Curvature Motion on Dual Hyperbolic Unit Sphere

Transcription

1 Journal of Applied Mathematics and Phsics Published Online Jul 4 in SciRes Curvature Motion on Dual Hperbolic Unit Sphere Zia Yapar Yasemin Sağıroğlu Karadeniz Technical Universit Trabzon Turke sagirogluasemin@gmailcom Received June 4; revised Jul 4; accepted Jul 4 Copright 4 b authors and Scientific Research Publishing Inc This work is licensed under the Creative Commons Attribution International License (CC BY) Abstract with dual signature ( ) In this paper we define dual curvature motion on the dual hperbolic unit sphere in the dual Lorentzian space ++ We carr the obtained results to the Lorent- zian line space b means of Stud mapping Then we make an analsis of the orbits during the dual hperbolic spherical curvature motion Finall we find some line congruences the famil of ruled surfaces and ruled surfaces in Kewords Dual Curvature Motion Dual Lorentzian Space Stud Mapping Introduction Dual numbers had been introduced b WK Clifford ( ) as a tool for his geometrical investigations After him E Stud (86-9) used dual numbers and dual vectors in his research on the geometr of lines and kinematics [] He devoted special attention to the representation of directed lines b dual unit vectors and defined the mapping that is known b his name There exists one-to-one correspondence between the vectors of dual unit sphere S and the directed lines of space of lines Hence a differentiable curve on the sphere S corresponds to a ruled surface in the line space []-[4] Ruled surfaces have been widel applied in surface design manufacturing technolog and simulation of rigid bodies [5] E Stud s mapping plas a fundamental role in the real and dual Lorentzian spaces [6] B this mapping a curve on a dual hperbolic unit sphere corresponds to a timelike ruled surface in the Lorentzian line space Corresponding author How to cite this paper: Yapar Z and Sağıroğlu Y (4) Curvature Motion on Dual Hperbolic Unit Sphere Applied Mathematics and Phsics Journal of

2 Z Yapar Y Sağıroğlu in other words there exists a one-to-one correspondence between the geometr of curves on and the geometr of timelike ruled surfaces in Similarl a timelike (spacelike) curve on a dual Lorentzian unit sphere S corresponds to a spacelike (timelike) ruled surface in the Lorentzian line space this means that there exists a one-to-one correspondence between the geometr of timelike (spacelike) curves on S and the geometr of spacelike (timelike) ruled surfaces in [7] Since the dual Lorentzian metric is indefinite the angle concept in this space is ver interesting For instance the dual hperbolic angle φ = ϕ + εϕ between two dual timelike unit vectors is a dual value formed with the (real) hperbolic angle ϕ between corresponding two directed timelike lines in the Lorentzian line space and the shortest Lorentzian distance ϕ between these directed timelike lines Real spherical curvature motion had been introduced b A Karger and J Novak [8] Also a dual spherical curvature motion has been defined b Z Yapar [9] In recent ears stud about the real spherical motion has been generalized to the Lorentz spherical motion [6] [7] [] [] In this work we consider the curvature motion on the dual hperbolic unit sphere of the dual Lorentzian space and the results are carried over to the Lorentzian line space b the E Stud s mapping Preliminaries and Definitions In this section we give a brief summar of the theor of dual numbers dual Lorentzian vectors and Stud s mapping Let be the -dimensional Minkowski space over the field of real numbers with the Lorentzian inner product given b ab = ab + ab ab where a = ( a a a) and = ( b b b) A vector a = ( a a a ) of b is said to be timelikeif aa < spacelike if aa > or a = and lighlike (null) if aa = and a The norm of a vector a is defined b a= aa Let a = ( a a a) and b = ( b b b) be two vec- tors in then the Lorentzian cross product of a and b is given b a b = ab ab ab ab ab ab ( ) If a and a are real numbers and ε = the combination A= a+ ε a is called a dual number where ε is a dual unit The set of all dual numbers forms a commutative ring over the real numbers field and is denoted b DD Then the set { a ( a a a) a i i } = = are called dual vec- is a module over the ring DD which is called a DD-module or dual space The elements of tors Thus a dual vector a can be written as a = a+ εa where a and a are real vectors at If A= a+ ε a and B = b+ εb with B then the division is given b A a + ε a a a ab = = + B b+ εb b b b Let f be a differentiable function with dual variable f is where f ( x) is the derivative of f ( ) ε X x x ( ) = ( + ε ) = ( ) + ε ( ) f X f x x f x x f x x Then it is eas to see that ( ε ) sh x + x = shx + εx chx = + ε Then the Maclaurin series generated b 89

