-
© 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
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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
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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
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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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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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On Darboux Helices in Euclidean 3-Space
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
On Darboux Helices in Euclidean 3-Space
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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On Darboux Helices in Euclidean 3-Space
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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On Darboux Helices in Euclidean 3-Space
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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On Darboux Helices in Euclidean 3-Space
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