334 Erciyes Üniversitesi Fen Bilimleri Enstitüsü Dergisi 24 (1-2) 334- 345 (2008) http://fbe.erciyes.edu.tr/ ISSN 1012-2354

PARALLEL (OFFSET) IN LORENTZIAN PLANE

Murat Kemal KARACAN a * , Bahaddin BÜKCÜ b

a Uşak Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, Uşak b GOP, Faculty of Sciences and Arts, Depratment of Mathematics, Tokat

ABSTRACT

In this study, we have defined parallel (offset) curves in the Lorentzian plane. Also, we have given the relations between of the of these curves.Then, we have defined and curves of a and we have given some theorems in lorentzian plane

Keywords: Plane curves; Involute and evolute curves; Parallel curves; Lorentzian plane.

LORENTZ DÜZLEMİNDE PARALEL DENGE EĞRİLERİ

ÖZET

Bu çalışmada, lorentz düzleminde paralel eğrileri tanımladık. Ayrıca bu eğrilerin eğrilikleri arasındaki ilişkileri verdik. Sonra, lorentz düzleminde bir eğrinin involute ve evolute eğrilerini tanımladık ve bazı teoremler verdik.

Anahtar Kelimeler: Düzlem eğrisi; İnvolute ve evolute eğriler; Paralel eğriler; Lorentz düzlemi.

*E-posta: [email protected]

335 M. K. Karacan, B. Bükcü / Erciyes Üniversitesi Fen Bilimleri Enstitüsü Dergisi 24 (1-2) 334- 345 (2008)

1.INTRODUCTION

2 Let = {(),, 2121 ∈ RxxxxR } be a 2-dimensional vector space, = ( , xxx 21 ) and = ( , xxx 21 ) be two vectors in R 2 .The Lorentzian scalar product of x and y is defined by

, −= yxyxyx . L 2211

= 22 ,, xxRE is called Lorentzian plane.The vector x in E 2 is called a spacelike 1 ( 21 L ) 1 vector, null vector or a timelike vector if yx > 0, , yx = 0, or yx < 0, L L L respectively. For ∈ Ex 2 , the norm of the vector x defined by = , xxx and x is 1 L L called a unit vector if x = 1.Denote by sNsT )(),( the moving Frenet frame along the L { } curve α s)( .Then T and N are the tangent and the principal normal vector of the curve

α s)( respectively. Let be T spacelike and N timelike.This set of orthogonal unit vectors, known as the Frenet-Serret frame, has the following properties

κ ',' == κTNNT (1.1)

where κ of the curve α s)( respectively. Let be T timelike and N spacelike. This set of orthogonal unit vectors, known as the Frenet-Serret frame, has the following properties [4].

κ ',' == κTNNT (1.2)

2. PARALLEL (OFFSET) CURVES

Definition 2.1. A planar parametric curve α t)( is given by α = ()α α 21 ttt )(),()( , ∈ It where α1 and α 2 are differentiable functions of a parameter t . 336 Parallel (Offset) Curves In Lorentzıan Plane

2 Definition 2.2. Let α :)( → EIt 1 be a plane curve. For a planar curve α t)( with the well defined normalized orthogonal vector tN )( . We define the offset curve at the distance d as

β α += α tdNtt )()()( (2.1)

' ' where α = ( 2 αα 1 tttN )(),()( ).It is a displacement of the original curve in the direction of the normal vector. The unit normal vector is

' ' ( 2 αα 1 tt )(),( ) α tN )( = . ' 2 ' 2 2 αα 1 tt )(),(

Lemma 2.1. Let α be spacelike curve.The curve α s)( is part of a circle(in the lorentzian mean) if and only if κ > 0 is constant and τ = 0 .

