This paper presents the curvature theory of non-null cylindrical surfaces in Minkowski 3-space. The definition of the line of striction and generator trihedron for cylindrical surfaces in Minkowski 3-space are given. The derivation formulae and Darboux instantaneous rotation vectors of generator trihedrons which play important role in robot kinematics are found. Moreover, curvature theory of a Lorentzian circular cylinder is given as an example.Publisher's Versio

Kazaz, Mustafa

Uğurlu, Hasan Hüseyin

Şahiner, Burak

TWMS Journal of Applied and Engineering Mathematics

English

Isik University Academic Open Access

TWMS J. App. Eng. Math. V.6, N.1, 2016, pp. 22-29.ON THE CURVATURE THEORY OF NON-NULL CYLINDRICALSURFACES IN MINKOWSKI 3-SPACEBURAK SAHINER1, MUSTAFA KAZAZ1, HASAN HUSEYIN UGURLU3, §Abstract. This paper presents the curvature theory of non-null cylindrical surfacesin Minkowski 3-space. The definition of the line of striction and generator trihedronfor cylindrical surfaces in Minkowski 3-space are given. The derivation formulae andDarboux instantaneous rotation vectors of generator trihedrons which play importantrole in robot kinematics are found. Moreover, curvature theory of a Lorentzian circularcylinder is given as an example.Keywords: Curvature theory, Cylindrical surface, Darboux instantaneous rotation vector,Generator trihedron, Minkowski space.AMS Subject Classification: 53A351. IntroductionRuled surfaces which are generated by the motion of straight lines are widely used incomputer aided geometric design (CAGD), industrial areas, spatial mechanisms, physics,kinematics and many other areas [4, 5, 9, 10]. Thus, ruled surfaces become an importantissue in differential geometry and engineering mathematics.Curvature theory which is one of the most popular research areas of ruled surfaces in-vestigates the intrinsic geometric properties of the trajectory of points, lines, and planesembedded in a moving rigid body [9]. The results obtained from curvature theory can beapplied to the analysis and synthesis of planar, spherical and spatial mechanisms. Curva-ture theory of a ruled surface has been studied by several researchers. Freudenstein [3],studied on analysis of higher order path curvature in plane kinematics and used charac-teristic numbers to characterize plane curves at a point. Kirson [4] examined higher-ordercurvature theory of a ruled surface in his thesis. Schaff [10] applied the curvature theory ofa ruled surface to solve the kinematic problems associated with spatial closed-loop mech-anisms. McCarthy [5] derived both scalar and dual formulations to study the curvaturetheory of a ruled surface and presented relations between two formulations. Ryuh [9] pre-sented the curvature theory of a general ruled surface and special types of ruled surfacesreferred to as developables. He gave separate formulations for a cylindrical surface, a coneand a tangent surface.1 Department of Mathematics, Celal Bayar University, P.O.Box 45140, Manisa, Turkey.e-mail: burak.sahiner@cbu.edu.tr, mustafa.kazaz@cbu.edu.tr;3 Department of Secondary Education Science and Mathematics Teaching, Gazi University, P.O.Box06560, Ankara, Turkey.e-mail: hugurlu@gazi.edu.tr;§ Manuscript received: October 21, 2015; Accepted: December 15, 2015.TWMS Journal of Applied and Engineering Mathematics, Vol.6, No.1; c© Işık University, Departmentof Mathematics, 2016; all rights reserved.22B.SAHINER, M.KAZAZ, H.UGURLU: ON THE CURVATURE THEORY OF NON-NULL ... 