3 Z Yapar Y Sağıroğlu The norm X of a dual number ( ε ) ch x x chx x shx + = + ε x + ε = + ε ( > ) x x x x x X x x = + ε is defined b Then we can write X = x + εx = X = x + εxx x = + ( ) x X x ε x x The Lorentzian inner product of two dual vectors a = a+ εa ( ) ab = ab + ε ab + a b b = b+ εb is defined b where ab is the Lorentzian inner product of the vectors a and b in the Minkowski -space A dual vector a is said to be timelike if aa < spacelike if aa > and lightlike (or null) if aa = where is a Lorentzian inner product with signature ( ) ++ The set of all dual Lorentzian vectors is called dual Lorentzian space and it is denoted b { a a εa aa } = = + : The Lorentzian cross product of dual vectors a and where a b is the Lorentzian cross product in Lemma Let abcd Then [] ) a b = b a ) a ba = ; and a bb = ) a bc = det ( abc ) 4) a bc d = ac bd + ad bc b ( ) is defined b a b = a b+ ε a b+ a b a ba = Let a = a+ εa Then a is said to be dual timelike unit vector (resp dual spacelike unit vector) if the vectors a and a satisf the following properties: aa = (resp aa = ) aa = The set of all dual timelike unit vectors (resp all dual spacelike unit vectors) is called the dual hperbolic unit sphere (resp dual Lorentzian unit sphere) and is denoted b (resp S ) [6] (See []-[6] for Lorentzian basic concepts) Theorem (E Stud Map) [6] There exists one-to-one correspondence between directed timelike (resp spacelike) lines of and an ordered pair of vectors ( aa ) such that aa = (resp aa = ) and aa = Definition A directed timelike line in ma be given b two points on it p and q If λ is an non-zero constant the parametric equation of the line is q = p+ λ In this case the vector given b = p = q is called the moment of the vector with respect to the origin This means that the direction vector of the timelike line and its moment vector are independent of the choice of the points pqr on the line However the vector and are not independent of one 8

4 Z Yapar Y Sağıroğlu another Also the satisf the following properties: and { ( )} = = Let e e e timelike denote the dual hperbolic unit sphere the center of dual orthonormal sstem at respectivel where we have and e = e + εe i i i i e e = e e e = e e e = e e e = e e e = e e e = e [7] In this case the orthonormal sstem { e e e } and the is the sstem of the space of lines A ruled surface is a surface generated b the motion of a straight line in This line the generator of the surface This follows the following definition Definition A ruled surface is said to be timelike if the normal of surface at ever point is spacelike and spacelike if the normal of surface at ever point is timelike [7] Let x and denote two different points at x The hperbolic angle φ has a value φ = ϕ + εϕ which is a dual number where ϕ and ϕ are the hperbolic angle and the minimal Lorentzian distance between directed lines x and respectivel and φ denote the dual hperbolic angle ( ) Dual Curvature Motion on the Dual Hperbolic Unit Sphere Let us consider a fixed dual orthonormal frame R = { u u u ( timelike) } and represent this frame b the dual hperbolic unit sphere H Consider the dual hperbolic spherical motion of a hperbolic spherical segment AB of constant such that its endpoints move along circles one of them being a great circle which ling on the plane ( u u ) on H Let a circle with radius which is perpendicular to the great circle be given in a plane which is parallel to the plane ( u u ) Its center is on the vector u and with distance from the plane ( u u ) The segment AB moves so that A B The position vectors of the endpoints of segment AB are chosen as the vectors v (timelike) and v (spacelike) of the moving frame R The vector v is then defined b the relation v = ( v v ) As the parameter of motion we choose the dual hperbolic angle φ of the timelike vectors u and v Let us denote the dual hperbolic angle of the vectors v and u b α Then v = u + uch α + u shα where the vector v = OA is spacelike Further we have where v is timelike It must be v = ush φ + u chφ ( v v = ie ie shφ = shαchφ or shα = tanhφ where ( φ) u timelike) shφ shαchφ = φ ϕ εϕ = + Then chα ( tanh φ) = + Thus we obtain v = ( v v ) = chφ( + tanh φ) u + tanh ( + tanh ) chφ shφ φ u shφ φ u = + + tanh + tanhφ v u u u v = shφu + chφu () 8

