Dual Split Quaternions and Screw Motion in Minkowski 3-Space* L
Iranian Journal of Science & Technology, Transaction A, Vol. 30, No. A3 Printed in The Islamic Republic of Iran, 2006 © Shiraz University
DUAL SPLIT QUATERNIONS AND SCREW MOTION IN MINKOWSKI 3-SPACE*
L. KULA AND Y. YAYLI**
Ankara University, Faculty of Science, Department of Mathematics 06100 Tandogan, Ankara, Turkey Email: [email protected], [email protected]
Abstract – In this paper, two new Hamilton operators are defined and the algebra of dual split quaternions is developed using these operators. It is shown that finite screw motions in Minkowski 3-space can be expressed by dual-number ( 33× ) matrices in dual Lorentzian space. Moreover, by means of Hamilton operators, screw 3 motion is obtained in 3-dimensional Minkowski space R1 .
Keywords – Dual split quaternions, semi-orthogonal dual matrix, screw-motion
1. INTRODUCTION
Quaternion algebra, enunciated by Hamilton, has played a significant role recently in several areas of physical science; namely, in differential geometry, in analysis and synthesis of mechanism and machines, simulation of particle motion in molecular physics and quaternionic formulation of equation of motion in the theory of relativity. Agrawal [1] gave some algebraic properties of Hamilton operators. Also, quaternions have been expressed in terms of 44× matrices by means of these operators. 3 The purpose of this paper is to develop the screw motion of a line in R1 . Firstly, we defined dual split quaternions and gave some algebraic properties of dual split quaternions. Also, we discussed Hamilton operators and their properties. Moreover, by means of dual-number ( 33× ) matrices, screw motion is 3 expressed in Minkowski 3-space. The last section is devoted to give the screw motion, in R1 , using the properties of the Hamilton operators defined in this paper.
2. DUAL NUMBERS AND DUAL SPLIT QUATERNIONS
A brief summary of dual numbers and dual split quaternions is presented in this section for easy reference and 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 aa+ є * , where a and a* are real numbers and є is the dual symbol subjected to the rules
єєєєєєє≠ 0, 000,== 11, == 2 =0.
A split quaternion q is an expression of the form
qa=+01 aiajak + 2 + 3,
∗Received by the editor December 11, 2004 and in final revised form November 14, 2006 ∗∗Corresponding author
246 L. Kula / Y. Yayli
where aaa012,, and a3 are real numbers, and ijk, , are split quaternionic units which satisfy the non- commutative multiplication rules
ijk222= −1, == 1 (1) ij==-, ji k jk == --,- kj i ki == ik j
(See [2] for split quaternions). Similarly, a dual split quaternion Q is written as
QA=++01 AiAjAk 2 + 3.
As a consequence of this definition, a dual split quaternion Q can also be written as
Qq=+є q*
*** * * where qa=++01 aiajak 2 + 3 and qaaiajak=+01 + 2 + 3 are, respectively, real and dual split quaternion components. The multiplication of split quaternionic units with a dual symbol is commutative; i.e. єєii= , and so on. Therefore, it is a matter of indifference whether one writes Ai1 or iA1 , and so on. Owing to the properties of the eight units equality, additions and subtraction of dual split quaternion are governed by the rules of ordinary algebra. The three dual split quaternionic units (ij, and 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 (1) and the definition of the scalar product. For this reason ij, and k are identified as an orthogonal triad of unit vectors in Minkowski 3-space ( R3 with 3 3 the metric tensor guv(), = − uv11++ uv 22 uv 33, uv,R∈ is called Minkowski 3-space and denoted by R1 [3]). It is useful, therefore, to define the following terms: Dual number part of Q :
SAQ = 0 .
Dual vector part of Q :
VAiAjAkQ =++12 3.
Hamiltonian conjugate of Q :
QAAiAjAk= 012− ()++ 3
= SVQQ− =+qqє *.