Proof. Suppose α is part of a circle. α is a plane curve, so τ = 0 .Also by definition, for all s , α )( =− rps .Squaring both sides gives

αα )(,)( =−− rpsps 2 . L

If we differentiate this expression, we get

α psT =− 0)(,2 or α psT =− 0)(, . L L

If we differentiate again, then we obtain

α TTpsT =+− ,0,)(,' L L ακ psN =+− ,01)(, (2.2) L ακ psN −=− .1)(, L 337 M. K. Karacan, B. Bükcü / Erciyes Üniversitesi Fen Bilimleri Enstitüsü Dergisi 24 (1-2) 334- 345 (2008)

This means, in particular, that κ > 0 and α psN ≠− 0)(, . Now differentiating (2.2) L produces

dκ α )(, +− ακ )(,' +− κ TNpsNpsN = 0, ds L L L dκ α )(, +− 2 ακ psTpsN =− .0)(, L ds L

Since α psT =− 0)(, by above , we have L

dκ α psN =− .0)(, ds L

dκ Also, α psN ≠− 0)(, by above, so .This means, of course, that κ > 0 is constant. L ds Suppose now that κ > 0 is constant. To show α s)( is part of a circle we must show that each α s)( is a fixed distance from a fixed point. For the standard circle, from any point on the circle to the center we proceed in the normal direction a distance equal to the 1 radius. That is, we go −= NrN .We do the same here. Let γ s)( denote the curve κ

1 αγ )()( −= sNss ).( κ α

Since we want γ to be a single point, the center of the desired circle, we must have γ s = 0)(' .Computing, we obtain 338 Parallel (Offset) Curves In Lorentzıan Plane

1 αγ )(')(' −= ' sNss )( κ α 1 −= κTT κ = 0

Hence γ s)( is a constant p . Then we have

11 α )( Nps =−=− , α κκ so p is the centre of a circle α s)( of radius .

Theorem 2.2. Let α t)( be regular plane curve and spacelike curve. Then the parallel curve

β t)( is a regular at those t for which + dκα ≠ 01 . Furthermore , its curvature is given by

κα κ β = . 1+ dκα

The unit tangent vector of the offset curve is

1+ dκα Tβ = Tα . 1+ dκα

The unit normal vector of the offset curve is

()1+− dκα N β = Nα . 1+ dκα

Proof. Since the curve is planar, τ = 0. We can assume that α has unit speed.Then

β α += α sdNss )()()( and using Eq (2.1 ),we have

αβ += ' sdNss )()(')(' α ()1+= κ Td αα and 339 M. K. Karacan, B. Bükcü / Erciyes Üniversitesi Fen Bilimleri Enstitüsü Dergisi 24 (1-2) 334- 345 (2008)

β )('' κ ' ()1++−= κκ NdTds αα ααα α ()1+= κκκ Nd ααα

since κα = tan tcons . Hence

()ββ '','det κα κ β = 3 = . β ' 1+ dκα L

Then the parallel curve β t)( is a regular at those t for which ( + dκα ) ≠ 01 .Furthermore , its curvature is given by

κα κ β = . 1+ dκα

The unit tangent vector of the offset curve is

β ' 1+ dκ T == α T . β β ' 1+ dκ α L α

The unit normal vector of the offset curve is

()1+− dκα ββ eTN zL =∧= Nα . 1+ dκα

where ez = ()1,0,0 .

Theorem 2.3. Let α t)( be regular plane curve and timelike curve. Then the parallel curve

β t)( is a regular at those t for which 1+ dκα ≠ 0 .Furthermore, its curvature is given by

− κα κ β = . 1+ dκα

The unit tangent vector of the offset curve is

1+ dκα Tβ = Tα . 1+ dκα

The unit normal vector of the offset curve is 340 Parallel (Offset) Curves In Lorentzıan Plane

()1+− dκα N β = Nα . 1+ dκα

3. AND OF THE PLANE CURVE

Definition 3.1. The center of the osculating circle is called the center of curvature, the radius of the osculating circle is called the radius of curvature of the given curve at the given point [1].

