ON THE DETERMINATION OF A DEVELOPABLE TIMELIKE RULED SURFACE. Mustafa KAZAZ, Ali ÖZDEMİR, Tuba GÜROĞLU

Transcription

1 SDÜ FEN EDEBİYAT FAKÜLTESİ FEN DERGİSİ (E-DERGİ). 008, (), 7-79 ON THE DETERMINATION OF A DEVELOPABLE TIMELIKE RULED SURFACE Mustafa KAZAZ, Ali ÖZDEMİR, Tuba GÜROĞLU Department of Mathematics, Faculty of Science, University of Celal Bayar Muradiye Campus, 45047, Manisa, TURKEY, m_kazaz@hotmail.com. Received: 04 June 007, Accepted: January 008 Abstract: This paper gives a method for determining a developable timelike ruled surface by using dual vector calculus. A developable timelike ruled surface can be parameterized in the form mt (, u) = pt () + uxt () ( pt () is called the base curve of mt (, u )). The dual vectorial expression of a developable timelike ruled surface is obtained from the coordinates and the first derivatives of the base curve. Key words: Minkowski space, Developable surface, Timelike ruled surface. Mathematics Subject Classifications (000): 5C50, 5A04, 5A05. BİR AÇILABİLİR TİMELİKE REGLE YÜZEYİN BELİRLENMESİ ÜZERİNE Özet: Bu makalede dual vektör analizi kullanılarak bir açılabilir timelike regle yüzeyin belirlenmesi için bir metot verilmiştir. Bir açılabilir timelike regle yüzey mt (, u) = pt () + uxt () formunda parametreleştirilebilir ( pt () ye mt (, u ) nın dayanak eğrisi denir). Bir açılabilir timelike regle yüzeyin dual vektörel ifadesi koordinatlar ve dayanak eğrisinin ilk türevlerinden elde edildi. Anahtar kelimeler: Minkowski Uzayı, Açılabilir Yüzey, Timelike Regle Yüzey.. INTRODUCTION Ruled surfaces, particularly developable surfaces, have been widely studied and applied in mathematics and engineering. The generation and machining of ruled surfaces play an important role in design and manufacturing of products and many other areas (RAVANI & KU 99, ZHA 997). Dual numbers, dual vectors, dual inner products, dual cross product, dual angle, E. Study Mapping, etc. are the most important notions on the geometry of lines and kinematics. E. Study mapping plays a fundamental role between the real and dual Lorentzian spaces. By this mapping, there exist one-to-one correspondence between the vectors of dual unit sphere S and the directed lines of the space of lines E (STUDY 90). Therefore, the motion locus of a straight line in E can be described by that of a point on the surface of dual unit sphere S in dual space D. Then a ruled surface in E

2 M. KAZAZ, A. ÖZDEMİR, T. GÜROĞLU 7 corresponds to a unique dual curve on the surface of S (GUGENHEIMER 977). Therefore, the generation of a ruled surface can be converted into the determination of a unique corresponding spherical curve. The correspondence of E. Study s mapping states that there is a one-to-one correspondence between an oriented straight timelike line in -dimensional Minkowski space IR and a dual point on the surface of a dual hyperbolic unit sphere H % in - dimentional dual Lorentzian space D (UĞURLU & ÇALIŞKAN 996). Thus, a timelike ruled surface in IR corresponds to a unique dual curve on the surface of 0 H % 0. In this study, using methods given in (KÖSE 999), we determine a method of determination of a developable timelike ruled surface and obtain a linear differential equation of first order.. DUAL NUMBERS AND DUAL LORENTZIAN VECTORS In this section we give a brief summary of the theory of dual numbers and dual Lorentzian vectors. Let IR be a -dimensional Minkowski space over the field of real numbers IR with the Lorentzian inner product <, > given by < a, b > = ab + ab ab, where a = ( a, a, a ) and b = ( b, b, b) IR. A vector a = ( a, a, a) of IR is said to be timelike if < a, a > < 0, spacelike if < a, a > >0 or a = 0, and lightlike (or null ) if < a, a > = 0 and a 0 (O NEILL 98). The norm of a vector a is defined by a = < a, a >. Now let a = a, a, ) and ( a b = ( b, b, b ) be two vectors in IR. Then the Lorentzian cross product is given by a b = ( ab ab, ab a b, ab ab ) (AKUTAGAWA & NISHIKAWA 990). A dual number has the form ˆ λ : = λ + ελ, where λ and λ are real numbers, and ε stands for the dual unit which is subject to the rules: ε 0, ε = 0, 0ε = ε0 = 0, ε = ε = ε. We denote the set of dual numbers by D : D= { ˆ λ = λ+ ελ } λ, λ IR, ε = 0. Addition and multiplication are defined in D by ( λ + ε λ ) + ( β + εβ ) = ( λ + β ) + ε ( λ + β ), and ( λ + ελ ) ( β + ε β ) = λβ + ε ( λ β + λ β ), respectively. Then it is easy to show that ( D, +,.) is a commutative ring with unity. Now let f be a differentiable function. Then the Maclaurin series generated by f is f ( xˆ) = f ( x + ε x ) = f ( x ) + ε x f ( x ), where f ( x ) is the derivative of f.