The split quaternion multiplication is, in general, not commutative. If Q and P are the two dual split quaternions and let RQP= , then R is given by
RABABABABABABABABi= ()()00− 11++ 22 33 + 01 ++ 10 32− 23
+++ ()ABABABABjABABABABk02 20 31− 13+++ () 03 30 12− 21 (2) =+qpє ( qp** + q p ),
* * where PB=++01 BiBjBkp 2 + 3 =+є p, p and p are split quaternions. Norm of Q :
Iranian Journal of Science & Technology, Trans. A, Volume 30, Number A3 Autumn 2006 Dual split quaternions and screw motion in Minkowski 3-space 247
2222 NQQQQAAAAQ ===+ 0123−− 2222 * * * * =+()2()aaaa0123−− ++є aaaaaaaa 00112233−− =+qqє(). qq** + q q
Reciprocal of Q :
Q Q−1 = NQ = qqqq−−−11*1− є(),
2 where ()Nq ≠ 0.
Unit quaternion:
NQ = 1.
Scalar product of quaternionQPand
gQP(, )= gPQ (, )
= −−AB00 AB 11++ AB 22 AB 33 11 =+=+()()QP PQ QP PQ 22 =+gqp(, )є ((, gqp** ) + gq ( , p )).
Cross product of dual split vectors R and U:
1 RU∧ = (). RUUR− 2
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
T QAAAA= []0123 .
Thus it follows that
Qq=+є q*, where q and q* are 41× column vectors.
3. HAMILTON OPERATORS AND THEIR PROPERTIES + − In this section, two new operators HH and , called Hamilton’s operators, are defined and their properties + _ are discussed. If q is a split quaternion, then Hamilton operators HH and are, respectively, defined as
aaaa0123− + aaaa1032− Hq()= (3) aaaa − 2301 aaaa− 3210 and
Autumn 2006 Iranian Journal of Science & Technology, Trans. A, Volume 30, Number A3 248 L. Kula / Y. Yayli
aaaa0123 − _ aa10 − a 3 a 2 Hq()= . (4) aaa− a 23 0 1 aa − aa 32 1 0
A direct consequence of the above operators is the following identities:
+ _ HHI(1)== (1) ,
++ + Hi( )=== E12 ; Hj() E ; Hk() E 3,
__ _ Hi()=== F123 ; Hj ( ) F ; Hk ( ) F.
where I is a 44× identity matrix. Note that the properties of En and Fn ()n = 1, 2, 3 are identical to that of split quaternionic units ijk, , . + _ Since HH and are linear in their elements, it follows that
+++++ Hq()=+++ aH0123 (1) aHi () aHj () aHk () (5) =+aIo aE11 + aE 22 + aE 33 ,
_____ Hq()=+++ aH0123 (1) aHi () aHj () aHk () (6) =+aIo aF11 + aF 22 + aF 33 ,
++ + HQ()=+ Hq ()є Hq (* ), (7)
__ _ HQ()=+ Hq ()є Hq (* ). (8)
+ _ Using the definitions of HH and , the multiplication of the two dual split quaternions Q and P is given by + _ RHQPHPQ==() () . (9)
(See [1] for Hamilton operators in R4 ). Equation (9) is of central importance in proving several identities. + _ In the discussion to follow some theorems associated with these operators ( HH and ) are presented.
+ _ Theorem 3. 1. If Q and P are dual split quaternions, λ is a real number and HHand are operators as defined in equations (3) and (4), respectively, then the following identities hold: ++ __ i. QP=⇔ HQHP() = () ⇔ HQHP () = (). +++___ ii. HQ(+= P ) HQ () + HP (),( HQ += P ) HQ() + HP (). ++__ iii. HQ()λλ== HQHQ (), () λλ HQ (). +++___ iv. HQP()== HQHP ()(), HQP()()(). HPHQ
++ __ TT−I2 0 v. HQ()===εε HQ () , HQ() εεε HQ () , . 0 I2 ++__ −−−−11112 vi. HQ()==≠ HQ (), HQ()(), HQ ()0. NQ
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+ _ 22 22 vii. detHQNgQQHQNgQQ ( )== (QQ ) ( , ) , det ( ) == ( ) ( , ) . + _ viii. tr HQ( )= 4 A0 , tr HQ( )= 4 A0 . 1000 + _ 0100− ix. HQ()= LHQ ()T L, L = , L−12= LLLIT ==, 0010 0001
Proof: Identities (i), (ii) and (iii) follow from equations (7) and (8). Using the associative property of quaternion’s multiplication it is clear that the following identities hold:
()QP R== Q () PR QPR .
