Characterization of Curves Whose Tangents Intersect a Straight Line in Euclidean 3-Space

(1)

Çümen ve Tunçer / Uşak Üniversitesi Fen ve Doğa Bilimleri Dergisi 80-83 2019 (2)

*Corresponding author:

E-mail: haticecumen92@gmail.com

80 Uşak Üniversitesi Fen ve Doğa

Bilimleri Dergisi

Usak University Journal of Science and Natural Sciences

http://dergipark.gov.tr/usufedbid

Araştırma Makalesi / Research Article

Characterization of Curves Whose Tangents Intersect a Straight

Line in Euclidean 3-Space

Hatice ÇÜMEN1*_{, Yılmaz TUNÇER}2

1_{Uşak University, Science Instutite, Department of Mathematics, TURKEY} 2_{Uşak University, Science and Art Faculty, Department of Mathematics, TURKEY}

Geliş: 21 Kasım 2019 Kabul: 9 Aralık 2019 / Received: 21 Kasım 2019 Accepted: 9 Aralık 2019

Abstract

In this study, we investigated the space curves in Euclidean 3-space whose tangent lines at each point intersect a given straight line passing the origin and intersect a fixed point, and we gave some characterizations in these cases.

Keywords: Frenet frame, tanget vector, space curve.

1. Introduction

The space curves whose principal normals intersecting a given straight line were first investigated by G. Pirondini, and further considered by E. Cesaro [1]. The corresponding question in affine space had been introduced by B. Su in 1929, He classified the curves and gave some remarkable results in affine 3-space by using equi-affine frame [3].

Let __:___E3_{be unit speed curve and}



_T₍_s_),_N₍_s_),_B₍_s₎



_{is the Frenet frame of}__(s_)._T_(s₎_, )

(s

N and B(s) are called the unit tangent, principal normal and binormal vectors respectively. Frenet formulae are given by

(2)

81  

 

                                   s B s N s T s s s s s B s N s T 0 0 0 0 0 ) ( ) ( ) (     (1)

where 

 

s and 

 

s are called the curvature and the torsion of the curve (s). A space curve 

 

s is determined by its curvature 

 

s and its torsion 

 

s , uniqely [2, 4].

2. The Space Curves Whose Tangents Intersect a Fixed Line

Let __:___E3_{be a curve with arclength parameter and l be the line passing the origin.} We assume that the tangents lines intersect the fixed l directed constant and unit vector

uat each point of the curve ,then we can write the following relation

u s s T s s) ( ) ( ) ( ). (      (2)

where (s)(s).u ve (s) are the differentiable vector depending s so since (s) is a line then we quaranteed 0. By taking the first and the second derivatives of (2),

we get u s s N s s s T s)) ( ) ( ) ( ) ( ) ( ) ( 1 (     (3)











s s s



B s su s N s s s s s s T s s s ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 ) ( ) ( ) ( ) ( ) ( 2                                 (4)

by using (2) and (4). If the tangents of the curve 

 

s intersect a fixed point on l then,

0

 

 and also (s)0 and (s)sc . In this case,  is the involute of 

 

s . Conversely, 

 

s is involute of  , then 

 

s is a line intersecting a fixed point of fixed line

l , so following corollary is concerned.

Corallary 2.1: The tangents of the curve 

 

s intersect a fixed point if and only if  is the involute of  and 

 

s is a line.

If 0and 0 then from (4), we have

0 ) ( ) ( ) (  2  s  s s  (5) 0 ) ( ) ( ) ( ) ( 2 ) (s  s s  ss   (6) 0 ) ( ) ( ) (s s s   (7)

(3)

82

Thus, we can say that there is no solution in the case 0 for (s)0 by considering (6), so there is no curve whose tangent lines intersect a fixed line.

Let 0then from (3) and (4), we have

0 ) ( )) ( 1 ( ) ( )) ( ) ( ) ( (s  s2 s s  s s  (8) 0 ) ( ) ( ) ( ) ( )) ( ) ( ) ( ) ( 2 ) ( ( s  s s  ss s  s ss  (9) 0 ) ( ) ( ) (s s s   (10) It is clear from (10) that 

 

s has to be planar, from (8), we get the solution

        ds e c c s ds s ds s s s ) ( ) ( 2 ) ( ) ( 2 1 ) (      1 _{. (11)} Rewrite (11) in (9),









0 ) ( ) ( ) ( ) ( ) ( )) ( 1 ( ) ( ) ( ) ( ) ( 2 ) ( 2 2 _                  s s s s s s s s s s s c            (12) and the solution of (12) is,

             ds s i ds s i ds s i ds s i e e c c ds e ds e c s ) ( ) ( 2 1 ) ( ) ( 2 ) ( _ _    (13)

Here, (s) is the real solution iff c₂1,so the real solution of (12) is

) cos( 2 ) cos( 2 ) ( 1    s   dsc (14) and from (11), (s) is ds ds e c s    ( ) ₁ (15) where _(s)ds and





         ) ) cos( 2 )( cos( ) ( ) cos( ) ( ) ( cos ) ( 2 ) sin( ) ( 2 ) cos( ) cos( ) ( 2 ) sin( ) ( 4 1 2 1 2 2 1 2 c ds s s c s s c ds s s                (16) and c is an arbitrary constant. For any ₁ c and nonzero constant ₂ (s)in (13), (s) is

(4)

83

)) sin( ) cos( ( ) cos( ) sin( ) ( 2 1 2 s s c c s s c s            . (17)

Hence following corallary is concerned.

Teorem 2.1: Let 

 

s be a planar curve with non-constant curvature and the tangent lines at each point of 

 

s , intersect fixed line l then

) cos( 2 ) cos( 2 ) ( 1    s   dsc and ds e c s  _{ }ds ( ) ₁ where _(s)ds and





         ) ) cos( 2 )( cos( ) ( ) cos( ) ( ) ( cos ) ( 2 ) sin( ) ( 2 ) cos( ) cos( ) ( 2 ) sin( ) ( 4 1 2 1 2 2 1 2 c ds s s c s s c ds s s               

Corallary 2.2: If 

 

s is a planar curve with constant nonzero curveture and the tangent lines at each points of 

 

s intersect fixed line l ,then

)) sin( ) cos( ( ) cos( ) sin( ) ( 2 1 2 s s c c s s c s            and         ds e c c s ds s ds s s s ) ( ) ( 2 ) ( ) ( 2 1 ) (      1 _.

References

1. Cesàro E. Lezioni di geometria intrinseca. Presso l'Autore-Editore. Napoli, Italy, 1896. 2. Blaschke W. Differential geometrie II. Verlag von Julius springer. Berlin, 1923. 3. Su B. Classes of curves in the affine space. Tohoku Mathematical Journal, 1929; 31:

283-291.