ISBN 978-1-59973-464-4

VOLUME 1, 2016

MATHEMATICAL COMBINATORICS (INTERNATIONAL BOOK SERIES)

Edited By Linfan MAO

EDITED BY

THE MADIS OF CHINESE ACADEMY OF SCIENCES AND

ACADEMY OF MATHEMATICAL COMBINATORICS & APPLICATIONS, USA

March, 2016

Vol.1, 2016 ISBN 978-1-59973-464-4

MATHEMATICAL COMBINATORICS (INTERNATIONAL BOOK SERIES) Edited By Linfan MAO

(www.mathcombin.com)

Edited By

The Madis of Chinese Academy of Sciences and

Academy of Mathematical Combinatorics & Applications, USA

March, 2016

Aims and Scope: The Mathematical Combinatorics (International Book Series) is a fully refereed international book series with ISBN number on each issue, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail

to be covered are:

Smarandache multi-spaces with applications to other sciences, such as those of algebraic multi-systems, multi-metric spaces,---, etc.. Smarandache geometries;

Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and map enumeration; Combinatorial designs; Combinatorial enumeration;

Differential Geometry; Geometry on manifolds; Low Dimensional Topology; Differential Topology; Topology of Manifolds; Geometrical aspects of Mathematical Physics and Relations with Manifold Topology;

Applications of Smarandache multi-spaces to theoretical physics; Applications of Combi- natorics to mathematics and theoretical physics; Mathematical theory on gravitational fields; Mathematical theory on parallel universes; Other applications of Smarandache multi-space and combinatorics.

Generally, papers on mathematics with its applications not including in above topics are also welcome.

It is also available from the below international databases:

Serials Group/Editorial Department of EBSCO Publishing

10 Estes St. Ipswich, MA 01938-2106, USA

Tel.: (978) 356-6500, Ext. 2262 Fax: (978) 356-9371

http://www.ebsco.com/home/printsubs/priceproj.asp and

Gale Directory of Publications and Broadcast Media, Gale, a part of Cengage Learning

27500 Drake Rd. Farmington Hills, MI 48331-3535, USA

Tel.: (248) 699-4253, ext. 1326; 1-800-347-GALE Fax: (248) 699-8075

http: //www.gale.com Indexing and Reviews: Mathematical Reviews (USA), Zentralblatt Math (Germany), Refer- ativnyi Zhurnal (Russia), Mathematika (Russia), Directory of Open Access (DoAJ), Interna- tional Statistical Institute (ISI), International Scientific Indexing (ISI, impact factor 1.659), Institute for Scientific Information (PA, USA), Library of Congress Subject Headings (USA). Subscription A subscription can be ordered by an email directly to

Linfan Mao

The Editor-in-Chief of International Journal of Mathematical Combinatorics Chinese Academy of Mathematics and System Science

Beijing, 100190, P.R.China

Email: maolinfan@163.com

Price: US$48.00

Editorial Board (4th)

Editor-in-Chief

Linfan MAO Shaofei Du

Chinese Academy of Mathematics and System Capital Normal University, P.R.China Science, P.R.China Email: dushf@mail.cnu.edu.cn

and

Academy of Mathematical Combinatorics & Minodone thu

Applications, USA

Email: maolinfan@163.com

Chinese Academy of Mathematics and System Science, P.R.China Email: xdhu@amss.ac.cn Deputy Editor-in-Chief Yuanqiu Huang Hunan Normal University, P.R.China Guohua Song Email: hyqq@public.cs.hn.cn Beijing University of Civil Engineering and ; Architecture, P.R.China leer

Email: songguohua@bucea.edu.cn Mansneld-UanetstiyUe Email: hiseri@mnsfld.edu

Editors Xueliang Li

Nankai University, P.R.China Arindam Bhattacharyya Email: lxl@nankai.edu.cn Jadavpur University, India Guodong Liu Email: bhattachar1968@yahoo.co.in Huizhou University Said Broumi Email: Igd@hzu.edu.cn Hassan IT University Mohammedia W.B.Vasantha Kandasamy Hay El Baraka Ben M’sik Casablanca Indian Institute of Technology, India B.P.7951 Morocco Email: vasantha@iitm.ac.in Junliang Cai Ion Patrascu Beijing Normal University, P-R.China Fratii Buzesti National College Email: caijunliang@bnu.edu.cn Craiova Romania Yanxun Chang Han Ren Beijing Jiaotong University, P.R.China East China Normal University, P.R.China Email: yxchang@center.njtu.edu.cn Email: hren@math.ecnu.edu.cn Jingan Cui Ovidiu-Ilie Sandru Beijing University of Civil Engineering and pojitechnica University of Bucharest Architecture, P.R.China Romania

Email: cuijingan@bucea.edu.cn

li International Journal of Mathematical Combinatorics

Mingyao Xu

Peking University, P.R.China

Email: xumy@math.pku.edu.cn

Guiying Yan

Chinese Academy of Mathematics and System Science, P.R.China

Email: yanguiying@yahoo.com

Y. Zhang

Department of Computer Science

Georgia State University, Atlanta, USA

Famous Words:

There is no royal road to science, and only those who do not dread the fatigu- ing climb of gaining its numinous summits.