Definition 3.2. The locus of the centers of curvature of a curve is called the evolute of the

2 curve. The evolute is defined for arcs along which the curvature is not zero. If α : → EI 1 is a curve with nowhere zero curvature function κ and unit normal vector field Nα ,then

~ 2 the evolute can be parameterized by the mapping β : → EI 1 ,

~ 1 αβ )()( −= tNtt ).( κ α

Definition 3.3. Let α be a smooth parameterized curve, t a point of its domain. We say that α t)( is a singular point (or singular parameter) of the curve α if α t = 0)(' [1].

Theorem 3.1. Singular points of the parallel curves of a regular curve α sweep out the evolute of α .

Proof. Let α be spacelike curve.Since β = (1)(' + κ )Tds αα ,singular parameters are characterized by dκα =+ 01 .Then the corresponding singular points ~ 1 αβ )()( −= tNtt )( lie on the evolute of the curve.It is also easy to show that any κ α evolute point is a singular point of a suitable parallel curve. 341 M. K. Karacan, B. Bükcü / Erciyes Üniversitesi Fen Bilimleri Enstitüsü Dergisi 24 (1-2) 334- 345 (2008)

Theorem 3.2. The evolutes of a planar curve is helix.

Proof. Let the curve α s)( be spacelike and the evolute of the curve α s)( be planar curve.

Then = NT αβ . So,

α β += λ β sTss )()()(

β α −= λ α sNss )()()(

β β ()+== κ αα sTdsT )(1)(' .

1 1 Since ⊥ NT αβ , α sd ,0)(1 λκ ==− . Hence,we have αβ )()( −= Nss α .The planar κα s)( κ α evolute of the curve α s)( is the locus of centre of curvature.If the nonplanar evolute of the curveα s)( is β s)( , then we have

β α −= λ β sTss )()()(

αβ λκ +−= TNTT βββ

β )( λκ NsT ββ =− 0

α ±= NT β .

If we derivate the equation , ∧= NTTf ,we have αβ L α L

= κ ,' +∧ ,κ ∧+∧ (κ NTNNTNTNf ) αββ L α L ααβ L αα L αα L = κ , ∧ NTN αββ L α L

= κ β (),,det NTN ααβ

= κ β 0. = .0

So, we find that f is constant.Hence,we have , NTT =∧∠ tcons .tan αβ L α L

342 Parallel (Offset) Curves In Lorentzıan Plane

Since the angle between the speed vector of the curve β s)( and the normal of plane of the curve α s)( is constant, β s)( is a helix.

2 Definition 3.4. Let α : → EI 1 be a unit speed curve with unit tangent vector field T . An involute ˆ ˆ of the curve α is a curve β of the form αβ ( −−= ) α sTsls )()( ,where l is a given real number. A curve has many involutes corresponding to the different choices of the length l of the thread.

Theorem 3.3. Let α be a unit speed curve with κ > 0 ,

ˆ αβ ()−−= α sTsls )()( an involute of it such that l is greater than the length of α . Then the evolute of βˆ is α .

Proof. Let be T spacelike and N timelike.We have

ˆ ()−= κβ sNssl )()(' αα ˆ 2 β '' ()()()−= κ α α )(')( −+ κ αα sTsslsNssl ).()(

ˆ Computing the curvature κˆβ of β ,

det ˆ ,' ββ ˆ '' ()− 2 κ 3 ssl )( 1 κˆ = ( ) = α = . β 3 3 3 − sl βˆ' ()− κα ssl )( () L

Thus, the evolute of βˆ is

⎛ 1 ⎞ βˆ − ⎜ ⎟ α () () =−+−−= α ssTslsTslsT ).()()()( ⎜ ˆ ⎟ βˆ α α ⎝ κ β ⎠ 343 M. K. Karacan, B. Bükcü / Erciyes Üniversitesi Fen Bilimleri Enstitüsü Dergisi 24 (1-2) 334- 345 (2008)

Theorem 3.4. Suppose that the regular curves γ 1 and γ 2 have regular evolutes. Then, γ 1 and γ 2 are parallel if and only if their evolutes are the same.

