We study the some algebraic properties of matrix associated to Hamilton operators which is defined for semiquaternions. The kinematic mapping corresponding to these operators in semi-Euclidean 4-space is the same as the kinematic mapping of Blaschke and Grünwal

Mehdi Jafarı

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173   Short Article   Kinematic Mapping in Semi-Euclidean 4-Space Mehdi JAFARI* Department of Mathematics, University College of Science and Technology Elm o Fan, Urmia, IRAN * Corresponding author’s e-mail address: mj_msc@yahoo.com    ABSTRACT We study the some algebraic properties of matrix associated to Hamilton operators which is defined for semi-quaternions. The kinematic mapping corresponding to these operators in semi-Euclidean 4-space is the same as the kinematic mapping of Blaschke and Grünwald. Keywords: Hamilton operators, Quasi-elliptic geometry, Semi-quaternion   Dört Buyutlu Semi-Oklidean Uzayda kinematik dönüşümler ÖZET Semi-kuaterniyonların Hamilton opratorlarına karşılık gelen matrislerin bazı cebirsel özeliklerin araştırdık. Bu opratorlara karşılık gelen dönüşümler kinematığı dört boyutlu semi oklıd uzayında, Blaşke ve Grünwald dönüşümler kinematığı aynıdır.. Anahtar Kelimeler: Hamilton operatöleri, Kuasi eliptik geometri, Semi kuaterniyon    I. INTRODUCTION  UATERNIONS was first introduced by William R. Hamilton as a successor to complex numbers. The quaternions have been used in various areas of mathematics. A brief introduction of the semi-quaternions is provided in [5]. Dyachkova [1] has showed that the set of all invertible elements of semi-quaternions with the quaternion product is a Lie group. Also, she considered the degenerate scalar product0 0 1 1, .q p a b a b   Accordingly, the semi-quaternions algebra with this product has the                               4-dimensional semi-Euclidean space structure with rank 2 semi-metric. In [2], the algebraic properties of semi-quaternions are studied and De-Moivre's and Euler's formulas for these quaternions are given. Q Received: 13/10/2014, Revised: 09/01/2015, Accepted: 09/01/2015  Düzce University  Journal of Science & Technology Düzce University Journal of Science and Technology, 3 (2015) 173-179  174 De Moivre's formula implies that there are uncountably many unit semi-quaternions satisfying for 1nq   for 2.n   The matrix associated with a semi-quaternion is studied and by De-Moivre’s formula the n  th power of such a matrix can be obtained [3]. In this paper, after a review of some fundamental properties of the semi- quaternions, algebraic properties of Hamilton operators of these quaternion are studied. By these operators, we get the kinematic mapping of Blaschke and Grünwald. The corresponding geometry is quasi-elliptic geometry.  II. PRELIMINARIES Definition1. The group of motion of the Euclidean plane is denoted by 2.OA If we choose a Cartesian coordinate system, then a motion 2OA  has the form  M. ,x x b with cos sinM . (1)sin cos       In homogeneous coordinate, Equ. (1) reads 1 0( . ) , = .MTx A x Ab       The kinematic mapping of Blashke and Grüwald is a correspondence between points of real projective three-space P3 and planar Euclidean motions. It is defined as: 2 20 33 2 20 1 2 3 3 0 0 3 22 21 3 0 2 0 3 3 00 0d P 2( ) 2 .2( ) 2d dd d d d d d d d OAd d d d d d d d             Note that the image that image of a point with coordinate 1 2(0, , ,0)d d  is not a Euclidean motion. We therefore call the line 0 3 0x x   the absolute line and consider the kinematic mapping defined in projective space without the absolute line. It is an elementary exercise to verify that a rotation with angle   and center ,m mx y corresponds to the point (1, , , cot )2m mx y   and that the translation x x b corresponds to the point 2 1(0, , ,2).b b  This is illustrated in Fig. 1, which shows an affine part of projective three-space[4].  