By Karl Marx, a German revolutionary .

Math. Combin. Book Ser. Vol.1(2016), 1-7

N*C*— Smarandache Curve of

Bertrand Curves Pair According to Frenet Frame

Stleyman Senyurt , Abdussamet Caliskan and Unzile Celik

(Faculty of Arts and Sciences, Department of Mathematics, Ordu University, Ordu, Turkey) E-mail: senyurtsuleyman@hotmail.com, abdussamet65@gmail.com, unzile.celik@hotmail.com

Abstract: In this paper, let (a,a*) be Bertrand curve pair, when the unit Darboux vector of the a* curve are taken as the position vectors, the curvature and the torsion of Smaran- dache curve are calculated. These values are expressed depending upon the a curve. Besides,

we illustrate example of our main results.

Key Words: Bertrand curves pair, Smarandache curves, Frenet invariants, Darboux vec-

tor.

AMS(2010): 53A04.

§1. Introduction

It is well known that many studies related to the differential geometry of curves have been made. Especially, by establishing relations between the Frenet Frames in mutual points of two curves several theories have been obtained. The best known of the 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 curve Partner. If a* = a+ AN, A = const., then (a,a*) are called Bertrand curves pair. If @ and a* Bertrand curves pair, then (T,T*) = cos@ = constant, [9], [10]. The definition of n-dimensional Bertrand curves in Lorentzian space is given by comparing a well-known Bertrand pair of curves in n- dimensional Euclidean space. It shown that the distance between corresponding of Bertrand pair of curves and the angle between the tangent vector fields of these points are constant. Moreover Schell and Mannheim theorems are given in the Lorentzian space, [7]. The Bertrand curves are the Inclined curve pairs. On the other hand, it gave the notion of Bertrand Representation and found that the Bertrand Representation is spherical, [8]. Some characterizations for general helices in space forms were given, [11].

A regular curve in Minkowski space-time, whose position vector is composed by Frenet frame vectors on another regular curve, is called a Smarandache curve [14]. Special Smarandache curves have been studied by some authors. Melih Turgut and Stha Yilmaz studied a special

case of such curves and called it Smarandache T Bz curves in the space Ej ([14]). Ahmad T.Ali

1Received August 9, 2015, Accepted February 2, 2016.

2 Siileyman Senyurt , Abdussamet Galigkan and Unzile Celik

studied some special Smarandache curves in the Euclidean space. He studied Frenet-Serret invariants of a special case, [1]. Senyurt and Caliskan investigated special Smarandache curves in terms of Sabban frame of spherical indicatrix curves and they gave some characterization

of Smarandache curves, [4]Ozcan Bektag and Salim Yiice studied some special Smarandache

curves according to Darboux Frame in E®, [3]. Kemal Tasképrii and Murat Tosun studied special

Smarandache curves according to Sabban frame on S$? ((2]). They defined NC-Smarandache curve, then they calculated the curvature and torsion of NB and TNB- Smarandache curves together with NC-Smarandache curve, [12]. It studied that the special Smarandache curve in terms of Sabban frame of Fixed Pole curve and they gave some characterization of Smarandache curves, [12]. When the unit Darboux vector of the partner curve of Mannheim curve were taken as the position vectors, the curvature and the torsion of Smarandache curve were calculated.

These values were expressed depending upon the Mannheim curve, [6].

In this paper, special Smarandache curve belonging to a curve such as N*C* drawn by Frenet frame are defined and some related results are given.

§2. Preliminaries

The Euclidean 3-space be inner product given by

C5 \oa?+a3+23

where (%1,22,2%3) E®. Let a: I be a unit speed curve denote by {7,N,B} the moving Frenet frame . For an arbitrary curve a E®, with first and second curvature, « and T

respectively, the Frenet formulae is given by [9], [10].

T’ =kN N'’=-«T+7B (2.1) B'=-1TN

Figure 1 Darboux vector

For any unit speed curve a: I > E?, the vector W is called Darboux vector defined by

W =7TT +B. (2.2)

N*C*— Smarandache Curve of Bertrand Curves Pair According to Frenet Frame 3

If we consider the normalization of the Darboux, we have

F K j = 2.3 sine = Ta 88? = TT ee) and C =sinyT + cos vB, (2.4)

where Z(W, B) = .

Definition 2.1([9]) Let a : I and a* : I be the C?— class differentiable unit speed two curves and let {T(s), N(s), B(s)} and {T*(s), N*(s), B*(s)} be the Frenet frames of

the curves a and a*, respectively. If the principal normal vector N of the curve a is linearly

dependent on the principal vector N* of the curve a*, then the pair (a, a*) is said to be Bertrand

curves pair.