23The research area of the curvature theory of a ruled surface has also been studied inMinkowski space. Ayyıldız and Yücesan [1] derived the scalar and dual Lorentzian for-mulations of the curvature theory of line trajectories for non-null curves in the Lorentzianspace. Ersoy and Tosun [2] gave the trajectory null scroll in 3-Minkowski space. Önderand Uğurlu [7] gave the Frenet frames and invariants of timelike ruled surfaces and of itsdirecting cone.Cylindrical surfaces which are developable surfaces are used in many applications inthe research areas of robotics and engineering. A robot end-effector, for example, canmove on a cylindrical surface. In order to determine the differential properties of themotion, curvature theory of cylindrical surface can be used. Since a cylindrical surfaceis an exception to a general ruled surface, the curvature theory of a cylindrical surfacemust be developed separately. In this study, we give the curvature theory of a non-nullcylindrical surface in Minkowski 3-space using the technique in [9]. The line of strictionare defined, and generator trihedrons, derivation formulae and Darboux instantaneousrotation vectors of the trihedrons are given in detail.2. PreliminariesIn this section we give a brief summary of basic concepts in Minkowski 3-space.Minkowski 3-space denoted by IR31 is vector space with the Lorentzian inner product〈x , y〉 = x1y1 + x2y2 − x3y3where x , y ∈ IR3 [6].Let x = (x1, x2, x3) be an arbitrary vector in IR31. If 〈x , x 〉 > 0 or x = 0, then x iscalled spacelike, if 〈x , x 〉 < 0, then x is called timelike, if 〈x , x 〉 = 0 and x 6= 0, then x iscalled null (lightlike) vector [6].Let x = (x1, x2, x3) and y = (y1, y2, y3) be two vectors in IR31. The Lorentzian vectorproduct of x and y can be defined by [12]x × y = (−x2y3 + x3y2, x1y3 − x3y1, x1y2 − x2y1).Let x and y be timelike vectors in IR31, then there is a real number θ ≥ 0 such that〈x , y〉 = −‖x‖ ‖y‖ cosh θ and this number is called the hyperbolic angle between thevectors x and y [8].Let x and y be spacelike vectors in IR31 and they span a timelike vector subspace, thenthere is a real number θ ≥ 0 such that |〈x , y〉| = ‖x‖ ‖y‖ cosh θ and this number is calledthe central angle between the vectors x and y [8].Let x and y be spacelike vectors in IR31 and they span a spacelike vector subspace, thenthere is a real number θ ≥ 0 such that 〈x , y〉 = ‖x‖ ‖y‖ cos θ and this number is called thespacelike angle between the vectors x and y [8].Let x be a spacelike vector and y be a timelike vector in IR31, then there is a real numberθ ≥ 0 such that |〈x , y〉| = ‖x‖ ‖y‖ sinh θ and this number is called the Lorentzian timelikeangle between the vectors x and y [8].A curve α = α(s) in Minkowski 3-space can be spacelike, timelike or null (lightlike) ifits velocity vector α′(s) is spacelike, timelike or null (lightlike), respectively [6].A surface in Lorentzian space IR31 is called a timelike surface if the induced metric onthe surface is a Lorentzian metric, i.e., normal vector on the surface is a spacelike vector[13].24 TWMS J. APP. ENG. MATH. V.6, N.1, 20163. Curvature theory of a cylindrical surface in Minkowski 3-spaceIn this section, we give the curvature theory of a cylindrical surface which includes thedefinitions of line of striction, generator trihedrons and their derivative formulae, theDarboux instantaneous rotation vectors.A cylindrical surface is a special ruled surface and can be represented byX (t, v) = α(t) + v Rwhere α is the directrix, R is a constant vector called ruling, t and v are real-valued pa-rameters [11]. Since the ruling is a constant vector, normalization based on differentiationof a ruling can not be used for a cylindrical surface. Therefore, normalization based ondifferentiation of the directrix will be used [9]. This normalization can be achieved byusing the equations(t) =∫ t0∥∥∥∥dα(t)dt∥∥∥∥ dt (1)where s is the arc-length parameter of the directrix.The definition of line of striction used for a general ruled surface is not valid for acylindrical surface. Thus, line of striction of a cylindrical surface can be defined as thelocus of points on the surface which are at the shortest distance from the origin of theCartesian reference frame along rulings [9]. From this definition, line of striction of acylindrical surface is unique and it is perpendicular to the ruling, i.e.,〈c(s),R〉 = 0 (2)where c(s) represents line of striction of the cylindrical surface. By differentiating equation(2), we have 〈c′(s),R〉= 0 (3)Equation (3) means that tangent vector of line