Cv25578584

A. Lakshmi, S. N. Hasan / International Journal of Engineering Research and Applications
(IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 5, September- October 2012, pp.578-584
Spinors as a tool to compute orbital ephemeris
A. Lakshmi1 and S. N. Hasan2
1
Department of Mathematics, Osmania University, Hyderabad -500007
2
Deparment of Astronomy, Osmania University, Hyderabad – 500007.

ABSTRACT
In this paper, the Spinors of the physical 3-D Let i1 , i2 ,...., im be the elements of S in the ascending
Euclidean spaces of Geometric Algebras over a finite order. We define
dimensional vector space are defined. Action of Spinors eS  ei1 ei2 ........ eim and e  1L .
on Euclidean spaces is discussed. The technique for the
calculation of ephemeris using Spinors is illustrated. In We shall identify e{i} with ei .
this process sequences of Spinors in their half angle form Note that if  is a permutation of {1, 2, ….., m}, then
as well as their matrix form are used and compared. It is
shown that when large number of rotations is involved, ei (1) ei ( 2 ) ........e i = (-1) m ei1 ei2 ........eim
(m)
Spinor matrix methods are advantageous over the
Spinor half angle method. m stands for order of permutation.
 1L
2
ei … and ei e j  e j ei if i  j
Keywords - Ephemeris, Euclidean space, Euler angles,
Rotations, Spinors. C (E ) = ⊕ A k where Ak = { ∑ ses }
a
s
I. INTRODUCTION s =k
Hestenes [1] defined Spinors as a product of two
vectors of i-plane and also proved that they can be treated dim A k = nC k and dim C (E ) = 2 n .
as rotation operators on i-plane. Spinors are also defined as
The operation ‘Geometric Product’ of vectors denoted by
elements of a minimal left ideal [2], [3]. Rotation operators 
play a key role in the determination of orbits of celestial ab , is defined as
bodies such as Satellites and Spacecrafts. Geometric     
Algebra develops a unique and coordinate free method in ab = a . b + a ∧ b ………….. (1)
this regard. There are various techniques using Quaternions,         
Matrices and Euler angles. Geometric Algebra unifies all b a = b . a + b ∧ a = a . b - a ∧b ……….. (2)
these systems into a unique system called Spinors.      
Geometric Algebra not only integrates the above mentioned Here a.b = ab 0 is the scalar part and a ∧b = ab is
2
systems but also establishes the relation among these the bivector part. Elements of Geometric Algebra are called
systems and thus providing a clear passage to move from multivectors as they are in the form A = A 0 + A 1 + ….
one system to the other.
In this paper we study different parameterizations used for + A n . A multivector is said to be even (odd) if
representing Spinors and rotations. We apply the Euler A r = 0 whenever r is odd (even) [5].
angle parameterization of Spinors to solve the problem of
finding the orbital ephemeris of celestial bodies [4]. A k ∈ A k , denotes the k - vector part of the multivector
A.
II. GEOMETRIC ALGEBRA There exists a unique linear map † : C(E) → C(E) that
Let E be an n - dimensional vector space over R,
the field of real numbers and g be a symmetric, positive takes A to A † satisfying
definite, bilinear form g : E × E → R denoted by (i) eS  ei1 ei2 ........ eim then
    
g ( x, y) = x. y x, y  E .There exists a unique Clifford
m (m-1)

Algebra (C (E ),  ) which is a universal algebra in which E eS† = ei m ei m-i ........ei 2 ei1 = (-1) 2 eS
is embedded. Clifford algebra is also called Geometric (ii) (AB)† = B†A †
Algebra as all elements and operation used in it can be ‘ † ’ is called the reversion operator.
interpreted geometrically. We shall identify E with  (E) .
We choose and fix an orthonormal basis 2.1 Euclidean nature of Geometric Algebra
Bn = {e1, e2 ,...., en } for E . 2.1.1 Definition Norm of a multivector To every
Let N   ,2,...., n, n = dim E and S ⊆ N
1 multivector, A  C (E) the magnitude or modulus of A is
1
†
defined as A  A A 2 . With this definition of norm,
0