The relations between the Frenet frames {T(s), N(s), B(s)} and {T*(s), N*(s), B*(s)} are as follows: T* =coséT +sin6B

N*=N (2.5) B* =—sindT + cos OB. where Z(T,T*) =0

Theorem 2.2((9], [10]) The distance between corresponding points of the Bertrand curves pair ES

an IE? is constant.

3. For the curvatures and the

Theorem 2.3((10]) Let (a,a*) be a Bertrand curves pair in

torsions of the Bertrand curves pair (a,a*) we have

Aw = sin? 0 r = constant A(1— AK)’ (2.6) F sin? 0 oe

73

Theorem 2.4([9]) Let (a,a*) be a Bertrand curves pair in E°. For the curvatures and the

torsions of the Bertrand curves pair (a,a*) we have

Ke as = «cos6—7TsinJd,

(2.7)

a = «sin@+7cos0.

By using equation (2.2), we can write Darboux vector belonging to Bertrand mate a*.

W* =7*T* +B". (2.8)

4 Stileyman Senyurt , Abdussamet Caligkan and Unzile Celik

If we consider the normalization of the Darboux vector, we have C* = siny*T™* + cosy" B*. (2.9)

From the equation (2.3) and (2.7), we can write

*

T Ksind + 7cosé

sing” = Sooo = Bin(ve + 9), (2.10) |W*| || ee. K* &cos@ 7 sind Soule 458) C= oo = ool + 9); |W*| || where ||W*|| = V«*? + 7*? = ||W|| and Z(W*, B*) = y*. By the using (2.5) and (2.10),

the final version of the equation (2.9) is as follows:

C* = sinyT + cos yB. (2.11)

§3. N*C*— Smarandache Curve of Bertrand Curves Pair According to Frenet Frame

Let (a, a*) be a Bertrand curves pair in E? and {T*, N*, B*} be the Frenet frame of the curve a* at a*(s). In this case, N*C* - Smarandache curve can be defined by

i * * Y(s) = (NT +0"), (3.1)

Solving the above equation by substitution of N* and C* from (2.5) and (2.11), we obtain

sinyT’ + N+cosyB

ae 32 v(s) a (3.2)

The derivative of this equation with respect to s is as follows, pee Cee eee Eee (3.3)

ds J2

and by substitution, we get

Ty = Cerone one (3.4)

V IW? 20'|WI + 9?

dsy [WIP -2¢'IWI +e? 7a | een Sa (3:9)

In order to determine the first curvature and the principal normal of the curve ~(s), we

where

N*C*— Smarandache Curve of Bertrand Curves Pair According to Frenet Frame 5

formalize

V2] (wi cos 6 + w3 sin @)T + woN + (—w} sin @ + w3 cos 0) B]

Ti,(s) = 5 : [IW I2 2v/IWI] + 9]

; (3.6)

where

wi = (—Kcosd + Tsind + y" cos(y + 4)) (|| WI? 2y'||WI| + v”) (— Kcos8 +rsin# + y’ cos(y + 4) (|WIWIl' e"|W I] e'IIWII + v'e")

w2 = (—||WIl? + o/IWI) (IW? 29] + 9”)

w3 = (ksind +7 cos y' sin(y + 4)) (|| WI? 2¢" |W] + y’”) (Ksind +7 cos 6 y'sin(y + 4)) (|WIWIl' e"IWIl ¢'IIWII + v'e")

The first curvature is

2(w4? + W2 + w37)

Ky = \|T%, ll, ky = |: 3 [IW 2 26 WI] +e?)

The principal normal vector field and the binormal vector field are respectively given by

[(w1 cos @ + w3 sin 0)T + w2N + (—w1 sind + w3 cos 6) B] LN See (3.7) VW1* + W2* + W3 wa | 2x sin 0 cos @ + 7(sin? 6 cos? 6) + y’ siny|T +w1[Ksin@ + 7 cos 6 vy’ sin(y + @)] N + w2[27 sin 6 cos 6

+k(sin? 6 cos? 0) + y! cos y] B SS (3.8) (IW? 2¢'|| WI + y’?) wi? + we? + w32)

The torsion is then given by

det (yo, py")

OS TW Ae _ ¥2(0n + od + up) Ty = Oe + + Pe

where n = (y' cos(y + 0) KcosO + rsin6)” + (kcos@ T sin @)||W||?

