334 Erciyes Üniversitesi Fen Bilimleri Enstitüsü Dergisi 24 (1-2) 334- 345 (2008) http://fbe.erciyes.edu.tr/ ISSN 1012-2354 PARALLEL (OFFSET) CURVES 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 curvatures of these curves.Then, we have defined involute and evolute curves of a curve 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 R= {() x1,, x 2 x 1 x 2 ∈ R} be a 2-dimensional vector space, x= ( x1, x 2 ) and x= ( x1, x 2 ) be two vectors in R 2 .The Lorentzian scalar product of x and y is defined by x, y= x y − x y . L 1 1 2 2 E2= R 2 ,, x x is called Lorentzian plane.The vector x in E 2 is called a spacelike 1 ( 1 2 L ) 1 vector, null vector or a timelike vector if x, y > 0 , x, y = 0 or x, y < 0 L L L respectively. For x∈ E 2 , the norm of the vector x defined by x= x, x and x is 1 L L called a unit vector if x = 1.Denote by T( s ), N ( s ) 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 TNNT','=κ = κ (1.1) where κ curvature 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]. TNNT','=κ = κ (1.2) 2. PARALLEL (OFFSET) CURVES Definition 2.1. A planar parametric curve α()t is given by α(t )= ()α1 ( t ),α 2 ( t ) , t∈ I 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():t I→ E1 be a plane curve. For a planar curve α()t with the well defined normalized orthogonal vector N() t . We define the offset curve at the distance d as β ()()()t=α t + dNα t (2.1) ' ' where Nα ( t )= (α2 ( t ), α1 ( t )).It is a displacement of the original curve in the direction of the normal vector. The unit normal vector is ' ' (α2 (t ), α1 ( t )) Nα () t = . ' 2 ' 2 α2 (t ), α1 ( t ) 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()s− p = r .Squaring both sides gives α(),()s− p α s − p = r 2 . L If we differentiate this expression, we get 2T ,α ( s )− p = 0 or T,α ( s )− p = 0 . L L If we differentiate again, then we obtain T',α ( s )− p + T , T = 0, L L κN, α ( s )− p + 1 = 0, (2.2) L κN, α ( s )− p = − 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 N,α ( s )− p ≠ 0 . Now differentiating (2.2) L produces dκ N,()α s− p + κ N',() α s− p + κ N, T = 0 ds L L L dκ N,()α s− p + κ2 T, α ( s )− p = 0. L ds L Since T,α ( s )− p = 0 by above , we have L dκ N,α ( s )− p = 0. ds L dκ Also, N,α ( s )− p ≠ 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 rN= − N .We do the same here. Let γ ()s denote the curve κ 1 γ()()s= α s − N( s ). 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 γ'()'()s= α s − N' () s κ α 1 =TT − κ κ = 0 Hence γ ()s is a constant p . Then we have 1 1 α()s− p = − N = , κα κ 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 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 −()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 ()()()s=α s + dNα s and using Eq (2.1 ),we have ' β'()'()()s= α s + dNα s =()1 + dκα T α and 339 M. K. Karacan, B. Bükcü / Erciyes Üniversitesi Fen Bilimleri Enstitüsü Dergisi 24 (1-2) 334- 345 (2008) ' β ''()s= − dκα T α +()1 + dκα κ α N α =κα ()1 + d κα κ α N α since κα = constan t . Hence det()β ', β '' κα κ β = 3 = . β ' 1+ dκα L 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 β ' 1+ dκ T = = α T . β β ' 1+ dκ α L α The unit normal vector of the offset curve is −()1 + dκα Nβ= T β ∧L e z = Nα . 1+ dκα where ez = ()0,0,1 . 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. INVOLUTES AND EVOLUTES 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 : IE→ 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 :IE→ 1 , ~ 1 β()()t= α t − N( t ). 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 '(s )= ( 1+ dκα) T α ,singular parameters are characterized by 1+dκα = 0 .Then the corresponding singular points ~ 1 β()()t= α t − N() t 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 TNβ= α . So, α ()()()s=β s + λ Tβ s β ()()()s=α s − λ Nα s Tβ =β '( s ) =() 1 + dκ α T α ( s ) . 1 1 Since TNβ⊥ α , 1−dκα ( s ) = 0, λ = . Hence,we have β()()s= α s − Nα .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 ()()()s=α s − λ Tβ s TTNTβ= α −λκ β β + β Tβ () s−λκ β N β = 0 TNα = ± β .