DUAL SPLIT QUATERNIONS AND SCREW MOTION IN MINKOWSKI 3-SPACE * L. KULA AND Y. YAYLI **

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1 Iranian Journal of Science & echnology, ransaction A, Vol, No A Printed in he Islamic Republic of Iran, 6 Shiraz University DUAL SPLI UAERNIONS AND SCREW MOION IN MINKOWSKI -SPACE L KULA AND Y YAYLI Ankara University, Faculty of Science, Department of Mathematics 6 ogan, Ankara, urkey kula@scienceankaraedutr, yayli@scienceankaraedutr Abstract In this paper, two new Hamilton operators are defined the algebra of dual split quaternions is developed using these operators It is shown that finite screw motions in Minkowski -space can be expressed by dual-number ( matrices in dual Lorentzian space Moreover, by means of Hamilton operators, screw motion is obtained in -dimensional Minkowski space R Keywords Dual split quaternions, semi-orthogonal dual matrix, screw-motion INRODUCION uaternion algebra, enunciated by Hamilton, has played a significant role recently in several areas of physical science; namely, in differential geometry, in analysis synthesis of mechanism machines, simulation of particle motion in molecular physics quaternionic formulation of equation of motion in the theory of relativity Agrawal [] gave some algebraic properties of Hamilton operators Also, quaternions have been expressed in terms of 4 4 matrices by means of these operators he purpose of this paper is to develop the screw motion of a line in R Firstly, we defined dual split quaternions gave some algebraic properties of dual split quaternions Also, we discussed Hamilton operators their properties Moreover, by means of dual-number ( matrices, screw motion is expressed in Minkowski -space he last section is devoted to give the screw motion, in R, using the properties of the Hamilton operators defined in this paper DUAL NUMBERS AND DUAL SPLI UAERNIONS A brief summary of dual numbers dual split quaternions is presented in this section for easy reference to provide the necessary background for the mathematical formulations to be developed in this paper In this paper, a dual number A has the form a є a, where a a are real numbers є is the dual symbol subjected to the rules є, є = є =, є = є = є, є = A split quaternion q is an expression of the form q = a ai aj a k, Received by the editor December, 4 in final revised form November 4, 6 Corresponding author

2 46 L Kula / Y Yayli where a, a, a a are real numbers, i, j, k are split quaternionic units which satisfy the noncommutative multiplication rules i =, j = k = ij = - ji = k, jk = -kj = -, i ki = -ik = j ( (See [] for split quaternions Similarly, a dual split quaternion is written as = A Ai Aj A k As a consequence of this definition, a dual split quaternion can also be written as = q є q where q = a ai aj ak q = a a i a j a k are, respectively, real dual split quaternion components he multiplication of split quaternionic units with a dual symbol is commutative; ie єi = iє, so on herefore, it is a matter of indifference whether one writes Ai or ia, so on Owing to the properties of the eight units equality, additions subtraction of dual split quaternion are governed by the rules of ordinary algebra he three dual split quaternionic units ( i, j k are orthogonal unit vector with respect to the scalar product defined below Further, under a proper semi-orthogonal transformation, these units preserve the definition of split quaternionic units given in equations ( the definition of the scalar product For this reason i, j k are identified as an orthogonal triad of unit vectors in Minkowski -space ( R with the metric tensor g( u, v = uv uv uv, uv, R is called Minkowski -space denoted by R [] It is useful, therefore, to define the following terms: Dual number part of : S = A Dual vector part of : V = Ai A j Ak Hamiltonian conjugate of : = A ( Ai A j A k = S V = q єq he split quaternion multiplication is, in general, not commutative If P are the two dual split quaternions let R = P, then R is given by R = ( A B AB AB A B ( A B AB A B AB i ( A B AB A B AB j ( A B A B AB AB k = qp є( qp q p, ( where P = B Bi Bj Bk = p є p, p p are split quaternions Norm of : Iranian Journal of Science & echnology, rans A, Volume, Number A Autumn 6