of striction is perpendicular to the ruling.The line of striction relative to the directrix can be given asc(s) = α(s)− µ(s)R(s) (4)where µ is a real-valued parameter [9]. The distance from line of striction to directrixalong the ruling is µ‖R‖, where ‖R‖ represents the magnitude of ruling.By substituting equation (4) into equation (2), we have〈α(s)− µ(s) R(s),R(s)〉 = 0 (5)where µ can be obtained from equation (5) asµ =〈α(s),R(s)〉〈R(s),R(s)〉.Since all rulings of a cylindrical surface pass through line of striction, line of strictioncan be thought as a directrix of the cylindrical surface. Thus, a cylindrical surface can beexpressed asX (s, v) = c(s) + v R.A cylindrical surface in Minkowski 3-space can be classified according to the Lorentziancharacter of its ruling and its line of striction. Thus, the curvature theory of a cylindricalsurface should be studied in two cases: curvature theory of a spacelike cylindrical surfaceand curvature theory of a timelike cylindrical surface.B.SAHINER, M.KAZAZ, H.UGURLU: ON THE CURVATURE THEORY OF NON-NULL ... 253.1. Curvature theory of a spacelike cylindrical surface. A spacelike cylindricalsurface can be represented by the equationX (s, v) = c(s) + v Rwhere, c is a spacelike directrix and also the line of striction of cylindrical surface, rulingR is a constant spacelike vector, s is the arc-length parameter of directrix and v is areal-valued parameter.The generator trihedron of a spacelike cylindrical surface which is defined on line ofstriction consists of three orthonormal vectors: the generator vector q , the central normalvector h, and the central tangent vector a. Generator vector can be defined asq =R‖R‖where ‖R‖ is the magnitude of ruling. Since ruling R is a spacelike vector, generatorvector q is a spacelike vector. From equation (3), it is known that c′ is perpendicular tothe ruling, thus central tangent vector can be defined asa = c′ (6)Note that central tangent vector is tangent to the line of striction. Since line of strictionis a spacelike curve, central tangent vector is a spacelike vector. Central normal vectorcan also be defined ash = a × q . (7)Since generator vector and central tangent vector are spacelike vectors, central normalvector should be a timelike vector.From equation (6), the first-order positional variation of line of striction isc′ = a. (8)From the definition of a cylindrical surface, we haveR′ = q ′ = 0. (9)In order to determine the first-order derivative of central tangent vector a, second-orderpositional variation of line of striction should be determined. Differentiating equation (8)givesc′′ = a ′. (10)The line of striction is always perpendicular to constant ruling vector of cylindricalsurface and it is a planar curve. From curvature theory of a planar curve, we know thatsecond-order positional variation of line of striction is curvature of line of striction. Thecenter of curvature is on direction line of central normal vector. Therefore, the secondorder positional variation of line of striction can be expressed by the equationc′′ = κc h (11)where κc = ‖ c′′‖ is curvature of line of striction. By comparing equation (10) withequation (11), first-order derivative of central tangent vector can be given asa ′ = γ hwhere γ = κc called curvature of a spacelike cylindrical surface characterizes the shapeof cylindrical surface. For example, if γ = 0 then cylindrical surface is a plane or if γ =constant then cylindrical surface is a right circular cylinder. By differentiating equation(7) and using equation (9), the first-order derivative of central normal vector can also befound ash ′ = γ a.26 TWMS J. APP. ENG. MATH. V.6, N.1, 2016Thus, the first-order angular variation of generator trihedron can be expressed in matrixform asdds qha = 0 0 00 0 γ0 γ 0 qhaThe Darboux instantaneous rotation vector of generator trihedron of a spacelike cylindricalsurface can be obtained asw = −γ q .3.2. Curvature theory of a timelike cylindrical surface. The central normal vectorof a timelike cylindrical