5 Z Yapar Y Sağıroğlu Thus we have the orthonormal dual frame { v v v } Let this sstem be represented b moving hperbolic sphere H Then a dual hperbolic curvature motion H H takes place This motion will be called a dual hperbolic curvature motion Let X be a fixed point on the arc vv During the dual hperbolic curvature motion the point X draws an orbit on the fixed hperbolic sphere H Denote the dual hperbolic angles of vx Xv b Θ = θ+ εθ and Θ = θ + εθ respectivel Then it can be written where v shθ + v shθ v shθ + v shθ X = = () sh ( Θ Θ ) + sh =Θ +Θ = σ + εσ [5] From Equation () making the necessar calculations for X we have x shθ shθ shϕ shθ shθ shϕ shθ shσ shσ shσ shσ shσ x = + + () shϕ = + sh σ sh σ ( θshσchθ σ shθchσ) ( θchθshσ σ shθchσ) shθ shθ shϕ + ϕ ϕ + ( θshσchθ σ shθchσ) shσ shσch ϕ sh σ shθ shϕ ϕ + θ σ θ σ θ σ shσch ϕ sh σ + + ( sh ch sh ch ) shθ shϕ ch ϕ ϕ ( θ ch θ sh σ σ sh θ ch σ ) shσ sh σ where x and x are the real and dual parts of X also Since Ө and Ө are constant (ie θ θ θ θ are all constants) A = σ + εσ is constant Equations () and (4) depend onl two parameters ϕ and ϕ Thus Equations () and (4) represent a timelike congruence in (for more details on congruences see [] []) Let p denote the position vector of an arbitrar point P( ) of a directed timelike line of this timelike line congruence in Then we have Since ( ) ( ϕϕ ) ( ϕϕ ) u ( ϕϕ ) p = x x + x (5) are the coordinates of P making the necessar calculations we obtain ( θ θ ϕ) ( θ σ θ σ θ σ) ϕ shθ shθ = sh + sh sh + sh ch sh ch sh σ sh σ shθ λ ( θ chθ shσ σ shθ ) shθ shθ shϕ sh σ shσ + + sh θ shθ shθ shϕ shθ shθ shϕ sh θ sh ϕ sh σch ϕ sh σ sh σch ϕ sh σ = ϕ shθshθshϕ sh θch ϕ shθ θ θ σ σ θ σ sh σ sh σ + sh σ sh sh sh sh ch θ ϕ θ ϕ ϕ ( ch sh sh ch ) + ( θshσchθ σshθchσ) + ( θchθshσ σ shθchσ) sh ch sh σ ϕ shθsh ϕ shθ sh σ sh σ ( θchθshσ σ shθchσ) ( θshσchθ σ shθchσ) sh sh ch + sh σ shθ θ ϕ ϕ ( θ ch θ sh σ σ sh θ ch σ ) λ shσ σ (4) (6) (7) 8

6 Z Yapar Y Sağıroğlu and ch sh sh sh sh sh ϕ ϕ θ θ ϕ θ ϕ = + shθ sh σ ch ϕ ch ϕ shθ shϕ θ σ θ σ θ σ sh σ ( sh ch sh ch ) + shθ shϕ λ shθ shϕ ( θchθshσ σ shθchσ) + + shθ sh σ shσ If Θ= θ+ εθ = (ie θ = θ = ) then = θ ie have x = v Thus from Equations (6)-(8) we obtain = λshϕ From Equation (9) we have = ϕ = λ + = λ = ϕ (8) σ = θ σ = θ In this case from Equation () we which represents a line congruence Thus we have the following theorem Theorem During the dual hperbolic spherical curvature motion H H in the case of Θ= (hence x = v ) in Equation () the Stud map of the orbit which is drawn on the H b x = v is the congruence in + = λ () = ϕ If we take λ = ϕ = in the Equation (9) then we have (9) () + = () Thus we have the following theorem Theorem During the dual hperbolic spherical curvature motion H H in the case of λ the Stud map in of the orbit drawn on the H b x = v is the cone which is given b In addition if we take + = ϕ = cϕ (c = constant) then we have = c tanh which represents a right helicoid If Θ= θ + εθ = ie θ = θ = then =Θ ie we have x = v Thus from Equations (6)-(8) we obtain If we put ϕ ϕ λ = ϕ + ϕ = + λ ch ϕ shϕ shϕ = ϕ + λ ch ϕ = from Equation () we have = ϕ = σ = θ σ = θ In this case from Equation () λ () 8

7 Z Yapar Y Sağıroğlu From Equation (4) we have = λ = λ shϕ = λ (4) + = (5) which represents a cone whose axis is the vector Thus we have the following theorem Theorem During the dual hperbolic spherical curvature motion the orbit drawn on H b ϕ ϕ λ = ) represents a cone in the whose axis is the vector If we put ϕ ϕ λ = from Equation () we have From Equation (6) we have ϕ = ϕ = ch ϕ shϕ = ϕ ch ϕ v (if (6) + = (7) which represents a cone whose axis is the vector Thus we have the following theorem Theorem 4 During the dual hperbolic spherical curvature motion the orbit drawn on H b = ) represents a cone in the ϕ ϕ λ whose axis is the vector 4 Analsis of the Orbit of v during the Dual Hperbolic Spherical Curvature Motion Seperating real and dual parts of v from Equation () we have shϕ v = ch ϕ ch ϕ ch ϕshϕ shϕ ch ϕsh ϕ v = ϕ + ϕ ϕ ch ϕ + ch ϕ ch ϕ v (if Equations (8) and (9) have onl two parameters ϕ and ϕ Hence Equations (8) and (9) represent a line congruence in Let n denote the position vector of an arbitrar point N( ) of an oriented line of this congruence in then considering Equation (5) we have Since ( ) ( ϕϕ ) ( ϕϕ ) ( ϕϕ ) n = v v + uv () are the coordinates of N making the necessar calculations we obtain u = ϕ + ch ϕ sh ϕ+ ch ϕ = ϕ u ch ϕ shϕ shϕ = ϕ ( ch ϕ+ sh ϕ) u ch ϕ (8) (9) () 84