578 | P a g e

A. Lakshmi, S. N. Hasan / International Journal of Engineering Research and Applications (IJERA)
ISSN: 2248-9622 www.ijera.com
C(E) becomes a Euclidean algebra. The inverse of a non- 2.3.2 Definition Spinor i - space The Spinor
zero element of A of C (E) , is also a multivector, defined i - space ' S 3 ’ is defined as
†
A    
by A 1  2
. { S3 = R / R = x y, x , y ∈ i - space}
A
2.1.2 Definition k  sp a c e Every k- vector Ak
+ +
i
S 3 can also be denoted by C3 (E ) or C3 () . ‘ R ’ is a
multivector, has a scalar part ‘  ’ and a bivector part
determines a k  space .
‘  i1 +  i 2 +  i 3 ’. Spinors do not satisfy commutative
2.2 Euclidean space E3 property with respect to the operation ‘Geometric product’
Denote the Clifford Algebra constructed over a three in view of the above definition (2.2.1).
dimensional vector space E with C3 (E ) .
III. ACTION OF SPINORS ON EUCLIDEAN
For a trivector A3 , designate a unit trivector ‘ i ’ proportional
SPACE - ROTATIONS
to A3 . That is A3  A3 i . Spinors of Euclidean space also can be treated as
rotation operators on i - space of vectors. Rotation
‘ i ’ represents the direction of the space represented by A3 .
 operators can be constructed by considering the group
The set of all vectors x which satisfy the equation action of Spinors by conjugation (Hestenes 1986).

x  i  0 , is called the Euclidean 3- dimensional vector    
Consider a Spinor R = v u , where u and v are unit
space corresponding to ‘ i ’ and is denoted by ‘E3’. E3 is vectors of E. Define a rotation operator R on E as
also be called an i  space , the trivector i is called the  †
R  x   R xR  u v x v u .
    
pseudoscalar of the space as every other pseudoscalar is a
scalar multiple of it. Unlike rotations in two dimensions, rotations in three
Factorize i = 123 where 1, 2 and 3 are dimensions are more complex as (i) the operation to be
considered is the group action by conjugation, giving
orthonormal vectors. They represent a coordinate frame in
 similarity transformations and (ii) the axis about which the
the i  space . x  x1 1  x2 2  x3 3 is a rotation takes place is also to be specified. The resulting
parametric equation of the i  space . x1 , x2 and x3 are vector changes as the axis of rotation changes. This can be
 shown in the following examples.

called the rectangular components of vector x with Rotation of the vector x about the axis  1 , the axis
respect to the basis  1 ,  2 , 3  . i  space of vectors is

perpendicular to the plane  2 3 is represented by the
a 3 – dimensional vector space with the above basis.
2.2.1 Definition Dual of vector i1 = 1i =  23 ; bivector i1   1i   2 3 .

i 2   2i   3 1 ; i3   3i   1 2 Let x i  space of vectors and

x  x1 1  x2 2  x3 3  .
i1 i 2 = - i 3 = - i1 i 2 , i 2 i 3 = - i1 = - i 3 i 2 and
† 
i 3 i1 = - i 2 = - i1 i3 i1 x i1   3 2 x 2 3   3 2 x11  x2 2  x3 3  2 3  x11  x2 2  x3 3 
i
The set of bivectors in C3 (E ) (also denoted by C3 () )is a

3-dimensional vector space with basis i1, i2 , i3  . Every Rotation of the vector x about the axis 2, the axis

bivector B is a dual of a vector b that is B = b i = i b . perpendicular to the plane  3 1 is represented by the
bivector i2   2i   3 1 .
2.3 Spinors of Euclidean space C3 () i † 
2.3.1 Definition Spinor The Geometric product of two
i2 x i2  1 31x 31  1 3 x11  x2 2  x3 3  31  x2 2  x11  x3 3 
vectors in the i - space is called a Spinor and is denoted by
‘R’. Thus 3.1 Composition of Rotations

       
R = x y = x . y + x ∧ y∀ x , y ∈ E .
Two rotations R x  R1† xR1 and
1
 †

‘ R ’ is a multivector, has a scalar part ‘  = x . y ’ and
 R x  R2 xR2 can be combined to give a new rotation
2
     †
a bivector part ‘  i1 +  i 2 +  i 3 = x ∧ y ’. R x  R R x  R2 R1† xR1R2  R3 xR3
†
3 2 1
Note that the order of rotations R R is opposite to that
2 1
of the corresponding Spinor R1 R2 .