surface is a spacelike vector. There are two possible cases in thestudy of curvature theory of a timelike cylindrical surface. The ruling may be a spacelikevector or a timelike vector. In this section, we study the curvature theory of a timelikecylindrical surface by considering these two cases.A timelike cylindrical surface with spacelike (resp. timelike) ruling can be expressed byX (s, v) = c(s) + v Rwhere, c is a timelike (resp. spacelike) directrix and also line of striction, the ruling R isa constant spacelike (resp. timelike) vector, s is the arc-length parameter of directrix andv is a real-valued parameter.Generator trihedron of a timelike cylindrical surface can be defined as follows:q =R‖R‖, a = c′, h = −a × qwhere q is the generator vector, a is the central tangent vector, and h is the centralnormal vector. Note that q is a spacelike and a is a timelike vector in the case of timelikecylindrical surface with spacelike ruling and q is a timelike and a is a spacelike vector inthe case of timelike cylindrical surface with timelike ruling.In order to determine the derivation formulae of generator trihedron, we follow a waysimilar to the case of a spacelike cylindrical surface. The first-order angular variation ofgenerator trihedron of timelike cylindrical surface can be expressed in matrix form asdds qha = 0 0 00 0 −εγ0 −γ 0 qhawhere γ is the curvature of timelike cylindrical surface and ε = 〈q , q〉. The Darbouxinstantaneous rotation vector of generator trihedron of a timelike cylindrical surface canbe obtained asw = −γ q .Example 3.1. Let us consider the Lorentzian circular cylinder x21 − x23 = r2 , r > 0,as seen in Figure 1. The parametric form of this surface can be written as X (t, v) =(r cosh t , v, r sinh t). The directrix and the ruling of the Lorentzian circular cylinder areα(t) = (r cosh t, 0, r sinh t) and R = (0, 1, 0), respectively. From the definition of lineof striction of a timelike cylindrical surface, we obtain that the directrix of the Lorentziancircular cylinder is also line of striction. Note that line of striction c is a timelike curveB.SAHINER, M.KAZAZ, H.UGURLU: ON THE CURVATURE THEORY OF NON-NULL ... 27and the ruling R is a spacelike vector. By using equation (1), normalized parameter canbe found as s = r t.The vectors of generator trihedron of the Lorentzian circular cylinder can be found asq = R = (0, 1, 0),a = c′ =(sinhsr, 0 , coshsr),h = −a × q =(− cosh sr, 0 , − sinh sr),where q is the spacelike generator vector, a is the timelike central tangent vector and h isthe spacelike central normal vector. The derivatives of vectors of generator trihedron ofthe Lorentzian circular cylinder can be found asq ′ = (0, 0, 0) ,a ′ =(1rcoshsr, 0 ,1rsinhsr),h ′ =(−1rsinhsr, 0 , −1rcoshsr).The first-order angular variation of generator trihedron of the Lorentzian cylinder canalso be expressed in matrix form asdds qha = 0 0 00 0 −1/r0 −1/r 0 qha .The curvature of the Lorentzian cylinder is γ = 1/r. Thus, Darboux instantaneousrotation vector of the Lorentzian circular cylinder can be obtained as w = −1rq.Figure 1 A Lorentzian circular cylinder.28 TWMS J. APP. ENG. MATH. V.6, N.1, 20164. ConclusionsA cylindrical surface which is a special ruled surface has widely used in robotic science andengineering. Especially, in order to find the differential properties of the motion of a robotend-effector which moves on the cylindrical surface in Minkowski 3-space, the curvaturetheory of cylindrical surface can be important tool. Since a cylindrical surface is not ageneral ruled surface, its curvature theory should be developed separately. In this paper,the curvature theory of a cylindrical surface which can be used as a tool of the motion ofa robot end-effector in Minkowski 3-space is given in detail.References[1] Ayyıldız,N. and Yücesan,A., (2006), On the scalar and dual formulations of the curvature theoryof line trajectories in the Lorentzian space, Journal of the Korean Mathematical Society, 43, pp.1339-1355.