—(Kcos 6 7 sin A)y'||W|| \ = («cos d 7 8in 9)(y' cos(y + 8) Kcos8 + rsin6)’ + (—||W||? +y'||W|])! («sin + 7 cos 0)(K sin 8 + 7 cos@ gy! sin(y + 0)’ p = (—K sin 6 rc0s8)||W ||? + («sin @ + 7 cos A)y"||W || + (Ksind

Mu %

+7 cos ¢' sin(y + @))

6 Siileyman Senyurt , Abdussamet Galigkan and Unzile Celik

8 =—(—||WI|/? + ¢'||W|) (ksind + 7 cos ¢ sin(y + 6)

0 = —|(¢ cos(y + 8) 0088 + rsin8) (sin8 + 7.c08 0 ¢' sin(y +8)’ +(' cos(y +8) 0080 + 7sin8)' (sind + 7.cos6 ¢ sin(y + 6))|

= (¢' cos(y + 0) kcos6 + 7 sin 8) ( ||W||? + ¢’||W]).

Example 3.1 Let us consider the unit speed a curve and a®* curve: a(s) = —=(-—coss,—sins,s) and a*(s) = tea s, sin 8, s)- V2 V2

The Frenet invariants of the curve, a*(s) are given as following:

Ts) = at sin s,cos s, 1), N*(s) = (—coss, —sins,0) B*(s) = yn eth 1),C*(s) = (0,0,1) K*(s) = —s,7T*(s) =

In terms of definitions, we obtain special Smarandache curve, see Figure 1.

as as

* “8

Figure 2 N*C*-Smarandache Curve

References

[1] Ali A. T., Special Smarandache curves in the Euclidean space, International J. Math. Combin., 2(2010), 30-36.

[2] Bektag O. and Yiice S., Special Smarandache curves according to Darboux frame in Eu- clidean 3-space, Romanian Journal of Mathematics and Computer Science, 3(1)(2013), 48-59.

[3] Galigkan A. and Senyurt S., Smarandache curves in terms of Sabban frame of spherical

10 11

12

13

14

N*C*— Smarandache Curve of Bertrand Curves Pair According to Frenet Frame 7

indicatrix curves, Gen. Math. Notes, 31(2)(2015), 1-15.

Caligskan A. and Senyurt S., Smarandache curves in terms of Sabban frame of fixed pole curve, Boletim da Sociedade parananse de Mathemtica, 34(2)(2016), 53-62.

Caligskan A. and Senyurt S., N*C*- Smarandache curves of Mannheim curve couple ac- cording to Frenet frame, International J.Math. Combin., 1(2015), 1-13.

Ekmekci N. and Ilarslan K., On Bertrand curves and their characterization, Differential Geometry-Dynamical Systems, 3(2)(2001), 17-24.

Gorgiilii A. and Ozdamar E., A generalizations of the Bertrand curves as general inclined curve in E”, Commun. Fac. Sci. Uni. Ankara, Series Al, 35(1986), 53-60.

Hacisalihoglu H.H., Differential Geometry, Inénii University, Malatya, Mat. No.7, 1983 Kasap E. and Kuruoglu N., Integral invariants of the pairs of the Bertrand ruled surface, Bulletin of Pure and Applied Sciences, 21(2002), 37-44.

Sabuncuoglu A., Differential Geometry, Nobel Publications, Ankara, 2006

Senol A., Ziplar E. and Yayl Y., General helices and Bertrand curves in Riemannian space form, Mathematica Aeterna, 2(2)(2012), 155-161.

Senyurt S. and Sivas S., An application of Smarandache curve, Ordu Univ. J. Sci. Tech., 3(1)(2013), 46-60.

Tasképrii K. and Tosun M., Smarandache curves according to Sabban frame on S?, Boletim da Sociedade parananse de Mathemtica 3 srie., 32(1)(2014), 51-59, issn-0037-8712. Turgut M. and Yilmaz S., Smarandache curves in Minkowski space-time, International J.Math.Combin., 3(2008), 51-55.

Math. Combin. Book Ser. Vol.1(2016), 8-17

On Dual Curves of Constant Breadth

According to Dual Bishop Frame in Dual Lorentzian Space D}

Siiha Yilmaz', Yasin Unliitiirk? and Umit Ziya Savei? 1. Dokuz Eyliil University, Buca Educational Faculty, 35150, Buca-Izmir, Turkey

2. Kirklareli University, Department of Mathematics, 39100 Kirklareli, Turkey

3. Celal Bayar University, Department of Mathematics Education, 45900, Manisa-Turkey E-mail: suha.yilmaz@deu.edu.tr, yasinunluturk@klu.edu.tr, ziyasavci@hotmail.com

Abstract: In this work, dual curves of constant breadth according to Bishop frame are defined, and applications of their differential equations are solved for special cases in dual Lorentzian space D?. Some characterizations of closed dual curves of constant breadth ac- cording to Bishop frame are presented in dual Lorentzian space D? . These characterizations are made by obtaining special solutions of differential equations which characterize closed

dual curves of constant breadth according to Bishop frame in dual Lorentzian space D}.