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3.2 Proposition Product of two rotations is also a (iv) Spinors can be converted into other forms easily as and
rotation. when required.
Proof: As Geometric product of Spinors is a binary
operation, R3 is also a Spinor. 3.3 Representing a rotation using Euler angles
Rotations are orthogonal transformations. They transform
R3 R3  R2 R1† R1R2  1 as R1† R1  1  R2 R2
† † † one coordinate frame XYZ into another coordinate frame
xyz preserving the angle between them. Euler stated that
Hence R3 is also a unitary Spinor and it can be proved that every rotation can be expressed as a product of two or three
determinant of R is one (Hestenes 1986). As a rotations about fixed axes of a standard basis in such a way
3 that no two successive rotations have the same axis of
consequence of this R is a rotation. rotation. This theorem is known as ‘Euler’s theorem’. Thus
3 every Spinor can be divided further into a product of two or
Product of two rotations in three dimensional spaces is not three Spinors that represent rotations about base vectors.
commutative in general. Euler angles are widely used to represent rotations.
The rotations determined by R R and R R are To represent the rotation using Euler angles, we select an
1 2 2 1 axis for the first rotation among the three axes in the
considered as two different rotations as their sequential sequence. Then according to the rule that no two successive
order is not the same. rotations have the same axis of rotation, we will have two
axes to choose for the second rotation , and for the third,
3.2.1 Spinors of the i - space in half angle form again two options are there to choose a different one from
As a unit vector is treated as a representation of the the previous. Thus we get totally 3 x 2 x 2 = 12 sets of
direction of a vector, a unit bivector can be treated as a Euler angles. Hence, one can represent the same rotation
representation of an angle, which is a relation between two using different sets of Euler angles. If the axes chosen for
directions. the first rotation is same as that of third, such a sequence is
  called a Symmetric set or a Classical set of Euler angles.
i ˆ ˆ
Hence for x , y ∈ C3 () , let x, y be their directions which
are elements of the i - space . 3.4 Finding the Matrix form of the Euler angle sequence
From the definition of a Spinor of the i - space , the Spinor of Spinors
We consider the 3-2-1 symmetric sequence of rotations; the
ˆ ˆ ˆ ˆ ˆ ˆ
R = x y = x . y + x∧ y . Spinor that represents the required rotation is given as a
sequence of three rotations about the base vectors
ˆ
= cos (1 / 2) A + A sin (1 / 2) A is the half angle form of the  1, 2 , 3  is defined by
Spinor R .
R = R Q R ,
S = C + (E ) if dim E ≤3 . ' S ’ can be related to complex
numbers for dim E = 2 and it can be related to Quaternion  
Where R = e 1/ 2 i  3 = cos + 1 2 sin ,
Algebra for dim E = 3.
( )
ˆ 2 2
R = cos (1 / 2) A + A sin (1 / 2) A can also be written as
 
e1 / 2 A . Q = e(1/ 2) i  2 = cos +  31 sin ,
2 2
ˆ x y
Here A = ˆ ∧ ˆ , the bivector representing the plane of  
R = e(1/ 2) i 1 = cos +  2 3 sin
rotation and A gives the magnitude of the angle through 2 2
which the rotation takes place. The new set of axes after rotation are given by
3.2.2 Matrix form of a Spinor
There are different matrices to represent rotations. Instead,
ek  R k  R † k R
the use of Spinors to represent a rotation gives the matrix † † †
elements directly by the formula = R Q R  k R Q R
This can be converted into the matrix form by calculating
e jk =  j . ek =  j . (  k ) .
R
the elements of the matrix [e jk ] given as
There are also other forms such as Exponential form,
Quaternion form etc. The advantages in using Spinor e jk =  j .ek = R k
Algebra as a substitute for all the above algebras
† † †
(i) The coordinate free nature of Spinors. e1 = R1 = R Q R 1R Q R
(ii) The existence of Spinors in every dimension facilitating
to perform rotations in higher dimensional spaces. † † †

(iii) Representation of rotations using Spinors enables us to = R Q ( R 1R )Q R
find the magnitude of the angle of rotation as well as the
† †
orientation of the rotation which is not the case with the = R Q (1(cos + 12 sin ))Q R
matrix representation.

