekin University, Faculty of Science, Department of Mathematics,

18100 Cankiri, Turkey.

E-mail

: [email protected]

Author ρ

: Ankara University, Faculty of Science, Department of Mathematics, 06100 Ankara,

Turkey.

E-mail

: [email protected]

I.

Introduction

© 2012 Global Journals Inc. (US)

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Notes

In di¤erential geometry, a curve of constant slope or general helix in Euclidean3-space R3 is dened by the property that tangent makes a constant angle with axed straight line (the axis of general helix). Due to a classical result proved by

M.A. Lancert in 1802 in R3 is a general helix if and only if the ratio��is constant

along curve, where � and � 6= 0 denote the curvature and torsion, respectively.Using killing vector eld along a curve, Barros gave a similar result for curves in3-dimensional real space forms [3]. Several authers introduced di¤erent types ofhelices and investigated their properties. For instance, Izumiya and Takeuchi de-ned in [1] slant helices by the property that the principal normal makes a constantangle with a xed direction. Moreover, they showed that � is a slant helix in R3 ifand only if the geodesic curvature of the principal normal of a space curve � is aconstant function. Kula &Yayl¬investigated spherical images of tangent indicatrixof binormal indicatrix of slant helix and they have shown that spherical images arespherical helix [2]. On the other hand the second and the third auther introducedin [6] LC helices in 3-dimensional real space forms and study their main properties.The purpose of this paper is to introduce and study Darboux helices in R3:We

give a characterization of Darboux helices in terms of �&� . We give the relationsbetween darboux helices and slant helices. As a consequence, we observe thatDarboux helices coincide with slant helices. Finally, we show that curves of constantprecession are darboux helices.

II. Preliminaries

We now recall some basic concepts on classical di¤erantial geometry of spacecurves in Euclidean space. Let � : I � R ! R3 be a curve parameterized by arclenght and let fT;N;Bg denote the Frenet frame of the curve �:

Keywords and phrases : helices, slant helices, curves of constant precession, darboux vector.

angle with a fixed straight line. We completely characterize Darboux helices in terms of and thus prove that the class of Darboux helices coincide with the class of slant helices. In special, if we take ttttt = constant, the curves are curve of constant precession.

� & �t �+ 22

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rOn Darboux Helices in Euclidean 3-Space

III. Darboux Helices

Tp(s) = �(s):N(s)

Np(s) = �(s):T (s) + �(s):B(s)

Bp(s) = �(s):N(s)

where �(s) is the torsion of � at s.For any unit speed curve � : I � R! R3 dened a vector eld

C =(�T + �B)p�2 + �2

along � under the condition that �(s) 6= 0 and called it the

modied Darboux vector eld of � [1].

Let � be a curve in E3 with�

�6= 0 everywhere with nonzero curvature and

torsion � and � in E3: We say that � is a Darboux helix if its Darboux vectormakes a constant angle with a xed direction d, that is hW;di =constant along thecurve, where d is a unit vector eld in E3:

W = �T + �B

The direction of the vector d is axis of the Darboux helix. We can identify Darbouxhelices by the condition torsion and curvature. If �2 + �2 =constant, the darbouxhelices are the curves of constant precession. So, our curves are more general thanthe curves of constan precession. Although every general helice is a slant helice,the general helices are not darboux helices. Moreover, there is a relation betweendarboux helice and the surface of constant precession. The following result describesthe relation between darboux helice and the surface of constant precession.

Theorem 1. A normal conical surface is constant angle if and only if Generatingcurve � is a Darboux helix [5].

Theorem 2. Let � be a curve constant precession. If the conical surfaces constructinvolving the normal lines to the curve �;then the surface is a constant angle surfacewith the axis of d =W + �n [5].

Theorem 3. � is a Darboux helix if and only if ��(s) =�2 + �2

� 32

�21� ��

�0 functionis constant.

Proof. If the spherical indicatrix of the darboux vector W is a circle or a part ofcircle, then the curve � is a darboux helis. Let the parameter of the curve (c) be scand let Tc be the unit tanget vector of (c) : Let �c be the geodesic curvature of(c)in E3:

� (sc) = c (s) =�p

�2 + �2T +

�p�2 + �2

B

� (sc) = sin�T + cos�B

d�

dsc=dc

ds

ds

dsc

d�

dsc=��0cos�T �

0sin�B + � sin�N � cos�N

� dsdsc

[5]Özkald

iS,Yayl¬

Y.Constan

tangle

surfaces

andcurves

inE3:Intern

at¬onalElectron

icJou

rnal

ofGeom

etry.,4(1),70-78

(2011).

Ref.