Key Words: Dual Lorentzian space, dual curve, dual curves of constant breadth, Bishop

frame, differential equations.

AMS(2010): 53A35, 53A40, 53B25.

§1. Introduction

Bishop frame is used in engineering. This special frame has been particulary used in the study of DNA, and tubular surfaces and made in robot. Most of the literature on canal surfaces within the CAGD context has been motivated by the observation that canal surfaces with the rational spine curve and rational radius function are rational, and it is therefore natural to ask for methods which allow one to construct a rational parameterization of canal surface from its spin curve and radius function [8]. The construction of the Bishop frame is due to L. R. Bishop in [2]. That is why he defined this frame that curvature may vanish at some points on the curve. That is, second derivative of the curve may be zero. In this situation, an alternative frame is needed for non continously differentiable curves on which Bishop (parallel transport frame) frame is well defined and constructed in Euclidean and its ambient spaces [4, 18]. Curves of constant breadth have been studied in pure mathematics, optimization, mechan- ical engineering, physics and related directions. Basic properties of curves of constant breadth can be explained to someone without having any mathematical background knowledge. The existence of non-circular curves of constant breadth in the standard Euclidean plane has been

known since the time of Euler; e.g., the Reuleaux triangle was presented by Reuleaux to horn-

lReceived May 22, 2015, Accepted February 4, 2016.

On Dual Curves of Constant Breadth According to Dual Bishop Frame in Dual Lorentzian Space D} 9

blower, the founder of the compound steam-engine. In recent years, mathematical properties of the Reuleaux triangle have led to some very important applications. Since a curve of con- stant breadth can be freely rotated in a square always maintaining contact to all four sides of the square, a Reuleaux triangle can be used for drilling holes of maximum area into squares. Another application is given by the basic single-rotor Wankel engine. Its oval-shaped housing surrounds a three-sided rotor similar to a Reuleaux triangle. As the rotor rotates and orbitally revolves, each side of the rotor gets closer and farther from the wall of the housing, as also described above, in view of drilling holes into squares. A Reuleaux triangle is also used in the gear for driving a movie film [12].

In the classical theory of curves in differential geometry, curves of constant breadth have a long history as a research matter [8, 5, 9]. First it was introduced by Euler in [5]. Then Fujivara obtained a problem to determine whether there exist space curves of constant breadth or not, and he defined the concept ”breadth” for space curves on a surface of constant breadth [6]. Furthermore, Blaschke defined the curve of constant breadth on the sphere [3]. Reuleaux gave a method to obtain these kinds of curves and applied the results he had by using his method, in kinematics and engineering [14]. Some geometric properties of plane curves of constant breadth were given by Kose in [11]. And, in another work of Kése [10], these properties were studied in

the Euclidean 3-space E®. In Minkowski 3-space as an ambient space, some characterizations

of timelike curves of constant breadth were given by Yilmaz and Turgut in [17]. Also, Yilmaz dealt with dual timelike curves of constant breadth in dual Lorentzian space in [16].

Dual numbers were introduced by W. K. Clifford as a tool for his geometrical investigations. Then dual numbers and vectors were used on line geometry and kinematics by Eduard Study. He devoted a special attention to the representation of oriented lines by dual unit vectors and defined the famous mapping: The set of oriented lines in a three-dimensional Euclidean space “3 is one to one correspondence with the points of a dual space of triples of dual numbers 7].

In this paper, we study dual curves of constant breadth according to Bishop frame in

dual Lorentzian space D?. We give some characterizations of dual curves of constant breadth

according to Bishop frame in D?. Then we characterize these kinds of curves by obtaining

special solutions of their differential equations in D}.

§2. Preliminaries

Let E} be the three-dimensional Minkowski space, that is, the three dimensional real vector

space with the metric

(dx,dx) = —dx? + dx3 + da2,

where (21,22, 273) denotes the canonical coordinates in E?. An arbitrary vector x of E? is said

to be spacelike if (z,x) > 0 or x = 0, timelike if (x,x) < 0 and lightlike or null if (v7, 2) = 0

and x #0. A timelike or light-like vector in E? is said to be causal. For x E? the norm is defined by ||z|| = ./|(z,x)|, then the vector x is called a spacelike unit vector if (x,2) = 1 and

a timelike unit vector if (z,2) = —1. Similarly, a regular curve in E? can locally be spacelike,

10 Siiha Yilmaz, Yasin Unliitiirk and Umit Ziya Save

timelike or null (lightlike), if all of its velocity vectors are spacelike, timelike or null (lightlike), respectively [13].

Dual numbers are given with the set

V={F=ax+4+ Ea*;x,2* CE},

the symbol designates the dual unit with the property ¢? = 0 for 4 0. Dual angle is defined as 0 =0 +€6*, where @ is the projected angle between two spears and 6* is the shortest distance

between them. The set D of dual numbers is commutative ring the the operations + and -. The set

3=DxDxD={G=+fy';,¢* E*}

is a module over the ring D [15].

