GENERALIZED QUATERNION and ROTATION in 3-SPACE The

GENERALIZED QUATERNION AND ROTATION 3 IN 3-SPACE Eαβ

Mehdi JAFARI1 and Yusuf YAYLI2

Department of Mathematics, Faculty of Science Ankara University, 06100 Ankara, Turkey

Abstract: The paper explains how a unit generalized quaternion is used to represent a rotation of a 3 vector in 3-dimensional Eαβ space. We review of some algebraic properties of generalized quaternions and operations between them and then show their relation with the rotation matrix.

AMS Subject Classification: 15A33

Key words: Generalized quaternion, Quasi-orthogonal Matrix, Rotation

1. INTRODUCTION

The quaternions algebra were invented by W.R. Hamilton as an extension to the complex numbers. He was able to find connections between this new algebra and spatial rotations. The unit quaternions form a group that is isomorphic to the group SU(2) and is a double cover of SO(3), the group of 3-dimensional rotations. Under these isomorphisms the quaternion multiplication operation corresponds to the composition operation of rotations [22]. Kula and yayli [15] showed that unit split quaternions in ' determined a rotation in semi-Euclidean 4 space 2 . In[18], is demonstrated how timelike split quaternions are used to perform rotations 3 in the Minkowski 3-space 1 . Rotations in a complex 3-dimensional space are considered in [25] and applied to the treatment of the Lorentz transformation in special relativity.

A brief introduction of the generalized quaternions is provided in [19]. Also, this subject have investigated in algebra [21,23]. Recently, we studied the generalized quaternions, and gave some of their algebraic properties [9]. It is shown that the set of all unit generalized quaternions with the group operation of quaternion multiplication is a Lie group of 3-dimension. Their Lie algebra and properties of the bracket multiplication are looked for. Also, a matrix corresponding to Hamilton operators that is defined for generalized quaternions is determined a Homothetic 4 motion in  [13]. Furthermore, we showed how these operators can be used to described 4 rotation in  [12]. In this paper, we briefly review some fundamental properties of the

1 [email protected] 2 [email protected] generalized quaternions and also, investigate how unit generalized quaternions can be used to 3 described rotation in 3-dimensional space  .

2. PRELIMINARIES

Quaternions are hypercomplex numbers (which comprise the reals, complexes, quaternions, and Cayley numbers), and thus share many of the properties of the complexes. In particular, they work well for representing 3D rotations. The basic definition of a real quaternion given in [4,18,25] as follows:

Definition 2.1: A real quaternion is defined as    q a01  a 1 i  a 2 j  a 3 k    where a0,, a 1 a 2 and a3 are real number and 1,i , j , k of q may be interpreted as the four basic vectors of Cartesian set of coordinates; and they satisfy the non-commutative multiplication rules    i2 j 2  k 2  1        ij k   ji,, jk  i   kj and       ki j   ik, ijk   1.

The quaternion algebra H is the even subalgebra of the Clifford algebra of the 3-dimensional n Euclidean space. The Clifford algebra Cl(E)p Cl n p, p for the n -dimensional non-degenerate n vector space E p having an orthonormal base {e1 , e 2 ,..., en } with the signature (,)p n p is the associative algebra generated by 1 and {}ei with satisfying the relations ei e j e j e i  0 for i j 1,if i  1,2,..., p and 2 ei   . 1,if i p  1,..., n

The Clifford algebra Cl has the basis {e e ... e : 1 i  i  ...  i  n } that is the division n p, p i1 i 2 ik 1 2 k  algebra of quaternions H is isomorphic with the even subalgebra Cl3,0 of the Clifford algebra     Cl3,0 such that Cl3,0 has the basis {1,e2 e 3 j , e 1 e 3  k , e 1 e 2  i }. The conjugate of the quaternion   q Sq  V q is denoted by q , and defined as q Sq  V q . The norm of a quaternion 2 2 2 2 q (,,,) a0 a 1 a 2 a 3 is defined by q q q q  a0  a 1  a 2  a 3 and is denoted by Nq and say that q0  q Nq is unit quaternion where q  0. Unit quaternions provide a convenient mathematical notation for representing orientations and rotations of objects in three dimensions. One can    represent a quaternion q a0  a 1 i  a 2 j  a 3 k by a 2 2 complex matrix (with i ' being the usual complex imaginary);