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† † † †
= R Q (1 cos + 2 sin )Q R = R (Q 3Q ) R
†
† †
= R [(Q 1Q ) cos + (Q 2Q ) sin ]R
†
= R [3 (cos  + 31 sin )]R
†
= R [1(cos + 31 sin ) cos + 2 sin ]R
†
= R (3 cos + 1 sin ) R
†
= R (1 cos cos - 3 sin  cos + 2 sin ) R † †
= ( R 3 R ) cos + ( R 1R ) sin 
† † †
= ( R 1R ) cos cos - ( R 3R ) sin  cos + ( R 2 R ) sin  = 3 (cos  + 23 sin ) cos  + 1 sin 

= 3 cos  cos  - 2 sin  cos  + 1 sin 
= 1 cos  cos  - 3 (cos  + 23 sin ) sin  cos  + 2 (cos  + 23 sin ) sin 
= 1 sin  - 2 sin  cos  + 3 cos  cos 

= 1 cos  cos  - 3 cos  sin  cos  + 2 sin  sin  cos  + 2 cos  sin  + 3 sin  sin 
The matrix obtained is as follows

cos  cos  - cos  sin  sin 
= 1(cos  cos ) + 2 (sin  sin  cos  + cos  sin ) + 3 (sin  sin  - cos  sin  cos )
sin  sin  cos  + cos  sin  cos  cos  - sin  sin  sin  - sin  cos 
† † †
sin  sin  - cos  sin  cos  sin  cos  + cos  sin  sin  cos  cos 
e2 = R2 = R Q R 2 RQ R
† † † This sequence is used in aerospace applications and to find
= R Q ( R2 R )Q R orbital ephemeris.
† †
= R Q [2 (cos + 12 sin )]Q R IV EULER ANGLES AND EQUIVALENT
ROTATIONS
† †
The set of Euler angles that represent a particular
= R Q (2 cos - 1 sin )Q R rotation are not unique. Representing the same rotation by
† † †
two different Euler angle sequences gives Equivalent
= R (Q 2Q ) cos - (Q 1Q ) sin ) R rotations. In this paper we use sequences of Spinors in their
matrix form and also in their half angle form in place of
†
rotation sequences to obtain the relationship between the
= R (2 cos - 1(cos + 31 sin ) sin ) R
two sets of angles (i) orbital ephemeris set ( L ,  and
†
= R (2 cos - 1 cos sin  + 3 sin  sin ) R  ) and (ii) orbital parameters (  ,  and  ) set. Thus
established the equivalence between Spinor methods and
† † † Quaternion methods.
= ( R 2 R ) cos - ( R 1R ) cos sin  + ( R 3R ) sin  sin 

= 2 (cos  + 23 sin ) cos  - 1 cos  sin  + 3 (cos  + 23 sin ) sin  sin 

= 2 cos  cos  + 3 sin  cos  - 1 cos  sin  + 3 cos  sin  sin  - 2 sin  sin  sin 

= 1(- cos  sin ) + 2 (cos  cos  - sin  sin  sin ) + 3 (sin  cos  + cos  sin  sin )