Fig. 1. A planar rotation α with center m ( , )m mx y transforms p to q has the kinematic image point d.  175 II. EXPERIMENT A. SEMI-QUATERNIONS This section summarizes the essentials of the algebra of semi-quaternions. A semi-quaternion q  is an expression of the form 0 1 2 3q a a i a j a k     where 0 1 2, ,a a a  and 3a  are real numbers and , ,i j k  are quaternionic units satisfying the equalities 2 2 21, 0,, 0 ,i j kij k ji jk kj                and  .ki j ik     The set of all semi-quaternions is denoted bys . We express the basic operations in terms of , , .i j k The addition becomes as 0 1 2 3 0 1 2 30 0 1 1 2 2 3 3( ) ( )( ) ( ) ( ) ( )a a i a j a k b b i b j b ka b a b i a b j a b k                    and the multiplication as 0 1 2 3 0 1 2 30 0 1 11 0 0 12 0 3 1 0 2 1 33 0 2 1 1 2 0 3( )( )( )( )( )( ) .a a i a j a k b b i b j b ka b a ba b a b ia b a b a b a b ja b a b a b a b k                   Given 0 1 2 3q a a i a j a k    , 0a is called the scalar part of ,q denoted by   0,S q a  and 1 2 3a i a j a k  is called the vector part of ,q denoted by  1 2 3( ) .V q a i a j a k     If   0,S q   then q is called pure semi-quaternion. The set of all the pure semi-quaternions is denoted by .   The conjugate of q is 0 1 2 3 .q a a i a j a k     The norm of q is 2 20 1 .qN qq qq a a     If 1,qN  then q  is called a unit semi-quaternion. 176 The inverse of q  with 0,qN   is  1 1 .qq qN   Clearly 1 1 0 0 0 .qq i j k      Note also that qp pq  and 1 1 1( ) .qp p q    The algebra Hs  has the 4-dimensional semi-Euclidean space structure 42  with rank 2 semi-metric[2].   B. MATRICES ASSOCIATED WITH SEMI-QUATERNIONS We introduce the R-linear transformations representing left and right multiplication in H .s  Let q be a semi-quaternion. Then : H Hq s sL  and : H Hq s sR  are defined as follows: ( ) , , H .q q sL x qx R xq x    If 0 1 2 3q a a i a j a k     then;  0 1 2 3 0 1 2 31 0 3 2 1 0 3 20 1 0 11 0 1 0(1) , (1)( ) , ( )( ) 0 0 , ( ) 0 0( ) 0 0 , ( ) 0 0q qq qq qq qL a a i a j a k R a a i a j a kL i a a i a j a k R i a a i a j a kL j i a j a k R j i a j a kL k i a j a k R k i a j a k                                                          Therefore the matrix representations of the linear operators ,q qL R are, respectively  0 11 02 3 0 13 2 1 00 00 0(2)a aa aqa a a aa a a a        and  0 11 02 3 0 13 2 1 00 00 0. (3)a aa aqa a a aa a a a         The Euler’s and De-Moivre’s formulae for the matrix A are studied in [3].  It is shown that as the De Moivre’s formula implies, there are uncountably many matrices of unit quaternion satisfying 4nA I  for 2.n     Theorem 1. If q and p are two semi-quaternions, λ is a real number and qL and qR are operators as defined in equations (2) and (3), respectively, then the following identities hold: i. ( ) ( ) ( ) ( ).q p q p q p        177 ii. ( ) ( ) ( ), ( ) ( ) ( ).q p q p q p q p           iii. ( ) ( ), ( ) ( ).q q q q          iv. ( ) ( ) ( ), ( ) ( ) ( ).qp q p qp p q        v.    1 11 1( ) ( ) , ( ) ( ) , 0.qq q q q N          vi.    2 2det ( ) ( ) , det ( ) ( ) .q qq N q N     vii.    0 0( ) 4 , ( ) 4 .tr q a tr q a     Proof: Identities (i), (ii) and (iii) can be proved easily. Using the associative property of the quaternions multiplication, it is clear that following identities hold: ( ) ( )qp r q pr qpr   In terms of operator ,  the above identities can be written as ( ) ( ( ) )( )( ( ) ) ( ) ( )qp r q p rq p r q p r         and similarly, ( ) ( ( ) )( )( ( ) ) ( ) ( ) .qp r q p rp q r p q r         Since r  is arbitrary, the above relation employs equation (iv). Using the inverse property, we have 1 14qq q q I    and in terms of operator  ,   the above identities can be written as 1 14 41 14 4( ) ( ) ( ) ( ) ,( ) ( ) ( ) ( ) ,qq q q I Iq q q q I I             therefore, the above relation employs equation (v).  