a0 i'' a 1  i a 1  a 2  A    i'' a1  a 2 a 0  i a 3  or by a 4 4 real matrix

a0 a 1  a 2  a 3  a a a a  A  1 2 3 2  . a2 a 3 a 0 a 1    a3 a 2 a 1 a 0 

The Euler’s and De-Moivre’s formulae for the matrix A are studied in [10]. It is shown that as the De Moivre’s formula implies, there are uncountably many matrices of unit quaternion n satisfying AI 4 for n  2.

In geometry and linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a rigid body around a fixed point. There are at least eight methods used commonly to represent rotation, including: i) orthogonal matrices, ii) axis and angle, iii) Euler angles, iv) Gibbs vector, v) Pauli spin matrices, vi) Cayley-Klein parameters, vii) Euler or Rodrigues parameters, and viii) Hamilton’s quaternions [8]. But to use the unit quaternions is a more useful, natural, and elegant way to perceive rotations compared to other methods [18].

Theorem 2.2: All the rotation about lines through the origin in ordinary space form a group, homomorphic to the group of all unit quaternions [2].

If a simple rotation is only in the three space dimensions, i.e. about a plane that is entirely in space, then this rotation is the same as a spatial rotation in three dimensions. But a simple rotation about a plane spanned by a space dimension and a time dimension is a "boost", a transformation between two different reference frames, which together with other properties of spacetime determines the relativistic relationship between the frames. The set of these rotations forms the Lorentz group[1].

With Cartesian point coordinates in 3-space, a rotation in 3-space about the origin can be represented by the orthogonal matrix

r11 r 12 r 13  R  r r r  , 21 22 23  r31 r 32 r 33  T where RRI 3 and detR  1. It is known that unit quaternions can represent rotations about the origin. Wittenburg gives the following conversion formulae. For any unit quaternion q, the entries of the rotation matrix are

2 2 r112( aa 0  1 )  1, r 21  2( aaaar 12  03 ), 31  2( aaaa 13  02 ), 2 2 r122( aaaar 12  03 ), 22  2( aa 0  2 )1,  r 32  2( aaaa 23  01 ), 2 2 r132( aaaar 13  02 ), 23  2( aaaar 23  01 ), 33  2( aa 0  3 )1[7]. 

1 1 Example 2.3: For the unit real quaternion q  (1,  1,0), the rotation matrix is 2 2 1 1 1    2 2  2  1 1 1  R    , q 2 2  2  1 1  0  2 2 

 the axis of this rotation is spanned by the vector (1, 1,0), and the angle of rotation is   . 2 3 1 Similarity, for the timelike quaternion q   (1,0,0) with timelike vector part, the rotation 2 2 matrix is   1 0 0    1 3 R 0   , q 2 2    3 1 0  2 2 

2  so, this quaternion represents rotation through an angle   about the timelike axis i  1,0,0 [18]. 3

  3  Definition 2.4: Let u(,,),(,,). u1 u 2 u 3 v  v 1 v 2 v 3   If ,,    the generalized inner product of   u and v is defined by   u,. v   u1 v 1   u 2 v 2   u 3 v 3

It could be written  0 0    u, v uTT 0 0  v  u G v .   0 0     3 3 Also, if0,   0 , u, v is called the generalized Lorentzian inner product. We put   (,,) . 3 The vector product in  is defined by    i  j k   u v  u u u 1 2 3 v1 v 2 v 3    ()()(),uv2332  uvi   uvuvj 3113    uv 1221  uvk         where i j  k,, j  k  i k  i   j [9].

Special cases: 3 3 1. If    1, then  is an Euclidean 3-space  .

3 3 2. If 1,    1, then  is a Minkowski 3-space 1 [18].