† † †
e3 = R3 = R Q R3RQ R
† † †
= R Q ( R 3R )Q R
† †
= R Q (3 )Q R

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Figure 1: ephemeris sequence frame of reference (XYZ, NED frame of reference) through
L = Earth-Latitude of orbiting body the orbital parameters (  ,  and  ). This can be done by
    0 using 3-1-3 sequence of Euler angles.
 = Ephemeris path direction angle
  Earth-Longitude of orbiting body
0  Greenwich with respect to X axis
 = Angle to the orbit ascending node, also called ‘Right
Ascension of the ascending node’. It is the angle subtended
at the center of the Earth from the Vernal Equinox (positive
X axis) to the ascending node ‘N’.
  The angle of inclination. It is the angle between orbital
plane and equatorial plane.
  The argument of the latitude to orbiting body
The position vector of the object is given by the vector
OR . ‘P’ is the point on the surface of the Earth in the
radial direction of the object. The reference frame XYZ is
the Equatorial frame of reference that is the plane that
contains the Earth’s equator. X, Y axes are contained in the Figure 2: Orbital sequence
equatorial plane of the Earth. The Z axis is normal to this
XY plane such that XYZ frame forms a right handed frame 4.1.2 Orbital parameter sequence
of reference. ‘NOR’ is the orbital plane. The trajectory of In this application, the aircraft’s body frame xyz is related
the object is as indicated in the figure 1. ON is the line to the NED reference coordinate frame XYZ (refer fig 3)
segment in which the orbit trajectory and the reference defined as XY plane is the Tangent plane to the Earth
frame intersect each other. pointing towards North and East directions respectively. Z
axis points towards the centre of the Earth (NED frame of
4.1 Matrix method: reference). The positive x axis of the body frame is directed
along the longitudinal axis. The positive y axis is directed
4.1.1 Orbit ephemeris sequence along its right wing and the positive z axis is perpendicular
A tabulation of data of the Earth longitude and Latitude of to the xy plane such that xyz forms a right handed system.
a body, as a function of time is called the orbit ephemeris of These two frames are related by the heading and attitude
the body. sequence of rotations followed by a third and final rotation
In this sequence the body frame (xyz) of the object will be about the newest x axis which is the position vector of the
related to an inertial frame of reference (XYZ) through the spacecraft through an angle  as depicted by the figure 2.
orbital Ephemeris ( L ,     0 ,  ). This requires This requires the 3-1-3 symmetric sequence of rotations; the
the 3-2-1 symmetric sequence of rotations; the Spinor that Spinor that represents the required rotation is given as a
represents the required rotation is given as a sequence of sequence of three rotations about the base vectors is defined
three rotations about the base vectors is defined by by
† † †
†
R  R Q  LR   R Q R  k R Q L R ,
† †
R  R Q R   R Q R k RQ R ,
 
(1/ 2) i 3   Where R = e(1 / 2) i 3  = cos + 12 sin ,
Where R = e = cos + 1 2 sin , 2 2
2 2
 
(1 / 2) i 2 (- L) Q = e(1 / 2) i 1  = cos + 23 sin ,
Q(- L) = e 2 2
 
R  e1 / 2  i  3  cos   1 2 sin
,
(- L) (- L) L L
= cos + 31 sin = cos - 31 sin 2 2
2 2 2 2
The matrix obtained is
(1/ 2) i 1  
R = e = cos +  2 3 sin cos  cos  - sin  cos  sin  - sin  cos  cos  - sin  cos  sin  sin 
2 2
The matrix is obtained is as follows cos  sin  + cos  cos  sin  cos  cos  cos  - sin  sin  - cos  sin 
sin  sin  sin  cos  cos 
(cos L cos ) (- cos L sin ) - sin L
As the orbital sequence and ephemeris sequence represent
= (- sin  sin L cos  + cos  sin ) (cos  cos  + sin  sin L sin ) - sin  cos L the same body frame of reference, they can be equated to
(sin  sin  + cos  sin L cos ) sin  cos  - cos  sin L sin  cos  cos L get the relations between the orbital ephemeris in terms of
In this application we relate the body frame (xyz) of a the orbital elements. Thus we obtain
spacecraft or a near earth orbiting satellite to an inertial

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sin L   sin sin ⇒ cos  cos  = cos  cos  cos  - sin  cos  sin  = cos  cos(  + )....(1)
cos sin  - sin  sin  = cos  cos  sin  + sin  cos  cos  = cos  sin(  + )....( 2)
tan    cos tan
cos
sin  cos  = cos  sin  cos  + sin  sin  sin  = sin  cos(  - )...(3)
cos L sin  (tan  + tan  cos)
tan  = =
cos L cos 1 - tan  cos tan 
sin  cos  = cos  sin  sin  - sin  sin  cos  = sin  sin(  - )...( 4)
4.2 Spinor half angle method
Equating the orbit sequence with Ephemeris sequence
Dividing (3) by (1) and (2) by (4) gives
R Q L R  R Q R
cos(  -  )
, tan  = tan 
cos(  +  )
 