+ In terms of operator H , the above identities can be written as
+++ HQPR()= HHQPR (()) (10) ++ ++ ==HQ ( )( HPR ( ) ) HQ ( ) HPR ( ) . And similarly,
___ HQPR()= HHQPR (()) (11) __ __ == HP ( )( HQR ( ) ) HPHQR ( ) ( ) .
Since column R is arbitrary, the above relation employs equation (iv). Identities (v), (vi), (vii), (viii) and (ix) can be proved (7), (8) and (9).
+ _ Theorem 3. 2. Matrices generated by operators HH and commute, or mathematically this can be stated as
++__ HQ() HP ()= HP () HQ ().
Proof: This follows from equations (9), (10) and (11). + _ From equations (7) and (8), it is clear that the matrices HQ() and HQ () are 44× dual matrices. Following the usual matrix nomenclature, a matrix Aˆ T is called a semi-orthogonal dual matrix (sogdm) if AAˆˆTTεε== AA ˆˆ ε ε AI, where Aa≠ 0 + є * is a dual number and I is an identity matrix and
−I2 0 ε = . A matrix Aˆ is called semi-orthonormal dual matrix (sondm) if A = 1. 0 I 2
+ _ Theorem 3. 3. Matrices generated by operators HH and are semi-orthogonal matrices, i.e. ++ ++ −I2 0 i) HQ()TTεε HQ ()== HQ () ε HQ () ε N I , ε = . Q 0 I 2
__ __ −I2 0 ii) HP()TTεε HP ()== HP () ε HP () ε N I , ε = . P 0 I 2 + _ HQ() and HQ() are semi-orthonormal dual matrices if and only if Q is a unit dual split quaternion. Autumn 2006 Iranian Journal of Science & Technology, Trans. A, Volume 30, Number A3 250 L. Kula / Y. Yayli
Proof: Equation (i) follows from Theorem 3. 1. Equation (ii) is proved in the same way. If Q is a unit + dual split quaternion, then N = 1. This implies that HQ() is a semi-orthonormal dual matrix. The same _Q arguments apply for matrix HQ().
ˆ * Theorem 3. 4. Let A be a 44× semi-orthonormal dual matrix with Aa• kk=+• є ak • k , ( = 1, 2, 3, 4) as its k’ th column vector. Then there exists two vectors u and v (i.e. uvoo==0 ) such that
* auaavk• kkk=+• • , ( = 1, 2, 3, 4) (12) and vectors u and v are unique.
Proof: Semi-orthonormal dual matrix Aˆ can be written as
Aˆ =+A єA*.
Semi-orthonormality condition implies
AAεεTT== A εε A I, (13)
AAεε**TT+= A εε A 0. (14)
Define
AA*εεT = B. (15)
Equation (14) implies that matrix B is a skew-symmetric matrix ( BBT = −εε). Multiplying equation (15) by A from the right and using equation (13), one finds
ABA* = . (16)
Matrix B in equation (16) is unique. If not, then let D be the other matrix that implies equation (16). Then it follows that
ABADA* == or ()DBA− = 0.
Since A is a semi-orthonormal matrix it follows that BD= , i. e. B is unique. Matrix B can be written uniquely as
+ _ BHuHv=+() (), (17) where u and v are two vectors, i.e. uv00==0. Equation (12) follows using equations (9), (16) and (17). The following theorem is now obvious.
Theorem 3. 5. Any 44× semi-orthonormal dual matrix can be represented as
ˆ A = [a. 0++єєє ( ua . 0. a 0 v ) a . 1. ++ ( ua 1. a 1. v ) a 2 ++ ( ua . 2. a 2 v ) a . 3 ++ є ( ua . 3. a 3 v )]
where Aa= ()ik is a real semi-orthogonal matrix and u and v are two vectors. A, u and v are unambiguously determined by Aˆ . + _ For a dual vector U, the matrices HU() and HU () are skew-symmetric matrices. These two matrices + _ + _ are denoted, respectively, as VU() and VU () to emphasize the fact that VU() and VU () are generated + _ using the vector component of a quaternion only. Matrices VU() and VU () are written, respectively as Iranian Journal of Science & Technology, Trans. A, Volume 30, Number A3 Autumn 2006 Dual split quaternions and screw motion in Minkowski 3-space 251
0−UU U 12 3 ++UUU132 0 − VU()== HVU (()), (18) UU230 − U 1 UUU− 0 321
0 −UUU 123 __UUU1320 − VU()== HVU (()). (19) UU23− 0 U 1 UU− U0 32 1 + _ Corollary 3. 6. Matrices VU() and VU () satisfy the following identities:
++ −I2 0 i) VU()T = −εεε VU (),= , 0 I 2 and − _ −I2 0 ii) (())VUT = −εεε VU (),= . 0 I 2
++ ++ −I2 0 iii) VU()TTεε VU ()=== VU () ε VU () ε N I , ε VU() 0 I 2 and __ __ −I2 0 iv) VU()TTεε VU ()=== VU () ε VU () ε N I , ε . VU() 0 I 2 + _ 22nnn v) (VU ( ))= (− NVU() ) I= ( VU ( )) .