3 SDÜ FEN EDEBİYAT FAKÜLTESİ FEN DERGİSİ (E-DERGİ). 008, (), Let D be the set of all triples of dual numbers, i.e. % i { } D = a= ( A, A, A ) A D, i =,,. The elements of D are called as dual vectors. A dual vector a% may be expressed in the form a% = a+ε a, where a and a are the vectors of IR. Now let a% = a+ε a, b % = b+ εb D and ˆ λ = λ+ ελ D. Let us define a% + b% = a+ b+ ε ( a + b ), ˆ λa% = λa+ ε( λa + λa). Then D is a module together with these operations. It is called as D -module or dual space. The Lorentzian inner product of dual vectors a% = a+ε a and b % = b + ε b is defined by < ab %, % >=< ab, >+ ε ( < ab, >+< a, b> ), where < a, b > is the Lorentzian inner product of the vectors a and b in the Minkowski -space IR (UĞURLU&ÇALIŞKAN 996). A dual vector a% = a+ε a is said to be timelike if < aa, >< 0, spacelike if < aa, > > 0 or a = 0 and lightlike (or null) if < aa, >= 0 and a 0. The set of dual timelike, spacelike and lightlike vectors is called dual Lorentzian space and it is denoted by D, i.e. { ε, } D = a% = a+ a a a IR. The Lorentzian cross product of dual vectors a% and b% D is defined by a% b% = a b+ ε ( a b+ a b ), where a b is the Lorentzian cross product in IR (UĞURLU & ÇALIŞKAN 996). Lemma.. Let a%, b%, c%, d% D. Then we have < a% b%, c% >= det( a%, b%, c% ) a% b% = b% a%, ( a% b% ) c% = < a%, c% > b% +< b%, c% > a%, < a% b%, c% d% >= < a%, c% >< b%, d% >+< a%, d% > < b%, c% >, < a% b%, a% >= 0 ; and < a% b%, b% >= 0 (UĞURLU & ÇALIŞKAN 996). Let a% = a+ε a D. Then a% is said to be dual timelike (resp. spacelike) unit vector if the vectors a and a satisfy the following equations < a, a> =, < a, a > = 0 (resp. < a, a> =, < a, a >= 0). The set of all dual timelike unit vectors is called the dual hyperbolic unit sphere and is denoted by H % 0 : H% 0 = a% = a+ εa IR < a, a>=. { }