[2] Ersoy,S. and Tosun,M., (2008), On the trajectory null scrolls in 3-dimensional Minkowski space-timeE31 , Kyungpook Math. J., 48, pp. 81-92.[3] Freudenstein,F., (1965), Higher path-curvature analysis in plane kinematics, Journal of Engineeringfor Industry, 87 (2), pp. 184-190.[4] Kirson,Y., (1975), Curvature theory in space kinematics, Doctoral dissertation, University of Califor-nia, Berkley, Calif, USA.[5] McCarthy,J. M., (1987), On the scalar and dual formulations of curvature theory of line trajectories,Journal of Mechanisms, Transmissions, and Automation in Design, 109 (1), pp. 101106.[6] ONeill,B., (1983), Semi-Riemannian Geometry with Applications to Relativity, Academic Press, Lon-don.[7] Önder,M. and Uğurlu,H.H., (2013), Frenet frames and invariants of timelike ruled surfaces, Ain ShamsEngineering Journal, 4, pp. 507-513.[8] Ratcliffe,J.G., (2006), Foundations of Hyperbolic Manifolds, Springer, New York.[9] Ryuh,B.S., (1989), Robot trajectory planning using the curvature theory of ruled surfaces, Doctoraldissertation, Purdue University, West Lafayette, Ind, USA.[10] Schaaf,J.A., (1988), Curvature theory of line trajectories in spatial kinematics, Doctoral dissertation,University of California, Davis.[11] Struik,D.J., (1950), Lectures on Classical Differential Geometry, Dover Publications, New York.[12] Turgut,A., (1995), 3-Boyutlu Minkowski Uzaynda Spacelike ve Timelike Regle Yzeyler, Doctoral the-sis, Ankara University, Ankara.[13] Beem,J.K. and Ehrlich,P.E., (1981), Global Lorentzian Geometry, Marcel Dekker, New York.Burak Sahiner graduated in Department of Mathematics from Ege University in2008 and received his M.Sc. degree in mathematics from the same University in2010. He completed his Ph.D. degree in mathematics from Celal Bayar Universityin 2013. The author has been working as a research assistant at Department ofMathematics, Faculty of Arts and Sciences, Celal Bayar University, Manisa, Turkeysince 2010. His research interests include differential geometry, differential line ge-ometry, dual geometry, kinematics, Lorentzian and dual Lorentzian geometry.B.SAHINER, M.KAZAZ, H.UGURLU: ON THE CURVATURE THEORY OF NON-NULL ... 29Mustafa Kazaz is an associated professor in the Department of Mathematics, Facultyof Arts and Sciences, Celal Bayar University, Manisa, Turkey. The author graduatedfrom Karadeniz Technical University, Department of Mathematics in 1987 and receivedhis M.Sc. degree in mathematics from the same University in 1992. He completedhis Ph.D. degree in mathematics from Southampton University, England in 1998. Heworked as a research assistant at Karadeniz Technical University and Celal Bayar Uni-versity. Then, he worked as an assistant professor at Celal Bayar University. He hasbeen a member of Department of Mathematics, Celal Bayar University where he be-came an associated professor in 2011. His research interests comprise: Differential geometry, Lorentziangeometry, dual geometry, dual Lorentzian geometry, group theory and its applications to geometry.Hasan Huseyin Ugurluis a professor in the Department of Mathematics, Faculty of Ed-ucation, Department of Secondary Education Science and Mathematics Teaching, Math-ematics Teaching Program, Gazi University, Ankara, Turkey. The author graduated fromAnkara University in 1983. He received his M.Sc. degree and Ph.D. degree from GaziUniversity in 1987 and 1992, respectively. He worked at Gazi University as a Faculty ofScience member from 1984 to 1993. Also, he worked at Celal Bayar University as a Fac-ulty of Arts and Sciences member from 1993 to 2004. He has been a member of Facultyof Education at Gazi University since 2004. His research interests comprise: Differentialgeometry, line geometry, dual geometry, Lorentzian and dual Lorentzian geometry

On the curvature theory of non-null cylindrical surfaces in Minkowski 3-space

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On the curvature theory of non-null cylindrical surfaces in Minkowski 3-space

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