++ 21nn+ vi) (())VU= (− NVU() ) VU () and __ 21nn+ (())VU= (− NVU() ) VU ().
4. THE SCREW MOTION OF A LINE IN MINKOWSKI 3-SPACE
Theorem 4. 1. (E. Study Theorem) 3 There exist one-to-one correspondence between directed timelike (resp. spacelike) lines of R1 and an ordered pair of vectors (,aa* ) such that aa,1= − (resp. aa,1= ) and aa,0* = [4].
Definition 4. 2. Let us consider a dual split vector
**** VQ =+ aєє a =()(). aiajak12 + + 3 + aiajak 1 + 2 + 3
The dual split vector is said to be timelike if aa,0< , spacelike if aa,0> or a = 0 , and null if aa,0= and a ≠ 0 , where , is a Minkowskian scalar product with signature (,− ++ , ).
Lemma 4. 3. Let QA=++01 AiAjAk 2 + 3 be a unit dual split quaternion.
Autumn 2006 Iranian Journal of Science & Technology, Trans. A, Volume 30, Number A3 252 L. Kula / Y. Yayli
2222 AAAA01+++ 232( AAAA 0312 − ) − 2( AAAA 0213 + ) 2222 (20) A =2(AAAAAAAA03 + 12 ) 0 −−+ 1 2 3 − 2( AAAA 01 + 23 ) 2222 2(−+A02AAA 13 ) 2( AAAAAAAA 01 − 23 ) 0 −+− 1 2 3 −100 TT is a semi-orthonormal dual matrix i.e. AAεε= A εε A = I , where ε = 010 . 3 By the matrix in (20) we have the following theorem. 001
Theorem 4. 4. Let Q be a unit dual split quaternion.
If VQ is a spacelike dual split vector, then that the matrix A defined in Lemma 4.3 can be written as
A =+I sinh Φ+−+SS**2 ( 1 cosh Φ )( ) (21) (,Φ S* ) 3
ΦΦ where Φ =+ϕϕє * is a dual hyperbolic angle, cosh==A , sinh − AAA222++, 220 123 0 SS32− * An * S = SSSn310,− n = ,13≤≤, and S is a skew-symmetric matrix. Φ −SS 0 sinh 21 2
If VQ is a timelike dual split vector, then the matrix A defined in Lemma 4.3 can be written as
A =+I sin Θ+−TT**2 (1 cos Θ )( ) , (22) (,Θ T* ) 3
0 TT32− * ΘΘ222 where Θ =+θθє is a dual angle, cos==A0123 , sin AAA−−, * An , 22 T = TTTn310,− n = ,13≤≤ Θ −TT 0 sin 21 2 and T* is a skew-symmetric matrix.
Proof: Let VQ be a spacelike dual split vector. Then, we have
ΦΦ Q =+cosh sinh S* 22
A establishing (21), where S* =++Si S j Sk and Sn= n ,1≤≤ 3. 12 3 n Φ sinh 2
If VQ is a timelike dual split vector, then
ΘΘ Q =+cos sin T* 22
A establishing (22), where T* =++Ti T j Tk and Tn= n ,1≤≤ 3. 12 3 n Θ sin 2 The matrices A and A satisfy the following: (,Φ S* ) (,Θ T* ) X1 * For a unit dual split vector Xx=+є x = X2 , we write X 3
Iranian Journal of Science & Technology, Trans. A, Volume 30, Number A3 Autumn 2006 Dual split quaternions and screw motion in Minkowski 3-space 253
A XX=Φ+Φ∧+−Φcosh sinhSSS*** X 1 cosh , X , (23) (,Φ S* ) ( ) ( ) and
A XX=Θ+Θ∧−−Θcos sinTTT*** X 1 cos , X . (24) (,Θ T* ) ( ) ( )
If X,0,S* = we have
A XX=Φ+Φ∧cosh sinh S* X. (,Φ S* ) ( )
If X,0,T* = we have
A XX=Θ+Θ∧cos sin T* X. (,Θ T* ) ( )
Theorem 4. 5. i. The matrix
R =+I sinhϕϕSS**2 +−+ ( 1 cosh )( ) (25) ()ϕ ,S* 3 yield a finite rotation of a line about the spacelike line S* with the hyperbolic angle ϕ in Minkowski 3- space. ii. The matrix
R =+I sinhϕϕSS**2 +−+ ( 1 cosh )( ) (26) ()ϕ ,S* 3 yield a finite rotation of a line about the timelike line T* with the angle θ in Minkowski 3-space.