4 M. KAZAZ, A. ÖZDEMİR, T. GÜROĞLU 75 Theorem. (E. Study s Mapping) The dual timelike unit vectors of the dual hyperbolic unit sphere H % 0 are in one to one correspondence with the directed timelike lines of the Minkowski -space IR (UĞURLU & ÇALIŞKAN 996).. THE DUAL VECTOR FORMULATION By using the dual vector representation, the Plücker vectors x and p x of a timelike line L can be collected into a single dual timelike vector %x = x+ ε p x= x+ ε x, where x is the direction vector of L and p is the position vector of any point on L. A timelike ruled surface mt (, u) = pt () + uxt () is written as the dual vector function %x () t given by () () ε ( ) t = x t + p() t x t = x() t + εx () t. () Since the spherical image of x( t ) is a timelike unit vector, the dual vector %x () t also has unit magnitude: <, >=< x+ ε p x, x+ ε p x> =< x, x>+< εx, p x>+ ε < p x, p x> =< x, x>=. Thus, a timelike ruled surface can be represented by a dual curve on the surface of a dual hyperbolic unit sphere. The dual arc-length of a timelike ruled surface %x () t is defined by t d st %() =. () 0 The integrant of Equation () is the dual speed, % δ, of %x ( t ) and is, dp < x > % d δ = = + ε, or <, > % δ = + ( ). ε = δ + ε () The curvature function dp <, x > <, > = = (4) is the well-known distribution parameter of the timelike ruled surface.

5 SDÜ FEN EDEBİYAT FAKÜLTESİ FEN DERGİSİ (E-DERGİ). 008, (), eg f The Gaussian curvature of a surface is given by K =, where E =< m,, EG F t mt > F =< mt, mu >, G =< mu, mu > are the first fundamental coefficients and mt mu mt mu mt mu e=< mtt, >, f =< mtu, >, g =< muu, > are the second mt mu mt mu mt mu fundamental coefficients. Then the relation between the Gaussian curvature K and the distribution parameter of a timelike ruled surface mt (, u ) is given by (KASAP et al. 005) K =. (5) ( + u ) If K is zero everywhere, that is, is zero everywhere then the timelike ruled surface is said to be developable. 4. THE DETERMINATION OF A DEVELOPABLE TIMELIKE RULED SURFACE % i i i The dual coordinates x = x + ε x ( i =,, ) of an arbitrary point %x of the dual hyperbolic unit sphere H % 0, centered at the origin, may be expressed as = x+ ε x = sinh % θ cos % ϕ, = x + ε x = sinh % θ sin % ϕ, (6) = x+ εx = cosh % θ where % θ = θ + εθ and % ϕ = ϕ+ εϕ are dual hyperbolic and spacelike angles with θ IR and 0 ϕ π, respectively. Since ε = ε = =0, according to the Taylor series expansion from Equation (6), we obtain = x+ εx = sinh( θ + εθ )cos( ϕ+ εϕ ) = (sinhθ + εθ cosh θ)(cosϕ εϕ sin ϕ) = sinhθ cos ϕ+ ε( θ coshθcosϕ ϕ sinhθsin ϕ), = x + εx = sinh( θ + εθ )sin( ϕ+ εϕ ) = (sinhθ + εθ cosh θ)(sinϕ+ εϕ cos ϕ) = sinhθ sin ϕ+ ε( θ coshθsinϕ+ ϕ sinhθcos ϕ), and = x+ ε x = cosh( θ + εθ ) = coshθ + εθ sinh θ. Then we obtain the real parts as x = sinhθ cos ϕ, x = sinhθ sin ϕ, (7) x = cosh θ, and the dual parts as