Definition 2.5: A generalized quaternion q is an expression of form    q a0  a 1 i  a 2 j  a 3 k    where a0,, a 1 a 2 and a3 are real numbers and i,, j k are quaternionic units which satisfy the equalities    i2 ,,, j 2    k 2           ijk   ji , jk = i =  kj ,

and     ki= j = ik ,  ,    .

The set of all generalized quaternions are denoted by  . A generalized quaternion q is a sum of a     scalar and a vector, called scalar part, Sq  a0 , and vector part Vq  a1 i  a 2 j  a 3 k. Therefore

 is form a 4-dimensional real space which contains the real axis  and a 3-dimensional real 3 3 linear space  , so that,      .

Special cases:

1.    1, is considered, then  is the algebra of real quaternions.

2. 1,    1, is considered, then  is the algebra of split quaternions.

3. 1,   0, is considered, then  is the algebra of semi-quaternions. 4.  1,   0, is considered, then  is the algebra of split semi-quaternions.

5. 0,   0, is considered, then  is the algebra of 1 4 quaternions[20].

The multiplication rule for generalized quaternions is defined as       qp  SSq p  VV q , p  SV q p  SVVV p q  p  q where   SaSbVVq0,,,, p  0 q p  ab 1 1   ab 2 2   ab 3 3      VVq p ( ab23  abi 32 ) +  ( ababj 3113  ) +( ab 12  abk 21 ) . It could be written

a0 a 1   a 2   a 3   b 0  a a a  a   b  qp  1 0 3 2   1 . a2 a 3 a 0  a 1   b 2      a3 a 2 a 1 a 0   b 3 

Obviously, quaternion multiplication is associative and distributive with respect to addition and subtraction, but the commutative law does not hold in general.

Corollary 2.6:  with addition and multiplication have all the properties of a number field expect commutativity of the multiplication. It is therefore called the skew field of quaternions.   The conjugate of the quaternion q Sq  V q is denoted by q, and defined as q Sq  V q . The norm of 2 2 2 2 a quaternion q (,,,) a0 a 1 a 2 a 3 is defined by Nq  q q  q q  a0  a 1   a 2   a 3 , and say that q0  q N q is a unit generalized quaternion where q  0. The set of unit generalized quaternions, G, with the group operation of quaternion multiplication is a Lie group of 3-dimension. The scalar   product of two generalized quaternions q Sq  V q and p Sp  V p is defined as   q,,. p S S  V V  S qp Also, using the scalar product we can defined an angle  between s q p q p   S() qp two quaternions q, p to be such cos   . NNq p

T Definition 2.7: A matrix A3 3 is called a quasi-orthogonal matrix if AA  and detA  1 where  0 0   0  0    0 0   and,.   The set of all quasi-orthogonal matrices with the operation of matrix multiplication is 3 called rotation group in 3-space  . 0 s3  s 2  Definition 2.8: A matrix S s0   s  is called a generalized skew-symmetric matrix if 3 3 3 1  s2 s 1 0   0 0  T SS   where  0  0  and,.      0 0  

RELATIONSHIP OF GENERALIZED QUATERNION TO ROTATION

3 In this section, we show that a unit generalized quaternion represents a rotation in 3-space  . Let q be a unit generalized quaternion. The map  acting on a pure quaternion :

3 3 1 :,()      q  q is a 3D vector, a length-preserving function of 3D vectors, a linear transformation and does not have   3     a reflection component. Since   span{,,} i j k and if q a0  a 1 i  a 2 j  a 3 k G then  2 2 2 2   ()(i a01   a   a 2   ai 3 )2(   aaaaj 1203  )2(   aaaak 1302  ),     ()2(j  aaaaia  )( 2   a 2   a 2   aj 2 )2(   aaaak  ), 1203 0 1 2 3 2301    2 2 2 2 ()2(k  aaaai1302  )2(  aa 01   aaja 23 )(  0   a 1   a 2   ak 3 ), so that the matrix representation of the map  is

2 2 2 2 aaa012     a 32  ( aaaa 1203  ) 2  ( aaaa 1302  )  2 2 2 2  M 2 (aaaa1203  ) a 01   a   a 2   a 3 2  ( aaaa 2301  )  . 2 2 2 2  2(aaaa1302 ) 2( aaaa 0123   ) aaa 012       a 3 

