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6 Quaternion
6.1 Introduction
We know that, complex numbers can be used to represent operations such as rotation in 2D.
Suppose that, I denote the pure imaginary number let i denote the pure imaginary number
i2 =-1 and we know that the euler’s identity ., eiθ =cosθ + i sinθ
such that
We can write the polar representation of a complex number c as
c=a + ib = reiθ where,
r=
a 2  b2
and θ = tan-1 b/a
If we rotate c about the origin by Φ to c’, then we can find c’ using rotation matrix or we can
use polar representation
c’ = rei(θ+Φ) = r eiθ eiΦ
Where eiΦ is rotation operator in complex plane and is alternative to transformations
However, in 3D it is difficult, because to specify a rotation about the origin we need to
specify
direction (a vector)
and
amount of rotation (a scalar), one Solution is, use a
representation that consists of both a vector and scalar such a representation is Quaternion.
6.2 Quaternion
Quaternion is an interesting mathematical concept with a deep relationship with the
foundations of algebra and number theory. It was discovered by Hamilton in the year
1843.Quaternions are an extension of complex numbers that most useful for representing the
orientation.
A quaternion consists of Scalar part and vector part , Let a be the quaternion and is defined
as
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a=(q0, q1, q2, q3)
Where q=(q1, q2, q3)
i.e.,
a = ( q0 , q)
so, we can call Quaternion is an extension to complex numbers
6.2.1 Operations on Quaternions
The operations among quaternions are based on the use of 3 complex numbers I, j and k with
the properties
i 2  j 2  k 2  ijk  1
i  jk  kj
j  ki  ik
k  ij   ji
These three complex numbers are analogous to the unit vectors in 3D, and we can write q as
q = q1i + q2j + q3k
Quaternion addition and multiplication can be defined using the dot and cross product of
vectors:
let b = ( p0, p) and a = (q0, q) are the two quaternion,
then
a + b = (p0 + q0 , p + q)
a*b = (p0q0 – q.p , q0p + p0q + q X p )
If p0=q0=0 then a*b = (q.p , q X p ) Note that
Quaternion multiplication is not commutative, so a * b ≠ b * a
a•b = (q0p0 + q•p)
We can also define the magnitude of quaternion as
a  q02  q12  q22  q32  q02  q.q
2
Multiplicative identity for the quaternion is (1 , 0 ) and the inverse of a quaternion is given
1
a 1 
( q0 ,  q )
by
2
a
a  q02  q12  q 22  q32  1
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unit quaternion is denoted by
6.2.2 Quaternion and Rotation
Quaternions can also be used to carry out the rotation operations. Suppose , we use the vector
part of the quaternion to represent the point in space, say
p = ( 0, p)
thus p = ( x, y ,z ) gives the location of point.
Consider the quaternion
r is unit quaternion.
Therefore

 

r   cos , sin v  where v is the unit length, so quaternion
2
2 


 

r 1   cos , sin v 
2
2 

If we consider the quaternion product of the quaternion p that represent a point with r, we get
the quaternion
p |  cos 2

2
p|  rpr 1 this is of the form (0 , p| ) where
p  sin2

2
( p.v)v  2 sin

2
cos

2
(v  p )  sin

2
(v  p )  v
p| is the representation of a point. We can say that p| is the result of rotating the point p by an
angle θ degree about the vector v.
If we compare the general rotation as we are discussed earlier, the result is same and if we
count the number of operations, quaternions are faster.
Consider an Example, the rotation about the z - axis by θ with a fixed point at the origin.
The desired unit vector v is (0 , 0 , 1 )
Then the quaternion is
r  cos

2
 sin

2
(0,0,1)
Rotation of an arbitrary point p = (x, y, z) yields the quaternion
p|  rpr 1  r (0, p)r |  (0, p| )
Where
p|  ( x cos   y sin  , x sin   y cos  , z )
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This is the expected result with fewer operations
If we consider a sequence of rotations about the coordinate axis that in matrix form yields the
R  Rx ( x ) R y ( y ) Rz ( z ) Instead of this, we can use product of the
matrix
corresponding quaternions to form rx.ry.rz (Quaternions can be used only for rotations )
6.2.3 Convert quaternion to matrix form.
We can use
p |  rpr 1
to find the elements of the homogeneous coordinate rotation
matrix embedded in M, using the following matrix
M=
2
2
1  2q 2  2q3
 2q q 2q q
0
3
 1 2
 2q 1 q 3 2q 0 q 2

0

if
2q 1 q 2 2q 0 q 3
1  2q12  2q32
2q 2 q 3 2q 0 q 1
0
2q 1 q 3 2q 0 q 2
2q 2 q 3 2q 0 q 1
1  2q12  2q 22
0
0
0
0

1

 

r   cos , sin
v
2
2 

Then, the rotation matrix R is


 

 

1  2 sin 2 (v y2  v z2 )
2 sin 2 v x v y  2 cos sin v z 2 sin 2 v x v z  2 cos sin v y

2
2
2 2
2
2 2

 
 
2 
2 
2
2
2 
1  2 sin (v x  v z )
2 sin v y v z  2 cos sin v x
2 sin v x v y  2 cos sin v z
2
2 2
2
2
2 2








2 sin 2 v v  2 cos sin v 2 sin 2 v v  2 cos sin v
1  2 sin 2 (v x2  v y2 )
x z
y
y z
x

2
2 2
2
2 2
2

0
0
0
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
0

0

0

1
4