�

�

�

� �

T (s) = �p(s) is a unit tangent vector of � at s:We dene the curvature of � by

For the derivatives of the frenet-serret formulae hold:�(s) =

�pp(s)

:


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Notes


Tc =d�

dsc= (�0 cos�T �0 sin�B)

ds

dsc

kTck = ��0cos�T �

0sin�B

� dsdsc

1 = �0 ds

dsc

ds

dsc=

1

�0

(1) Tc = cos�T sin�B

DTcTc =dTcdSc

ds

dsc

DTcTc =��0sin�T �

0cos�B + � cos�N + � sin�N

� 1�0

(2) DTcTc =�

sin�T cos�B +kwk�:

N

�Hence, from the equation (2), the geodesic curvature of (c) are computed as thefollowing.

�c = DTcTc

= sin�T cos�B +

kwk�0

N

(3) �c = DTcTc

=

s

1 +

�kwk�0

�2Therefore, we obtain

DTcTc = rTcTc

c(s)

(4) �2c = �2g + 1

by using the Gauss map

DTcTc = rTcTc

s(Tc); Tci c(s):

and from the equations (3) and (4), we have:

1 +

�kwk�0

�2= �2g + 1

(5) �g =kwk�0

On the other hand, taking the derivative of tan� =�

�;

�0: 1 + tan2 �

�=� ��

�0

(6) �0=

��2

�2 + �2

��

�0:

�

�

�

��

� �

�

�

�

h�

�

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Notes


Hence, by using the equations (5) and (6) , we get:

�g =

p�2 + �2��2

�2+�2

��

�0�g =

�2 + �2�32

�21

��

�0 ;where kwk =

p�2 + �2:The spherical indicatrix of (c) is a circle or a part of cir-

cle. Since the rst curvature of a circle is constant, we obtain �c =constant. So,�g =constant. If we denote �g with ��(s),

�g =�2 + �2

�32

�21

��

�0 = ��(s)and so, we have

�2 + �2�32

�21

��

�0 = ��(s)which is constant function.

Theorem 4. Let � : I ! E3 be a curve in E3. We assume that ��is not constant,

where � and � are curvature of �:Then,

� is a slant helice if and only if � is a Darboux helice

Proof. we assume that � is a slant helice. So we can write:

(7) �(s) =�2

(�2 + �2) 32

� ��

�0:

Similarly, if the curve � is a darboux helice

(8) ��(s) =�2 + �2

� 32

�21� ��

�0 :Consequently, we obtain:

�(s)��(s) = sbt

�(s) = sbt , ��(s) = sbt

From the previous Theorem, rstly we are going to nd the axis of the slanthelices since a slant helice is also a darboux helice.

3.1. The axis of Darboux helice. We rst assume that � is a slant helix. Let dbe the vector eld such that the function hN; di =cos�=constant. There exists a1and a3 such that

(9) d = a1T + a3B + cos�N:

Then, if we take the derivative of the equation (9) and by using frenet equation, wehave:

d0 = (a0 cos�:�)T + (a1� �a3)N + (a03 + cos�:�)B

since the system fT;N;Bg is linear independent, we get:

a01 cos�:� = 0

��

� ��

�

� �

�


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(10) a1� �a3 = 0

(11) a03 + cos�:� = 0

and from (10) and (9), respectively

(12) a1 =� ��

�:a3

(13) hd; di = a21 + a23 + cos2� = constant

By using the equalities (12) and (13), we obtain:

(14)� ��

�2a21 + a

23 + cos

2� = constant

and from the equation (14) we have��

�2+ 1

�a23 = m

2

where m2 is constant. So,

(15) a3 =mr

1 +� ��

�2 ;Taking the derivative in each part of the equation (15) and by using (13), we get:

(16)�2

(�2 + �2)32

:� ��

�0= constant

We deduce from that the curve � is slant helice when we have d . Conversely,assume that the condition (16) is satised. In order to simplify the computations,we assume that the function (16) is constant. Dene

(17) d =�p

�2 + �2T +

�p�2 + �2

B + cos�N

A di¤erentiation of (17) together the frenet equations gives d0 = 0, that is, d is aconstant vector. It can easily be seen that d0 = 0; that is d is a constant. On theother hand, hN; di =cos� and this means that � is a slant helix.Now, we are going to show that the darboux vector W = �T + �B makes a

constant angle with the constant direction

d =�p

�2 + �2T +

�p�2 + �2

B + cos�N:

The constant direction d is the axis of both the slant helice � and the darbouxhelice �:These axises coincide but the making angles of these helices with d aredi¤erent.

