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The set of hypercomplex numbers is defined as
| ![\begin{displaymath}
q= q_0+\sum_{l=1}^{n}i_lq_l \quad q_l \in \Re.\end{displaymath}](img1.gif) |
(1) |
Hypercomplex numbers
define a n+1-dimensional complex space with il orthonormal to im, for l
m. For all cases presented in this
paper l will be limited to 3. Such numbers are quaternions, which
can be represented as
q=q0+iq1+jq2+kq3, where i,j,k are imaginary numbers that
satisfying the following relations:
| ![\begin{displaymath}
ij=-ji=k,~~~\text{and}~~~i^2=j^2=k^2=-1.\end{displaymath}](img3.gif) |
(2) |
Table 1:
The multiplication table for quaternion algebra.
|
1 |
i |
j |
k |
1 |
1 |
i |
j |
k |
i |
i |
-1 |
k |
-j |
j |
j |
-k |
-1 |
i |
k |
k |
j |
-i |
-1 |
The multiplication table for quaternion unit vectors is shown in
Table 1.
With these definitions, quaternionic addition between two quaternions,
q and p,
can be defined as
| ![\begin{eqnarray}
q+p~=~(q_0+iq_1+jq_2+kq_3)+(p_0+ip_1+jp_2+kp_3)\nonumber\ =~(q_0+p_0)+i(q_1+p_1)+j(q_2+p_2)+k(q_3+p_3),\end{eqnarray}](img4.gif) |
|
| (3) |
and multiplication as
| ![\begin{eqnarray}
qp~=~(q_0+iq_1+jq_2+kq_3)(p_0+ip_1+jp_2+kp_3)\nonumber\ =~(q_0...
...\nonumber\ +~k(q_0p_3+q_3p_0+q_1p_2-q_2p_1).~~~~~~~~~~~~~~~~~~~~~\end{eqnarray}](img5.gif) |
|
| |
| |
| |
| (4) |
Notice that multiplication in equation 4, is not
commutative due to the quaternionic algebra rules defined in table
1.
Quaternions are often separated into two parts, q0 and
, respectively called the scalar and vector
part of the quaternion.
Using this definition, the conjugate of q,
, is
| ![\begin{displaymath}
\bar{q}=q_0-\textbf{q}=q_0-iq_1-jq_2-kq_3\end{displaymath}](img8.gif) |
(5) |
and the norm of a quaternion is defined by
| ![\begin{displaymath}
\mid\mid q \mid\mid =\sqrt{q\bar{q}}=\sqrt{{q_0}^2+{q_1}^2+{q_2}^2+{q_3}^2}.\end{displaymath}](img9.gif) |
(6) |
It is useful to formulate a polar representation of the
quaternion, as this will be the primary notation throughout this
paper. For any complex number,
, the argument or
phase-angle is defined as atan2(b,a). If
is written it
the form
, then
is the
phase (argument) of
, denoted arg(
)=
.Quaternions contain three complex subfields and, correspondingly, three
phases that are the projections onto the i, j, or k-complex plane,
| ![\begin{eqnarray}
\phi=\text{arg}_{\text{i}}(\textbf{z})=\text{atan2}(q_1,q_0)~\...
...)~\ \psi=\text{arg}_{\text{k}}(\textbf{z})=\text{atan2}(q_3,q_0).\end{eqnarray}](img14.gif) |
(7) |
| (8) |
| (9) |