For any @ =a+€a*, b=b+ €b* D®, if the Lorentzian inner product of @ and b is defined

by <G,b >=< a,b > +€(< a*,b>+<a,b* >),

then the dual space together with this Lorentzian inner product is called the dual Lorentzian

space and denoted by D? [1]. For @ 4 0, the norm ||@]| of @ is defined by

Ill =V<%e >.

A dual vector @ = w + €w* is called dual spacelike vector if (6,@) > 0 or @ = 0, dual timelike vector if (@,@) <0 and dual null (lightlike) vector if (@,@) = 0 for & # 0. Therefore, an arbitrary dual curve which is a differential mapping onto D?, can locally be dual spacelike,

dual timelike or dual null if its velocity vector is dual spacelike, dual timelike or dual null,

respectively. Also, for the dual vectors a,b D3, Lorentzian vector product of these dual

vectors is defined by Gx b=ax b+ E(a* x b+ax b*)

where a x 6 is the classical cross product according to the signature (+,+,—) [1].

The dual arc length of the curve ¢ from t; to t is defined as t t t s= JP Ollat=fle'Olldtt+e f (t,e)dt=st Es", ty ty ty

where ¢ is a unit tangent vector of y(t). From now on we will take the arc-length s of y(t) as the parameter instead of t [9].

Let @: I C E— D} be a dual spacelike curve with the arc-length parameter s. The Bishop

derivative formula of dual spacelike curve @ is expressed as

fs kiN RH Res Ni oa! -ekT, (1) ha bP

On Dual Curves of Constant Breadth According to Dual Bishop Frame in Dual Lorentzian Space D3 11

where (7, e) =1, (™,™) =e=H41, (No, No) = —e and ki, ko are Bishop curvatures. Here

n~

ky =R(s)cosh6(s), kh = &(s)sinh@(s)

Let @: I C E> D} be a dual timelike curve with the arc-length parameter s. The Bishop

derivative formula of dual spacelike curve @ is expressed as

T' = kN, + koNo,

Ni= kf, (2) NS= keT, where (P,P) a -1, (1, ™) 1, (No, No) = 1 and ki, ko are Bishop curvatures. Here do yo T= a and % = ig - ig}. Thus, Bishop curvatures are defined by ((1], [2])

n~ n~ n~

k= K(s) cosh@(s), ko = &(s) sinh 6(s)

§3. Main Results

In this section, we give some characterizations of dual spacelike (timelike) curves of constant

breadth according to Bishop frame in the dual Lorentzian space D?. First, we give the definition

of dual spacelike (timelike) curves of constant breadth in D}. Then we characterize these kinds

of curves by obtaining special solutions of their differential equations in D}.

Definition 3.1 Let (C) be a dual spacelike (timelike) curve with position vector @ = ((s) in

3. If (C) has parallel tangents in opposite directions at corresponding points G(s) and A(sq) and the distance between these points is always constant, then (C1) is called a dual spacelike (timelike) curve of constant breadth. Moreover, a pair of dual curves (C,) and (C2) for which the tangents at the corresponding points @(s) and G(sq), respectively, are parallel and in opposite directions, and the distance between these points is always constant are called a dual (timelike) curve pair of constant breadth.

3.1 Dual Spacelike Curves of Constant Breadth According to Dual Bishop Frame

Let @ = G(s) be a simple closed dual spacelike curve in D?. We consider a dual spacelike curve

in the class I’ as in [6] having parallel tangents fe y and T, in opposite directions at the opposite points ¢ and @ of the curve according to Bishop frame. A simple closed dual spacelike curve of constant breadth having parallel tangents in opposite directions at opposite points can be

12 Siiha Yilmaz, Yasin Unliitiirk and Umit Ziya Savci

represented with respect to dual Bishop frame by the equation @ = G+ FT +N, + AN2, (3)

where 7,6 and \ are arbitrary functions of s. Differentiating both sides of (4), we get

da ds Oy se «a 3 a ee A= te ——S = (dk 1- + (Fkit+—)M, + (-Fko + —)No. | dsq ds (7s ephatenkate LE SA ae ees ae = 2) Considering T a ai y by the definition 3.1, we have the following system of equations dy dsq —_—= k —1-— Ts cok t+ edko ae dd iz We YR1, (5) dX ~ —= Ykp. ds ae

If we call @ as the angle between the tangent of the curve C' at point @ with a given direction

do dO and taking fe = TF, a = T* into account, the equation (5) turns into Sa d ak, «k ol, 541 FO); do T ak = 7, (6) T do) 1 1 where f(0) = =+ Ges Tis) a ky is k Let K, = +, Ky = = and using the system of ordinary differential equations (6), we have ae

= the following dual third order differential equation with respect to ¥ as;

OF, ie KR, «dk OY eRe RDO 4 ae(R dk _p oy do? do do do

fet Se he At ace he P10) +e k,d0)—— -¢« Kodé —=_ (4 id0) We Cia: a de de

=0 We can give the following corollary.