Since � is a slant helice, hN; di =cos� =constant

d =�p

�2 + �2T +

�p�2 + �2

B + cos�N

�

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Notes


d =W

kWk+ cos�N

hd;W i = kdk : kWk :cos�

hd;W i =p1 + cos2�: kWk :cos�

hW;W ikWk

=p1 + cos2�: kWk :cos�

cos� =1p

1 + cos2�

Since cos� =constant , cos� is constant.

3.2. Curves of constant precession. A unit speed curve of constant precessionis dened by the property that its (Frenet) Darboux vector revolves about a xedline in space with angle and constant speed. A curve of constant precession ischaracterized by having

�(s) = $ sin(�(s);

�(s) = $ cos(�(s));

where $i0, � and are constant[4].If � is a curve of constant precession ;� is a slant helix [?]From the axis of the Darboux helice,

d =�p

�2 + �2T +

�p�2 + �2

B + cos�N

and

(18) d =W

kWk+ cos�N

where W = �T + �B:From (18),

p�2 + �2:d =W +

p�2 + �2:cos�N

By taking $ = kWk =p�2 + �2 , $:d = A and $:cos� = � :

A =W + �:N

If kWk =constant, the darboux helice � a curve of constant precession. We deducefrom that [4] is true.

Remark 1. All characterizations given for these slant helices can be given for thesedarboux helices.

Theorem 5. Let � be a unit speed curve in E3and let � be a slant helice (darbouxhelice). The curvatures �; � of the curve � satisfy the following non-linear equationsystem. �

�p�2 + �2

�� = 0;

��p

�2 + �2

�� = 0

Proof. Since � is a slant helice (darboux helice), the axis of � :

(19) d =�p

�2 + �2T +

�p�2 + �2

B + cos�N;

� �


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where �; � are curvatures of �:Taking the derivative in each part of the equation(19), we get

d�=

��p

�2 + �2

��T +

��p

�2 + �2

��B + �( �T + �B) = 0

since the systemfT;Bg is linear independent,��p

�2 + �2

�� = 0;

��p

�2 + �2

��+ �� = 0

Conclusion 1. If we take �2+�2 =constant, then the curve � is a curve of constantprecession [4].

So, the following theorem can be given.

Theorem 6. A necessary and su¢ cient condition that a curve be of constant pre-cession is that �(s) = $ sin(�(s); �(s) = $ cos(�(s)):up to reection or phase shiftof arclength, for constants $ and �.

Proof. Since A0= 0,

�� T + ��+ ��

�B = 0

and uniqueness of solutions of pairs of linear equations imply that A0= 0 if and

only if �(s) = $ sin(�(s); �(s) = $ cos(�(s)):The following example is related to darboux helices.

Example 1. Let the curve �(s) be a curve parametrized by the vector function:

�(s) = (

p5 + 1

5p5Sin(

p5 1

2s)

p5 1

5 +p5Sin(

p5 + 1

2s);

p5 + 1p5 5

Cos(

p5 1

2s) +

p5 1

5 +p5Cos(

p5 + 1

2s);

4p5Sin(

s

2))

where s 2 [0; 10�]:Then, �(s) is a darboux helix (or a curve of constant precession),where �(s) = Sin

p52 s and �(s) =Cos

p52 s:The curve is rendered in the following

gure.

Figure 1. The darboux helix � (s)

[4]Scoeld,P.D.Curvesofconstantprecession.Am.Math.Montly102,531-537,(1995).

Ref.

�

�

��

� �

� �

�

�

�

�

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Notes


References Références Referencias

Conclusion 2. All helices are slant helices. The slant helices which are not helicesare dened as Darboux helices. The Darboux helices are more general than thecurves of constant precession.

[1] Izumiya, S and Tkeuchi, N. New special curves and developable surfaces, Turk J. Math., 28,153-163, (2004).

[2] Kula, L and Yayl¬Y. On slant helix and its spherical indicatrix. Applied Mathematics andcomputation, 169, 600-607, (2005)

[3] Barros, M. General helices and a theorem Lancert. Proc. Amer. Math. Soc.,125(5),1503-1509(1907).

[4] Scoeld, P.D. Curves of constant precession. Am. Math. Montly 102, 531-537, (1995).[5] Özkaldi S, Yayl¬Y. Constant angle surfaces and curves in E3:Internat¬onal Electronic Journal

of Geometry., 4(1), 70-78 (2011).[6] Senol A, Yayl¬Y., LC helices in space forms, Chaos, Solitons& Fractals, 42 (4), 2115-2119

(2009).

On Darboux Helices in Euclidean 3-SpaceAuthorsKeywords and phrasesI. IntroductionII. PreliminariesIII. Darboux HelicesReferences Références Referencias

Documents

On Darboux Helices in Euclidean 3-Space - Global Journalshelices in terms of and thus prove that the class of Darboux helices coincide with the class of slant helices. In special,