3 Dual split quaternions screw motion in Minkowski -space 47 N = = = A A A A є = ( a a a a ( a a a a a a a a = qq є( qq q q Reciprocal of : = N = q є( q q q, where ( N Unit quaternion: q N = Scalar product of quaternion P gp (, = gp (, = AB AB AB AB = ( P P = ( P P = gqp (, є((, gqp gq (, p Cross product of dual split vectors R U: R U = ( RU UR From the computation point of view, the above operations are expressed in terms of matrix operations It is clear that a dual split quaternion is essentially a tetrad of dual numbers In this paper, these tetrad of dual numbers are also written in a column matrix form as hus it follows that where q q are 4 column vectors = [ A A A A ] = q є q, HAMILON OPERAORS AND HEIR PROPERIES In this section, two new operators H H, called Hamilton s operators, are defined their properties are discussed If q is a split quaternion, then Hamilton operators H H are, respectively, defined as a a a a a a a a Hq ( = a a a a ( a a a a Autumn 6 Iranian Journal of Science & echnology, rans A, Volume, Number A

4 48 L Kula / Y Yayli a a a a a a a a Hq ( = a a a a (4 a a a a A direct consequence of the above operators is the following identities: H( = H( = I, Hi ( E; Hj ( E; Hk ( E = = =, Hi ( F; Hj ( F; Hk ( F = = = where I is a 4 4 identity matrix Note that the properties of E n n of split quaternionic units i, j, k Since H H are linear in their elements, it follows that F (,, n = are identical to that H( q = a H( a H( i a H( j a H( k = ai ae ae ae, o H( q = a H( a H( i a H( j a H( k = ai af af af, o (5 (6 H ( Hq ( Hq ( = є, (7 H ( = Hq ( є Hq ( (8 Using the definitions of by H H, the multiplication of the two dual split quaternions P is given R = H( P = H( P (9 4 (See [] for Hamilton operators in R Equation (9 is of central importance in proving several identities In the discussion to follow some theorems associated with these operators ( H H are presented heorem If P are dual split quaternions, λ is a real number defined in equations ( (4, respectively, then the following identities hold: i ii iii iv v vi = P H( = H( P H( = H( P H ( P = H ( HP (, H ( P = H ( HP ( H( λ = λh(, H( λ = λh( HP ( = HHP ( (, HP ( = HPH ( ( I H ( = ε H ( ε, H ( = ε H ( ε, ε = I H ( = H (, H ( = H (, ( N H H are operators as Iranian Journal of Science & echnology, rans A, Volume, Number A Autumn 6

5 Dual split quaternions screw motion in Minkowski -space 49 vii det H( = ( N = g(,, det H( = ( N = g(, viii tr H ( = 4A, tr H ( = 4 A ix H ( = LH ( L, L =, L = L = L, L = I Proof: Identities (i, (ii (iii follow from equations (7 (8 Using the associative property of quaternion s multiplication it is clear that the following identities hold: ( P R = ( PR = PR In terms of operator H, the above identities can be written as And similarly, HPR ( = HHPR ( ( HPR ( = HHPR ( ( = H ( ( HPR ( = H ( HPR ( = HP ( ( HR ( = HPHR ( ( ( ( Since column R is arbitrary, the above relation employs equation (iv Identities (v, (vi, (vii, (viii (ix can be proved (7, (8 (9 heorem Matrices generated by operators as H H commute, or mathematically this can be stated H ( HP ( = HP ( H ( Proof: his follows from equations (9, ( ( From equations (7 (8, it is clear that the matrices H ( H ( are 4 4 dual matrices Following the usual matrix nomenclature, a matrix A ˆ is called a semi-orthogonal dual matrix (sogdm if ˆ ˆ ˆ ˆ A εa ε = A εa ε = AI, where A є a is a dual number I is an identity matrix I ε = I A matrix Aˆ is called semi-orthonormal dual matrix (sondm if A = heorem Matrices generated by operators H H are semi-orthogonal matrices, ie I i H ( εh ( ε = H ( εh ( ε = NI, ε = I ii ( ( ( I HP εhpε = HPεHP ( ε = NPI, ε = I H ( H ( are semi-orthonormal dual matrices if only if is a unit dual split quaternion Autumn 6 Iranian Journal of Science & echnology, rans A, Volume, Number A