6 M. KAZAZ, A. ÖZDEMİR, T. GÜROĞLU 77 x x x = θ coshθ cosϕ ϕ sinhθsin ϕ, θ coshθsinϕ ϕ sinhθcos ϕ, = + = θ sinh θ. Now let us consider a dual curve ( t) = x( t) +ε x (t) on the dual hyperbolic unit sphere corresponding to a timelike ruled surface mt (, u) = pt () + uxt () in ( ( ) xt %() = xt () + εx(t) = sinh θ()cos t ϕ() t, sinh θ()sin t ϕ() t, cosh θ() t ) IR. Then we write + ε θ ( t) cosh θ( t)cos ϕ( t) ϕ ( t) sinh θ( t) sin ϕ( t), θ ( t) cosh θ( t) sin ϕ( t) + ϕ ()sinh t θ() t cos ϕ(), t θ () t sinh θ(). t Since x () t = p() t x() t, we have the following system of linear equations in p, p and p (where p, p and p are the coordinates of p( t ) ): pcoshθ + psinh θ sin ϕ = θ cosh θ cosϕ ϕ sinh θ sinϕ pcosh θ p sinh θ cos ϕ = θ cosh θ sinϕ+ ϕ sinh θ cosϕ psinhθsinϕ psinh θ cosϕ= θ sinhθ The matrix of coefficients of unknowns p, p and p is the Lorentzian skewsymmetric matrix 0 cosh θ sinhθsinϕ coshθ 0 sinhθcosϕ sinhθsinϕ sinhθcosϕ 0 and therefore its rank is with θ () t IR. Also the rank of the augmented matrix 0 coshθ sinhθcosϕ θ coshθcosϕ ϕ sinhθsinϕ coshθ 0 sinhθcosϕ θ coshθsinϕ+ ϕ sinhθcosϕ sinhθsinϕ sinhθcosϕ 0 θ sinhθ is. Hence this system has infinite solutions given by p = ( p+ ϕ ) tanh θ cosϕ+ θ sin ϕ, p = ( p+ ϕ ) sinh θ sinϕ θ cos ϕ, (0) p = p. Since p () t can be chosen arbitrarily, then we may take p () t = ϕ () t. In this case, Equation (0) reduces to p = θ sin ϕ, p = θ cos ϕ, () p = ϕ. The distribution parameter of the timelike ruled surface given by Equation (9) is dθ dθ dϕ dϕ <, > dϕ + sinh θ + θ sinh cosh θ θ = = () dθ dϕ + sinh θ (8) (9)

7 SDÜ FEN EDEBİYAT FAKÜLTESİ FEN DERGİSİ (E-DERGİ). 008, (), If this ruled surface is a developable one, then = 0 and then Equation () becomes dθ dθ dϕ dϕ dϕ + sinh θ + θ sinhθcoshθ = 0. () Dividing the Equation () by sinh θ gives dθ d dϕ dϕ dϕ (coth θ) θ cothθ = 0. Setting dϕ θ dϕ dϕ yt () = coth θ, At () =, and Bt () =, dθ dθ we are lead to a linear differential equation of first order dy At () y Bt () 0. = (4) In the case that the hyperbolic angle θ (t) and spacelike angle ϕ(t) are both constant, this equation is identically zero, that is, the ruled surface ( t) is a timelike cylinder. Let p( t ) be a curve. Then we can find a developable timelike ruled surface such that its base curve is the curve p( t ). In fact from Equation (), we have p tan ϕ =, θ =± p + p, and ϕ = p p Now only ϕ (t) remains to be determined. The solution of the linear differential Equation (4) gives cothθ. This solution includes an integral constant. Therefore we have infinitely many developable timelike ruled surfaces such that its base curve is p() t. It is to be noted that θ ( t) has two values; when we use the minus sign we obtain the reciprocal of the timelike ruled surface ( t) obtained by using the plus sign for a given integral constant. REFERENCES AKUTAGAWA K, NISHIKAWA S, 990. The Gauss map and space-like surfaces with prescribed mean curvature in Minkowski -space, Tohoku Mathematical Journal, 4, GUEGENHEIMER H, 977. Differential Geometry, Dover, New York, pp.78. KASAP E, AYDEMİR İ, KURUOĞLU N, 005. Erratum to: Ruled surfaces with timelike rulings [Appl. Math. Comput. 47(004) 4 48], Applied Mathematics and Computation. 68, KÖSE Ö, 999. A Method of the Determination of a Developable Ruled Surface, Mechanism and Machine Theory, 4, O NEILL B, 98. Semi-Riemannian Geometry, With Applications to Relativity, Academic Press Inc., London, pp.468. RAVANI B, KU TS, 99. Bertrand offsets of ruled and developable surfaces, Computer Aided Geometric Design. Elsevier, (), STUDY E, 90. Geometrie der Dynamen, Leipzig.

8 M. KAZAZ, A. ÖZDEMİR, T. GÜROĞLU 79 UĞURLU HH, ÇALIŞKAN A, 996. The Study Mapping for Directed Spacelike and Timelike Lines in Minkowski -Space IR,Mathematical and Computational Applications, (), ZHA XF, 977. A New Approach to Generation of Ruled Surface and its Applications in Engineering, International Journal of Advanced Manufacturing Technology,, 55-6.