Special cases:

1) For the case   1, we have M(M) H for real quaternion H. MH is a orthogonal matrix, then the map  corresponds to a rotation in 3. If we take the rotation axis to be

S ( s1 , s 2 , s 3 ), then we have 2 MISSH 3 sin  (1  cos  ) ,

where S is a skew-symmetric matrix, 0 s3 s 2     s0  s . 3 1  s2 s 1 0 

2) For the case 1,    1, we have M(M) H ' for split quaternion H'. MH ' is a semi- 3 orthogonal matrix, then the map  corresponds to a rotation in 1 . If we take the rotation axis

to be C (,,), c1 c 2 c 3 then we have 2 MH 'ICC 3  sinh  (  1  cosh  ) ,

where C is a skew-symmetric matrix, i.e. CCT    and

0 c3 c 2  1 0 0  C c0  c  ,   0 1 0  [16]. 3 1    c2 c 1 0  0 0 1 

Theorem 3.1: Let q be a unit generalized quaternion and,,    then the matrix M can be written as

2 M( ,S )I 3  sin S  (1  cos  )S ,

  where  is a elliptic angle, such that cosa , sin  a2   a 2   a 2 , and S is a generalized 20 2 1 2 3 skew-symmetric matrix.    Proof: Every unit generalized quaternion q a0  a 1 i  a 2 j  a 3 k can be written in polar form

  qcos  S sin , 2 2

then the matrix M can be written as

         cos2 (s 2   s 2   s 2 )sin 2 2  ( s s sin 2  s cos sin ) 2  ( s s sin 2  s cos sin ) 21 2 3 2 1 2 2 3 2 2 1 3 2 2 2 2            M2 ( s s sin2  s cos sin ) cos 2  (   s 2   s 2   s 2 )sin 2 2  (  s s sin 2  s cos sin )  1 22 3 2 2 2 1 2 3 2 2 3 2 1 2 2    2    2   2  2 2 2 2   2(s1 s 3 sin s 2 cos sin ) 2(s2 s 3 sin s 1 cos sin ) cos  (   s 1   s 2   s 3 )sin 2 2 2 2 2 2 2 2 

    (s2 s 2  s 2  1)sin 2 2  s s sin 2   s sin 2  s s sin 2   s sin 1 2 32 1 2 2 3 1 3 2 2    2 2 2 2 2  2  I 2 s s sin   s sin (  s  s  s  1)sin 2  s s sin   s sin  4 1 22 3 1 2 3 2 2 3 2 1    2 2  2 2 2 2   2s1 s 3 sin s 2 sin  2  s 2 s 3 sin  s 1 sin  (   s 1   s 2   s 3  1)sin 2 2 2 

 with used of 2sin2  1  cos and s2  s 2   s 2 1, we have 2 1 2 3

2 2 0 s3  s 2     s 3   s 2  s 1 s 2  s 1 s 3    M I sin  s 0   s   (1  cos  )  s s   s2   s 2  s s , 3 3 1   1 2 1 3 2 3    2 2  s2 s 10   s 1 s 3  s 2 s 3   s 1   s 2 

so, proof is complete.

Theorem 3.2: Let q be a unit generalized quaternion. Then M is a quasi-orthogonal matrix, i.e.

 0 0  T MM  and detM  1 where  0  0  .   0 0  

  Corollary 3.3: Let qcos  S sin be a unit generalized quaternion. If ,,    the linear map 2 2  ()  q  q1 represents a rotation of the original vector  by an angle  around the axis S in 3- 3 space  .

 1 1 1 1 Example 3.4: Let ,,    for the unit generalized quaternion q   ( , ,0), the rotation 2 2   matrix is 1 1    ()     2 4 2  2  1 1   R   (),    q 2 2 4  2  1 1 1 1  ()    2 2 2 4 

1 1  the axis of this rotation is spanned by the vector ( , ,0), and the angle of rotation is   .   2

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