R  e1 / 2  i  3  cos  1 2 sin ; tan  cos(  -  )
2 2 ⇒ = ....(5)
L L tan  cos(  +  )
Q L  e1 / 2  i  2  L   cos   31 sin
2 2 1 sin(  +  )
- tan  =
  tan  sin(  -  )
R  e1 / 2  i  1  cos   2 3 sin ;
2 2
sin(  +  ) sin(  +  )
1 / 2  i  1   ⇒ - tan  tan  = = ...(6)
R  e  cos  1 2 sin ; sin(  -  ) sin(  - )
2 2
  Multiplying (5) and (6) gives
Q = e (1 / 2) i  2 = cos + 1 2 sin and
2 2 cos(  -  ) sin(  +  ) tan(  +  )
  - tan 2  = = .....( 7)
R = e (1 / 2) i 3 = cos + 1 2 sin sin(  -  ) cos(  +  ) tan(  -  )
2 2
dividing (6) by (5) gives
RQ L R = RQ R ⇒
sin(  +  ) cos(  +  ) sin 2( +  )
RQ- L = RQ RR -1 = RQ RR- = RQ R- = RQ R - tan 2  = =
sin(  -  ) cos(  -  ) sin 2( -  )
⇒ R Q-L = R Q R
1 - tan 2 
but cos 2 =
To avoid half angles 1 + tan 2 
 = 2, L = 2,  = 2,  = 2,   =  = 2,  = 2 sin 2( -  ) + sin 2( +  ) tan 2
= = -
R Q- L = R Q R ⇒ R2Q-2 = R2 Q2 R2 sin 2( -  ) - sin 2( +  ) tan 2
Gives tan 2 = - cos 2 tan 2
(cos  +  2 3 sin )(cos  + 31 sin ) substituting   2 ,       2 ,  2
= (cos  + 1 2 sin )(cos  +  2 3 sin )(cos  + 1 2 sin ) gives

⇒ cos  cos  + 12 sin  sin  + 23 sin  cos  - 31 sin  cos  tan = - cos tan

tan  - tan 
[
= (cos  + 1 2 sin ) cos  cos  + 1 2 cos  sin  +  23 sin  cos  + 31 sin  sin  ] - cos tan  = tan  = tan( - ) =
1 + tan  tan 
After simplification we get
cos tan + tan 
= cos  cos  cos  + 1 2 cos  cos  sin  +  2  3 cos  sin  cos  +  31 cos  sin  sin  + tan =
1 - tan  cos tan
1 2 sin  cos  cos  - sin  cos  sin  + 1 3 sin  sin  cos  +  2  3 sin  sin  sin  Similarly we get the other relations.

= cos  cos  cos  - sin  cos  sin  + 12 (cos  cos  sin  + sin  cos  cos ) 4.3 Singularities in Euler sequences
+ 23 (cos  sin  cos  + sin  sin  sin ) + 31(cos  sin  sin  - sin  sin  cos ) A singularity occurs in every sequence of Euler angles. For
example in the 3-1-3 sequence,

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e sin  cos  sin 
tan  = 12 = = . Singularity occurs at
e11 cos  cos  cos 

= .
2
But Euler angles work well for small angles or Infinitesimal
rotations. Hence in the problems like finding the orientation
of the spacecraft and tracking an aero plane employment of
Euler angles is advantageous.

V. DISCUSSIONS
Even though there are many representations for
rotations in 3-D namely Quaternions, Spinors, Euler axis-
angle and sequences of Euler angles, each one can be
transformed into the other. Each system has its own merits
or demerits over the other systems. Singularities occur for
every set of Euler angles but they work well in infinitesimal
rotations. Comparatively, the other methods of using
Quaternions and Euler axis and angle for representing
rotations provide a better procedure as they need four
parameters to be defined and they can be converted into any
other convenient form depending upon the information
available. In the case of Spinors, whose parametric form is

given as s    i , the number of parameters reduce

further as the parameter  is not independent of  .

VI. CONCLUSIONS
The technique of using Spinors can replace the
conventional methods and also provide a richer formalism.
If large number of rotations is involved Spinor matrix
methods are advantageous over the Spinor half angle
method.

ACKNOWLEDGMENTS
I would like to thank Professor S. Uma Devi, dept
of Engineering Mathematics, Andhra University for her
valuable and encouraging suggestions while writing the
paper.

REFERENCES
Books:
[1] Hestenes D., New Foundations for Classical
Mechanics, D. Reidel Publishing, 1986.
[2] Hestenes D., Space-Time Algebra, Gordon and
Breach (1966).
[3] Lounesto P., Clifford Algebras and Spinors, 2nd
Edition, Cambridge University Press (2001).
[4] Kuipers J.B., Quaternions and Rotation Sequences,
Princeton University Press, Princeton, New Jersey
(1999).

Thesis:
Hasan S. N., Use of Clifford Algebra in Physics:
A Mathematical Model, M.Phil Thesis, University
of Hyderabad, India (1987).

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