Identities (vi),  and (vii) can be proved easily. Theorem 2. Let q be a unit semi-quaternion. Matrices generated by operators ( )q  and ( )q are semi-orthogonal matrices, i.e. i)  ( ) ( ) ,Tq q     ii)  1 0 0 00 1 0 0( ) ( ) , .0 0 0 00 0 0 0Tq q             Theorem 3. The map  (4,R): (H , ,.) (M , , )s      defined as 178 0 11 00 1 2 32 3 0 13 2 1 00 00 0( i j k) ,a aa aa a a aa a a aa a a a            is an isomorphism of algebras.  Proof: We first demonstrate its homomorphic properties. If 0 1 2 3 ,p a a i a j a k    0 1 2 3q b b i b j b k    are any two semi-quaternions then:    0 0 1 1 2 2 3 30 0 1 11 1 0 02 2 3 3 0 0 1 13 3 2 2 1 1 0 00 1 0 11 0 1 02 3 0 1 2 3 0 13 2 1 0 3( + ) ( ) ( )( + ) 0 0+ 0 0( + )( ) ( + )0 0 0 00 0 0 0p q a b a b i a b j a b ka b a ba b a ba b a b a b a ba b a b a b a ba a b ba a b ba a a a b b b ba a a a b b                                   2 1 0,b bp q           0 0 1 1 1 0 0 1 2 0 3 1 0 2 1 33 0 2 1 1 2 0 30 1 0 11 0 1 02 3 0 1 2 3 0 13 2 1 0 3 2 1( ) ( )( )( )0 00 00 0 0 00 0 0 0pq a b a b a b a b i a b a b a b a b ja b a b a b a b kA B i C j DkA BB AC D A BD C B Aa a b ba a b ba a a a b b b ba a a a b b b b                                   0.p q         Thus the map is a homomorphism. It is also one-to-one and onto and so is an isomorphism.    If q is a nonzero semi-quaternion, the mapping 1: ,qv x q xq  is called the inner automorphism defined by .q We embed K into 42 by letting 1 2 3 1 2 3 1 2 3( , , ) (0, , , ) .x x x x x x x x i x j x k       The matrix representation of the map qv  is 179 2 20 12 21 2 0 3 0 1 0 12 21 3 0 2 0 1 0 10 0M 2( ) 2 .2( ) 2a aa a a a a a a aa a a a a a a a          Lemma 1: qv is a linear mapping for all nonzeroq , and it transforms the subspace of vectorial quaternions onto itself. Proof: The linearity of x qx follows directly from Theorem 1. The argument x xp  for is similar. Composition of these two mappings for 1p q  gives qv ,so qv is linear.  III. RESULTS According to definition 1, the kinematic mapping correspond with qv is kinematic mapping of Blaschke and Grünwald. The corresponding geometry is not elliptic one, but so called quasi-elliptic geometry.    V. REFERENCES  [1] Dyachkova M., On Hopf bundle analogue for semi-quaternion algebra, 10th International conference DGA, Olomouc, Czech Republic, (2007) 45-47.  [2] M. Jafari, H. Molaei Some properties of matrix algebra of semi-quaternions, Accepted for publication in Konuralp Journal of Mathematics.   [3] H. Mortazaasl, M. Jafari A study on semi-quaternions algebra in semi-Euclidean 4-space, Mathematical science and application E-Notes, 1(2) (2013) 20-27.  [4] H. Pottman J. Wallner Computational line geometry, Springer-Verlag Berlin Heidelberg New York, 2000.  [5] Rosenfeld B., Geometry of Lie groups, Kluwer Academic Publishers, Netherlands, (1997).  [6] J.P. Ward, Quaternions and Cayley numbers algebra and applications, Kluwer Academic Publishers, London, (1997).  

Kinematic Mapping in Semi-Euclidean 4-Space

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Kinematic Mapping in Semi-Euclidean 4-Space

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