Corollary 3.1.1 The dual differential equation of third order given in (7) is a characterization

of the simple closed dual spacelike curve @ according to Bishop frame in D3.

Since position vector of a simple closed dual spacelike curve can be determined by solution of the equation (7), let us investigate solution of the equation (7) in a special case. Let Ki, Ko

On Dual Curves of Constant Breadth According to Dual Bishop Frame in Dual Lorentzian Space D3 13

n~

and f(0) be constants. Then the equation (7) turns to the following form

BF * a dy +e¢(K2— k?)— =0. (8) aps (Ky 2)

Solution of equation (8) yields the components

7 = A+ Bcos(\/K?2 K26) + Csin(,/ K? 20) ar {A+ Boos /R? 86) + Gsin(,/R? yo) Rido @) n= if {A+ Beos(/R? R26) + Gsin(\/R? R30) Rodi.

Corollary 3.1.2 Position vector of a simple dual spacelike closed curve with constant dual curvature and constant dual torsion according to Bishop frame is obtained in terms of the

values of 4, 5 and X as in the equation (9).

If the distance between opposite points of @ and @ is constant, then we can write that

|| Gl] = -7? + 6? + 9? = constant. (10) Differentiating (10) with respect to 0 gives

dy ~dd ~dr See

= = 0. (11) dd dd do

By virtue of (6), the differential equation (11) yields —~6K1(1 +e) +\Ro(1 —€) + f(0) =0,7 =0. (12) There are two cases for the equation (12), we study these cases as follows:

Casel. If Kk 1 =O and Ks = 0 then we find that the components 5 and \ are constants and

f(6) =0. Hence, Dual spacelike curves of constant breadth according to Bishop frame can be written as @=G+hT+ LM +13No, (13)

n~ n~

where ¥ = i,6= le A= Is; Licbls are constants.

n~

Case 2. If f() =0, then we have a relation among radii of curvatures as

1 1 =-==0. (14) E T

14

Siiha Yilmaz, Yasin Unliitiirk and Umit Ziya Savci

For this case, the equation (7) turns into

BF a dy Oka ky. oT + (RK? - K2) SL SeCk ile een do? do do do (15) +e(f Kid0)F K.d0)y7—— =0

The equation (15) is a characterization for the components. However, its general solution of

has not been found. Due to this, we investigate its solutions in special cases.

Let us suppose that Kk, =k,= 0, then we rewrite the equation (15) as

3 - =r (16)

By this way, we have the components as follows:

FH=Q +0460, 6 = constant, (17)

x = constant.

3.2 Dual Timelike Curves of Constant Breadth According to Dual Bishop Frame

Let ¢ = G(s) be a simple closed dual timelike curve in D3. We consider a dual timelike curve

in the class I as in [6] having parallel tangents - y and T, in opposite directions at the opposite

points @ and @ of the curve according to Bishop frame. A simple closed dual timelike curve

of constant breadth having parallel tangents in opposite directions at opposite points can be

represented with respect to dual Bishop frame by the equation

@=G+9T +6N, + XMo, (18)

where ¥, 6 and X are arbitrary functions of s. Differentiating both sides of (18), we get

oe oa = (FF 4 Khe +1) + Fk14 Oy, t (Fhe Ea (19) Considering T,=—-T y by the Definition 3.1, we have the following system of equations Hi Be fe, 3h 1, 2 = -7h, (20) 2 = —~YFkp.

If we call @ as the angle between the tangent of the curve C at point @ with a given direction

On Dual Curves of Constant Breadth According to Dual Bishop Frame in Dual Lorentzian Space D3 15

I

n pe and taking & Ts = = T* into account, we have (20) as follow; 8 Sa

dy ahiy. 265 A

—é mS A= os 0 )

op = f()

ky

eee ae (21) do if

B_ _sh

do ca

n~

where f(0) =

a ype

a Let ky = ae es = and using the system of ordinary differential equations (21), we

have the following dual third order differential equation with respect to ¥ as;

Bae.) tee) ods RG. ie Ke i - o (22) a~ nak ~ n~ .d?Ko d? f (0) —(| K,d0é Kodé —— ——=0 (f #id0)7 i (f Kod0)¥ ae @

We can give the following corollary.

Corollary 3.2.1 The dual differential equation of third order given in (22) is a characterization

of the simple closed dual timelike curve @ according to Bishop frame in D3.