6 5 L Kula / Y Yayli Proof: Equation (i follows from heorem Equation (ii is proved in the same way If is a unit dual split quaternion, then N = his implies that H ( is a semi-orthonormal dual matrix he same arguments apply for matrix H ( heorem 4 Let A ˆ be a 4 4 semi-orthonormal dual matrix with A k = a k є a k,( k =,,, 4 as its k th column vector hen there exists two vectors u v (ie u = v = such that vectors u v are unique Proof: Semi-orthonormal dual matrix ˆ A can be written as Semi-orthonormality condition implies Define o a k = ua k a k v, ( k =,,, 4 ( ˆ A = A є A o = =, ( AA ε ε A εaε I Equation (4 implies that matrix B is a skew-symmetric matrix ( B (5 by A from the right using equation (, one finds AA ε ε AεA ε = (4 A εa ε = B (5 A = εbε Multiplying equation = BA (6 Matrix B in equation (6 is unique If not, then let D be the other matrix that implies equation (6 hen it follows that or A = BA = DA ( D B A = Since A is a semi-orthonormal matrix it follows that B = D, i e B is unique Matrix B can be written uniquely as where A = ( a ik is a real semi-orthogonal matrix u v are two vectors A, u v are unambiguously determined by A ˆ For a dual vector U, the matrices HU ( HU ( are skew-symmetric matrices hese two matrices are denoted, respectively, as VU ( VU ( to emphasize the fact that VU ( VU ( are generated using the vector component of a quaternion only Matrices VU ( VU ( are written, respectively as Iranian Journal of Science & echnology, rans A, Volume, Number A Autumn 6 B = H( u H( v, (7 where u v are two vectors, ie u = v = Equation ( follows using equations (9, (6 (7 he following theorem is now obvious heorem 5 Any 4 4 semi-orthonormal dual matrix can be represented as A ˆ = [ a є ( ua a v a є ( ua a v a є ( ua a v a є ( ua a v ]

7 Dual split quaternions screw motion in Minkowski -space 5 U U U U U U VU ( = = HVU ( (, U U U U U U (8 U U U U U U VU ( = = HVU ( ( (9 U U U U U U Corollary 6 Matrices VU ( VU ( satisfy the following identities: I i VU ( εvu = ( ε, ε =, I I ii ( VU ( = εvu ( ε, ε = I I iii VU ( εvu ( ε VU ( εvu = ( ε = NVU ( I, ε = I iv I VU ( εvu ( ε = VU ( εvu ( ε = NVU ( I, ε = I v ( VU ( n = ( N n I = ( VU ( n VU ( n n vi ( VU ( = ( NVU ( VU ( n n ( VU ( = ( N VU ( VU ( 4 HE SCREW MOION OF A LINE IN MINKOWSKI -SPACE heorem 4 (E Study heorem here exist one-to-one correspondence between directed timelike (resp spacelike lines of R an ordered pair of vectors (, aa such that aa, = (resp aa, = aa, = [4] Definition 4 Let us consider a dual split vector V = a єa = ( a i a j a k є ( a i a j a k he dual split vector is said to be timelike if aa, <, spacelike if aa, > or a = aa, = a, where, is a Minkowskian scalar product with signature (,,, null if Lemma 4 Let = A Ai Aj Ak be a unit dual split quaternion Autumn 6 Iranian Journal of Science & echnology, rans A, Volume, Number A