8 Z Yapar Y Sağıroğlu In the case of ϕ = ϕ u from Equation () we have = u ch ϕ u = shϕ = u From Equation () we obtain which represents an one-parameter famil of cone in ϕ = + in the Equation () then we have If we put ln ( ) () = + ch ϕ () = + (4) which represents an elliptic cone whose axis is the vector Thus we have the following theorem Theorem 4 During the dual hperbolic spherical curvature motion the orbit drawn on H b v (if ϕ = ϕ = ln ( + ) u ) represents an elliptic cone whose axis is the vector in the In addition putting various values of parameters in the Equations () or () we have different line congruences or ruled surfaces in 5 Conclusion This paper presents the curvature motion on the dual hperbolic unit sphere We define the curvature motion on the dual hperbolic unit sphere of the dual Lorentzian space and the results are carried over to the Lorentzian line space b the E Stud mapping The orbits drawn on the fixed dual hperbolic unit sphere b unit dual vectors of an orthonormal base { v v v } are obtained During this carring we do an analsis of orbits the drawn b the vectors v v v of dual hperbolic unit sphere and then we get some line congruences the families of ruled surfaces and ruled surfaces in according to variables of parameters Moreover we find equations of these line congruences the families of ruled surfaces and ruled surfaces This motion and its results ma give a wa to define new motions and contribute to the stud of surface design manufacturing technolog robotic research and special and general theor of relativit and man other areas in -dimensional Lorentzian space References [] Stud E (9) Geometrie der Dnamen Mathematiker Deutschland Publisher Leibzig [] Guggenheimer HW (96) Differential Geometr McGraw-Hill New York [] Hacısalihoğlu HH (97) On the Pitch of a Closed Ruled Surface Mechanism and Machine Theor [4] Veldkamp GR (976) On the Use of Dual Number Vector and Matrices in Instantaneous Spatial Kinematics Mechanism and Machine Theor [5] Zha XF (997) A New Approach to Generation of Ruled Surfaces and Its Application in Engineering The International Journal of Advanced Manufacturing Technolog [6] Uğurlu HH and Çalışkan A (996) The Stud Mapping for Directed Spacelike and Timelike Line in Minkowski -Space R Mathematical and Computational Applications 4-48 [7] Yalı Y Çalışkan A and Uğurlu HH () The E Stud Maps of Circles on Dual Hperbolic and Lorentzian Unit Spheres H and S Mathematical Proceedings of the Roal Irish Academ A 7-47 [8] Karger A and Novak J (985) Space Kinematics and Lie Groups (Translated b Michal Basch) Gordon & Breach New York [9] Yapar Z (989) On the Curvature Motion Communications Facult of Sciences Universit of Ankara Series A

9 Z Yapar Y Sağıroğlu [] Uğurlu HH Çalışkan A and Kılıç O () On the Geometr of Spacelike Congruence Communications Facult of Sciences Universit of Ankara Series A [] Kazaz M Özdemir A and Uğurlu HH (9) Elliptic Motion on Dual Hperbolic Unit Sphere and Machine Theor H Mechanism [] Çalışkan A Uğurlu HH and Kılıç O () On the Geometr of Timelike Congruence Hadronic Journal Supplement 8-8 [] Akutagawa K and Nishikawa S (99) The Gauss Map and Spacelike Surfaces with Prescribed Mean Curvature in Minkowski -Space Tohoku Mathematical Journal [4] Birman GS and Nomizu K (984) Trigonometr in Lorentzian Geometr American Mathematical Monthl [5] O Neil B (98) Semi-Riemannian Geometr with Applications to Relativit Academic Press London [6] Yaglom IM (979) A Simple Non-Euclidean Geometr and Its Phsical Basis Springer-Verlag New York 86

10 Scientific Research Publishing (SCIRP) is one of the largest Open Access journal publishers It is currentl publishing more than open access online peer-reviewed journals covering a wide range of academic disciplines SCIRP serves the worldwide academic communities and contributes to the progress and application of science with its publication Other selected journals from SCIRP are listed as below Submit our manuscript to us via either submit@scirporg or Online Submission Portal