Since position vector of a simple closed dual timelike curve can be determined by solution of (22), let us investigate solution of the equation (22) in a special case. Let K,, Ky and £0) be constants. Then the equation (22) turns into the following form

ce anes a5, dy oo _ (Kh? 4 K2)2 =0. (23) do

Solution of equation (23) yields the components

F = A+ BelKi+k2)0 + Ce (Kit 526, f22f { Gi pee Ce (Kit Kayay Ridd, (24) Res { Rie Baer Ce Ki+ kD 0} Rd0

Corollary 3.2.3 Position vector of a simple dual timelike closed curve with constant dual curvature and constant dual torsion according to Bishop frame is obtained in terms of the values of 7, 6 and A in the equation (24).

16 Siiha Yilmaz, Yasin Unliitiirk and Umit Ziya Save

If the distance between opposite points of ¢ and @ is constant, then we can write that || Bl] = -7? + 6? + 9? = constant. (25)

Differentiating (25) with respect to 6 gives

adi 50 52 _ 9 (26) "(6 dg dQ

By virtue of (21), the differential equation (26) yields

7f(0) =0. (27) There are two cases for the equation (27), we study these cases as follows: Case 1. If 7 =0 then we find that the components 6 and X are constants.

Hence, Dual timelike curves of constant breadth according to Bishop frame can be written

as @=G4hT +N, +13No, (28) where ¥ = ,6= Ten = is; i ala are constants.

n~

Case 2. If f(0) =0, then we have a relation among radii of curvatures as

1 ae =O. (29) For this case, the equation (22) turns into BF aT + dk, = dk. Cet Ra a eae aay do (30) ee ~ ~ @K —(f Kidd} (f Kad) ==0

The equation (30) is a characterization for the components. However, its general solution has not been found. Due to this, we investigate its solutions in special cases.

Let us suppose that Kk, =k, = 0, then we rewrite the equation (30) as

BF if); (31) do? By this way, we have the components as follows: a 6 0 302, 6 = constant, (32)

r = constant.

On Dual Curves of Constant Breadth According to Dual Bishop Frame in Dual Lorentzian Space D3 17

References

1

oN D oO

10 11

12

13

14 15

16

17

18

N.Ayyildiz, A.C. Coken, and A.A. Yiicesan, A characterization of dual Lorentzian spherical curves in the dual Lorentzian space, Taiwanese J. Math. 11 (4) (2007), 999-1018. L.R.Bishop, There is More Than One Way to Frame a Curve, Amer. Math. Monthly 82 (1975), 246-251.

W.Blaschke, Konvexe Bereiche Gegebener Konstanter Breite und Kleinsten Inhalts. Math. Ann. 76 (1915), 504-513.

B.Bukctt and M.K.Karacan, The slant helices according to bishop frame of the spacelike curve in Lorentzian space, Jour. of Inter. Math. 12(5) (2009), 691-700.

L.Euler, De curvis trangularibus, Acta Acad Petropol, (1870), 1870.

M.Fujivara, On Space Curves of Constant Breadth, Tohoku Math. J. (5) (1963), 179-184. H.Guggenheimer, Differential Geometry, McGraw Hill, New York, 1963.

T.Korpinar, E.Turhan, On characterization of image-canal surfaces in terms of biharmonic image-slant helices according to Bishop frame in Heisenberg group Heis?, J. Math. Anal. Appl. 382 (1) (2011) 57-65.

O. Kose, 8. Nizamoglu, and M. Sezer, An explicit characterization of dual spherical curves, Doga Turk. J. Math., 12 (3) (1988), 105-113.

O. Kése, On Space Curves of Constant Breadth, Doga Turk. J. Math. (10) 1 (1986) 11-14. O. Kose, Some Properties of Ovals and Curves of Constant Width in a Plane, Doga Turk. J. Math. 8 (1984), 119-126.

H.Martini, Z.Mustafaev, A new construction of curves of constant width, Comp. Aid. Geom. Des. 25 (9) (2008) 751-755.

B.O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, 1983.

F.Reuleaux, The Kinematics of Machinery, Dover Publications, New York, 1963.

G.R. Veldkamp, On the use of dual numbers, vectors and matrices in instantaneous spatial kinematics, Mech. Math. Theory, 11 (1976), 141-156.

S.Yilmaz, Timelike dual curves of constant breadth in dual Lorentzian space, IBSU Sci. J. 2 (2008) 129- 136.

S.Yilmaz, M.Turgut, On the time-like curves of constant breadth in Minkowski 3-Space, Int. J. Math. Combin., 3 (2008) 34-39.

S.Yilmaz, Bishop spherical images of a spacelike curve in Minkowski 3-space, Int. Jour. of the Phys. Scien. 5(6) (2010) 898-905.

Math. Combin. Book Ser. Vol.1(2016), 18-26

On (r,m,k)-Regular Fuzzy Graphs

N.R.Santhimaheswari

Department of Mathematics

G.Venkataswamy Naidu College, Kovilpatti-628502, Tamil Nadu, India

C.Sekar Department of Mathematics

Aditanar College of Arts and Science, Tiruchendur,