8 5 L Kula / Y Yayli A A A A ( AA AA ( AA AA A = ( AA AA A A A A ( AA AA ( ( AA AA ( AA AA A A A A is a semi-orthonormal dual matrix ie A εa ε = A εa ε = I, where ε = By the matrix in ( we have the following theorem heorem 4 4 Let be a unit dual split quaternion If V is a spacelike dual split vector, then that the matrix A defined in Lemma 4 can be written as A = I (, sinh Φ ( cosh Φ( Φ S S S ( Φ Φ where Φ = ϕ є ϕ is a dual hyperbolic angle, cosh = A, sinh = A A A, S S An S = S S, Sn =, n, S is a skew-symmetric matrix Φ S sinh S If V is a timelike dual split vector, then the matrix A defined in Lemma 4 can be written as A = I (, sin Θ ( cos Θ(, ( Θ Θ Θ cos = A, sin = A A A, where Θ = θ є θ is a dual angle, is a skew-symmetric matrix An =, n =, n Θ sin, Proof: Let V be a spacelike dual split vector hen, we have Φ Φ = cosh sinh S An establishing (, where S = Si Sj Sk Sn =, n Φ sinh If V is a timelike dual split vector, then Θ Θ = cos sin An establishing (, where = i j k n =, n Θ sin he matrices A A satisfy the following: ( Φ, S ( Θ, X For a unit dual split vector X = x є x = X, we write X Iranian Journal of Science & echnology, rans A, Volume, Number A Autumn 6

9 Dual split quaternions screw motion in Minkowski -space 5 ( ( A X = cosh Φ X sinh Φ S X cosh Φ S, X S, ( ( Φ, S If X, S =, we have If X, =, we have ( ( A X = cos Θ X sin Θ X cos Θ, X ( Θ, (4 ( S A X = cosh Φ X sinh Φ X ( Φ, S cos sin (, ( A X = Θ X Θ X Θ heorem 4 5 i he matrix R = I sinh ϕ ( cosh ϕ( ( ϕ, S S S (5 yield a finite rotation of a line about the spacelike line S with the hyperbolic angle ϕ in Minkowski - space ii he matrix R = I sinh ϕ ( cosh ϕ( ( ϕ, S S S (6 yield a finite rotation of a line about the timelike line with the angle θ in Minkowski -space Proof: i In equation (5 the real part dual part may be interpreted separately he real part of R ( ϕ, S is A d( = I, sinh S ( cosh ( S (7 S ϕ ϕ ϕ the dual part of R ( ϕ, S is J = sinhϕs ( ( coshϕ( S S S S ϕ, S, (8 where S is a real part of S S is a dual part of S hus, equation (5 may be written Let X X = x x = X X є be a unit dual split vector In this case, R ( ϕ = A d, ( ϕ, S єj S ( ϕ, S (9 R X = A d( x є d, ( x x ( ϕ, ϕ S A J ϕ, S ( ϕ,, ( S S Autumn 6 Iranian Journal of Science & echnology, rans A, Volume, Number A

10 54 L Kula / Y Yayli where the vector A d( ϕ,s x is a finite rotation of a vector x about the spacelike axis S with the hyperbolic angle ϕ ii By a similar calculation, the real part of R is the dual part of R is ( θ, ( θ, d I A ( =, sin θ ( cos θ( ( θ J = sin θ ( cos θ(, ( ( θ, where is a real part of is a dual part of hus, equation (6 may be written d θ R ( θ = A, (, єj ( θ, ( For a unit dual split vector X X = x є x = X, we write X R X = A d( x є d (,, ( x x θ θ A J θ, ( θ,, (4 where A d( x is a finite rotation of a vector x about timelike axis θ with the angle θ, heorem 4 6 i he matrix yield a translation along the spacelike line S in Minkowski -space ii he matrix yield a translation along the timelike line in Minkowski -space I ( ϕ, S = є ϕ S (5 I ( θ, = є θ (6 Proof: i In the equation (5, the real part of X For a unit dual split vector X = x є x = X, we write X ( ϕ, S is I the dual part of ( ϕ, S is S ( X = x x ϕ ( ( S x ϕ, ϕ S є (7 Iranian Journal of Science & echnology, rans A, Volume, Number A Autumn 6