Proof. Let γ 1 be spacelike curve and let γ 1 andγ 2 parallel curve.In this case

γ γ += dγ 112 .

If the evolute of γ 1 is β1 and the evolute γ 2 is β 2 ,then we can write that 1 γβ −= N (3.1) 11 κ γ1 γ1

1 γβ −= N . (3.2) 22 κ γ 2 γ 2

We will show that β = β 21 .From the equations (3.1) and (3.2) ,we have 11 ()γγββ −+−=− NN 1212 γ1 γ 2 γ κκ γ 1 2 11 dN −+= NN . γ1 γ1 κκ γ 2 γ1 γ 2

Since the curves γ 1 and γ 2 are parallel, we can take to equal the normals of these curves i.e = = NNN .Therefore, we have 1 γγγ 2

⎛ 11 ⎞ ββ ⎜d −+=− ⎟N . (3.3) 12 ⎜ κκ ⎟ γ ⎝ 1 γγ 2 ⎠

On the other hand,let the curvature functions of the curves γ and γ be κ and 1 2 γ1 κ , respectively.Since γ 2

κ γ κ = 1 . (3.4) γ 2 1+ κ γ1 344 Parallel (Offset) Curves In Lorentzıan Plane

If we substitute the equation (3.2) in the equation (3.1),we have

⎛ 11 ⎞ ββ ⎜d −+=− ⎟N 12 ⎜ κκ ⎟ γ ⎝ 1 γγ 2 ⎠ ⎛ 11 −− dκ ⎞ ⎜d += γ1 ⎟N = 0 ⎜ κ ⎟ γ ⎝ γ1 ⎠

= ββ 12 .

The contrary proof of this theorem is obvious.Because it is trivial.

Theorem 3.5. If two involutes of a regular curve are regular, then they are parallel.

Proof. Let α t)( be spacelike curve.Let the frenet vectors of the curve β be { , NT } and let 1 1 ββ 1 the frenet vectors of the curve β be { , NT } and let the frenet vectors of the curve 2 2 ββ 2

α t)( be {, NT αα }.Let two involutes of the curve α t)( be β1 and β 2 ,respectively.Then, we have

β α (),)()( ∈−−= RlTtltt 1 1 α 1 (3.5) 2 αβ )()( ()2 α , 2 ∈−−= RlTtltt

From the equation (3.5),we have

β β α )()()( ( ) −−+=− α )( − ( − )TtltTtlttt 2 1 2 α 1 α 2 ββ 1 )()( ()−=− 12 Tlltt α .

Since = NT and = TN , we have 1 αβ 1 αβ 345 M. K. Karacan, B. Bükcü / Erciyes Üniversitesi Fen Bilimleri Enstitüsü Dergisi 24 (1-2) 334- 345 (2008)

β 2 β1 )()( ()−+= 12 Nlltt β 1 ββ ∈+= RddNtt .,)()( 2 1 β1

Hence, the curves β1 and β 2 are parallel.

REFERENCES

1. B. Csikós, Differential Geometry, Lecture Notes, on the web. 2. J.Oprea, Differential Geometry and its Applications, Prentica Hall, Inc, (1997). 3. A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, CRC Press, (1998). 4. T. Ikawa, Balkan J. Geom. Appl. 8, 2 (2003) 31. 5. N. M. Patrikalakis, Computational Geometry, Lecture Notes, Massachusetts Institute of Technology, (2003). 6. E. Hartmann, Geometry and Algorithms for Computer Aided Design, Darmstadt University of Technology, (2003). 7. A. Sabuncuoglu, Diferensiyel Geometri, Nobel Yayin Dagitim, Ankara, (2004).