Proof: i. In equation (25) the real part and dual part may be interpreted separately. The real part of R is ()ϕ ,S*
A dI=+sinhϕϕ S +−+ ( 1 cosh )( S )2 (27) ()ϕ ,S0 30 0 and the dual part of R is ()ϕ ,S*
*** J * =sinhϕϕSSSSS +−+( 1 cosh ) + , (28) ()ϕ ,S 00000( )
* * * where S0 is a real part of S and S0 is a dual part of S . Thus, equation (25) may be written
RA* = d + є J* . (29) (,ϕ S ) ()ϕ ,S0 ϕ ,S () X1 * Let Xx=+є x = X2 be a unit dual split vector. In this case, X3 * RA* Xdx=+є A dx + J* x, (30) (,ϕ S ) ()ϕϕ,,SS00 () ()ϕ ,S
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where the vector A d x is a finite rotation of a vector x about the spacelike axis S0 with the ()ϕ ,S0 hyperbolic angle ϕ . ii. By a similar calculation, the real part of R is (,θ T* )
A dI=+sinθθ T +− (1 cos )( T )2 (31) ()θ ,T0 30 0 and the dual part of R is (,θ T* )
J =+−+sinθθTTTTT*** (1 cos )( ) , (32) ()θ ,T* 00000
* * * where T0 is a real part of T and T0 is a dual part of T . Thus, equation (26) may be written
RA* = d + є J* . (33) (,θ T ) ()θ ,T0 ()θ ,T
X1 * For a unit dual split vector Xx=+є x = X2 , we write X 3
* RA* Xdx=+є A dx + J* x, (34) (,θ T ) ()θθ,,TT00 () ()θ ,T
where A dx is a finite rotation of a vector x about timelike axis T0 with the angle θ . ()θ ,T0
Theorem 4. 6. i. The matrix
T =+I єϕ **S (35) ()ϕ**,S 3 yield a translation along the spacelike line S* in Minkowski 3-space. ii. The matrix
T =+I єθ **T (36) ()θ **,T 3 yield a translation along the timelike line T* in Minkowski 3-space.
Proof: * i. In the equation (35), the real part of T ** is I3 and the dual part of T ** is ϕ S0 . ()ϕ ,S ()ϕ ,S X1 * For a unit dual split vector Xx=+є x = X2 , we write X 3
** T **Xx=+є x +ϕ ( S ∧ x) . (37) ()ϕ ,S ( 0 )
Iranian Journal of Science & Technology, Trans. A, Volume 30, Number A3 Autumn 2006 Dual split quaternions and screw motion in Minkowski 3-space 255
Therefore, the unit dual split vector T X is parallel to directed spacelike lines X of Minkowski 3- ()ϕ**,S space. ii. By a similar calculation, from equation (36) we write
** T **Xx=+є x +θ ( Tx ∧) . (38) ()θ ,T ( 0 )
Thus, the unit dual split vector T X is parallel to directed timelike lines X of Minkowski 3-space. ()θ **,T The following theorem is now obvious.
Theorem 4. 7. The dual matrix A * in equation (21) is written (,Φ S ) ART= . , (39) (,Φ S* ) ()()ϕϕ,,SS*** or
ATR= . (,Φ S* ) ()()ϕϕ**,,SS * and the dual matrix A in equation (22) is written (,Θ T* )
ART= . , (40) (,Θ T* ) ()()θθ,,TT*** or
ATR= . . (,Θ T* ) ()()θθ**,,TT *
Consequently, the screw motion of a line may be considered a combination of a finite rotation about the spacelike line (resp. timelike line) and translation along the spacelike line (resp. timelike line) in Minkowski 3-space.