11 Dual split quaternions screw motion in Minkowski -space 55 herefore, the unit dual split vector space X ( ϕ, S is parallel to directed spacelike lines X of Minkowski - ii By a similar calculation, from equation (6 we write ( є (8 X = x x θ ( ( x θ, hus, the unit dual split vector X ( θ, is parallel to directed timelike lines X of Minkowski -space he following theorem is now obvious heorem 4 7 he dual matrix A in equation ( is written S or ( Φ, A = R ( Φ, S ( ϕ, S ( ϕ, S, (9 A = R ( Φ, S ( ϕ, S ( ϕ, S the dual matrix A in equation ( is written or ( Θ, A = R ( Θ, ( θ, ( θ,, (4 A = R ( Θ, ( θ, ( θ, Consequently, the screw motion of a line may be considered a combination of a finite rotation about the spacelike line (resp timelike line translation along the spacelike line (resp timelike line in Minkowski -space Special cases: i he case that ϕ = : A R ( ϕ S = ( Φ, S, ii he case that ϕ = : A ( ϕ = ( Φ, S, S iii he case that θ = : A R ( θ = ( Θ,, iv he case that θ = : A ( θ = ( Θ,, Corollary 4 8 Autumn 6 Iranian Journal of Science & echnology, rans A, Volume, Number A

12 56 L Kula / Y Yayli i Let A A be a two screw motion about a spacelike axis (, S ( Φ, S angle Φ Φ In this case A A = A ( Φ, S ( Φ, S ( Φ Φ, S S with two dual hyperbolic ii Let A A ( Θ, Θ Θ In this case (, be a two screw motion about a timelike axis with two dual angles Θ A A = A ( Θ, ( Θ, ( Θ Θ, 5 HE SCREW MOION IN MINKOWSKI -SPACE Let us consider the transformation R = R, (4 where R f is a unit dual vector = A Ai Aj Ak is a unit dual split quaternion Since R f is a unit dual vector, R is also a unit dual vector Using the definition of H H, equation (4 can be written as or R = H( H( R = H( H( R f f f R = R f (4 where the matrix A is a semi-orthonormal matrix defined by equation ( he time derivative of R (denoted by a dot is f R = H ( HR ( H ( HR ( (4 Using equations (4 (4, from heorem, equation (4 can be written as or R = H( H( R f R = R W (44 where W is a skew symmetric matrix In terms of dual split quaternions, we write R = ( R R ( (45 Owing to the fact that is a unit dual split quaternion, it is clear that is a dual vector herefore, equation (45 can be written as a cross product form Also, the time derivative of R is given by R = ( R (46 Iranian Journal of Science & echnology, rans A, Volume, Number A Autumn 6

13 Dual split quaternions screw motion in Minkowski -space 57 R = W R, (47 where W is the dual split vector of the velocity screw of the line Since R is an arbitrary vector, equations (46 (47 lead to W = (48 or equivalently, From the equation (49, if g(, > is a spacelike dual split vector If ( W = H( = H( (49 or = (ie gqq (, > or q =, then the dual split vector W g, < (ie gqq (, <, then the dual split vector W is a timelike dual split vector In other words, the skew symmetric matrix W in equation (44 can be written as the form S or in equations (, ( Also or equivalently, = W = H( W = H( W Using equations (47 (48, the two components of W can be written as w = qq or w = H( q q v = ( qq q q or v = H( q q H( q q Since H H are semi-orthogonal operators, the formulation presented here provides a singularity free relation between W, its time derivative 6 CONCLUSIONS Split quaternion algebra of dual numbers has been formulated in terms of two operators Properties of these two operators are presented that establish a relationship between dual split quaternions their equivalent matrix operations hese properties are applied to develop kinematics equations of the screwmotion of a line a point in Minkowski -space REFERENCES Agrawal, O P (987 Hamilton operators dual quaternions in spatial kinematics Mech Mach heory Autumn 6 Iranian Journal of Science & echnology, rans A, Volume, Number A

14 58 L Kula / Y Yayli (6, Inoguchi, J (998 imelike Surfaces of Constant Mean Curvature in Minkowski -Space okyo J Math ( 4-5 O Neill, B (98 Semi-Riemannian Geometry New York, Academic Press 4 Yayli, Y Caliskan, A & Ugurlu, H H ( he E Study Maps of Circles on Dual Hyperbolic Lorentzian Unit Spheres H S Math Proc R Ir Acad (, 7-47 Iranian Journal of Science & echnology, rans A, Volume, Number A Autumn 6