Special cases: i. The case that ϕ * = 0:
AR= . (,Φ S* ) ()ϕ ,S* ii. The case that ϕ = 0 :
AT= . (,Φ S* ) ()ϕ**,S iii. The case that θ * = 0:
AR= . (,Θ T* ) ()θ ,T* iv. The case that θ = 0 :
AT= . (,Θ T* ) ()θ **,T
Corollary 4. 8.
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* i. Let A * and A * be a two screw motion about a spacelike axis S with two dual hyperbolic (,)Φ1 S (,)Φ2 S angle Φ and Φ . In this case 1 2
AA**. = A *. (,)ΦΦ12SS ( ,) ( Φ+Φ 12 ,) S
* ii. Let A * and A * be a two screw motion about a timelike axis T with two dual angles Θ1 (,)Θ1 T (,)Θ2 T and Θ . In this case 2
AA**. = A *. (,)ΘΘ12TT ( ,) ( Θ+Θ 12 ,) T
5. THE SCREW MOTION IN MINKOWSKI 3-SPACE
Let us consider the transformation
RQRQ= f , (41)
where Rf is a unit dual vector and QA=++01 AiAjAk 2 + 3 is a unit dual split quaternion. Since Rf is a unit dual vector, R is also a unit dual vector. + _ Using the definition of HH and , equation (41) can be written as
++__ RHQHQRHQHQR==() ()f () ()f or
10 RR= (42) 0 f where the matrix A is a 33× semi-orthonormal matrix defined by equation (20).
The time derivative of R (denoted by a dot “.”) is
++__ R=+ HQ() HQR ()f HQ () HQR ()f . (43)
Using equations (41) and (42), from Theorem 3.3, equation (43) can be written as
+ _ RHQQHQQR= ()− () or
10 RR = (44) 0 W where W is a 33× skew symmetric matrix. In terms of dual split quaternions, we write
RQQRRQQ= ()− (). (45)
Owing to the fact that Q is a unit dual split quaternion, it is clear that QQ is a dual vector. Therefore, equation (45) can be written as a cross product form
RQQR= 2()∧ . (46)
Also, the time derivative of R is given by
Iranian Journal of Science & Technology, Trans. A, Volume 30, Number A3 Autumn 2006 Dual split quaternions and screw motion in Minkowski 3-space 257
RWR = ∧ , (47) where W is the dual split vector of the velocity screw of the line. Since R is an arbitrary vector, equations (46) and (47) lead to
WQQ= 2 (48) or equivalently,
+ _ WHQQHQQ==2() 2(). (49)
. . From the equation (49), if gQQ(),0> or Q = 0 (i.e. gqq(, )> 0 or q = 0 ), then the dual split vector W is a spacelike dual split vector. If gQQ(),0< (i.e. gqq(, )< 0), 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 T* in equations (21), (22).
Also
1 QWQ = 2 or equivalently,
11+ _ QHWQHQW ==() () . 22
Using equations (47) and (48), the two components of W can be written as
_ wqq= 2 or
_ _ wHqq= 2() and
** vqqqq0 =+2() or ++ ** vHqqHqq0 =+2() () . + _ Since HH and are semi-orthogonal operators, the formulation presented here provides a singularity free relation between W, Q and 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 and their equivalent matrix operations. These properties are applied to develop kinematics equations of the screw- motion of a line and a point in Minkowski 3-space.
REFERENCES
1. Agrawal, O. P. (1987). Hamilton operators and dual quaternions in spatial kinematics. Mech. Mach. Theory. Autumn 2006 Iranian Journal of Science & Technology, Trans. A, Volume 30, Number A3 258 L. Kula / Y. Yayli
22(6), 569-575. 2. Inoguchi, J. (1998). Timelike Surfaces of Constant Mean Curvature in Minkowski 3-Space. Tokyo J. Math. 21(1) 140-152. 3. O’Neill, B. (1983). Semi-Riemannian Geometry. New York, Academic Press. 4. Yayli, Y. Caliskan, A. & Ugurlu, H. H. (2000). The E. Study Maps of Circles on Dual Hyperbolic and Lorentzian 2 2 Unit Spheres H0 and S1 . Math. Proc. R. Ir. Acad. 102(1), 37-47.
Iranian Journal of Science & Technology, Trans. A, Volume 30, Number A3 Autumn 2006