Biomedical Engineering Reference
In-Depth Information
2
4
3
5
Q
1I
Q
1II
Q
1III
Q
1IV
Q
1V
Q
1VI
Q
2I
Q
2II
Q
2III
Q
2IV
Q
2V
Q
2VI
Q
3I
Q
3II
Q
3III
Q
3IV
Q
3V
Q
3VI
Q
¼
Q
4I
Q
4II
Q
4III
Q
4IV
Q
4V
Q
4VI
Q
5I
Q
5II
Q
5III
Q
5IV
Q
5V
Q
5VI
Q
6I
Q
6II
Q
6III
Q
6IV
Q
6V
Q
6VI
2
4
3
5
p
Q
1II
Q
1III
p
Q
1I
Q
1III
p
Q
1I
Q
1II
Q
1I
Q
1II
Q
1III
p
Q
2II
Q
2III
p
Q
2I
Q
2III
p
Q
2I
Q
2II
Q
2I
Q
2II
Q
2III
p
Q
3II
Q
3III
p
Q
3I
Q
3III
p
Q
3I
Q
3II
Q
3I
Q
3II
Q
3III
¼
p
Q
2I
Q
3I
p
Q
2II
Q
3II
p
Q
2III
Q
3III
Q
2II
Q
3III
þ
Q
3II
Q
2III
Q
2I
Q
3III
þ
Q
3I
Q
2III
Q
2I
Q
3II
þ
Q
3I
Q
2II
p
Q
1I
Q
3I
p
Q
1II
Q
3II
p
Q
1III
Q
3III
Q
1II
Q
3III
þ Q
3II
Q
1III
Q
1I
Q
3III
þ Q
3I
Q
1III
Q
1I
Q
3II
þ Q
3I
Q
1II
p
Q
1I
Q
2I
p
Q
1II
Q
2II
p
Q
1III
Q
2III
Q
1II
Q
2III
þ Q
2II
Q
1III
Q
1I
Q
2III
þ Q
2I
Q
1III
Q
1I
Q
2II
þ Q
1II
Q
2I
(A.167)
To see that
Q
is an orthogonal matrix in six dimensions requires some algebraic
manipulation. The proof rests on the orthogonality of the three-dimensional
Q
:
Q
T
¼ Q
T
;) Q
Q
¼ 1
Q
T
Q
T
Q
¼
Q
¼
1
:
(A.168)
In the special case when
Q
is given by
2
3
cos
a
sin
a
0
4
5
;
Q
¼
sin
a
cos
a
0
(A.169)
0
0
1
Q
has the representation
2
4
p
cos
3
5
cos
2
sin
2
a
a
0
0
0
a
sin
a
p
cos
sin
2
cos
2
a
a
0
0
0
a
sin
a
0
0
1
0
0
0
Q
¼
0
0
0
cos
a
sin
a
0
0
0
0
sin
a
cos
a
0
p
cos
p
cos
cos
2
sin
2
a
sin
a
a
sin
a
0
0
0
a
a
(A.170)
It should be noted that while it is always possible to find
Q
given
Q
by use of
(A.167), it is not possible to determine
Q
unambiguously given
Q
. Although
Q
is
uniquely determined by
Q
, the reverse process of finding a
Q
given
Q
is not unique
in that there will be a choice of sign necessary in the reverse process. To see this
nonuniqueness note that both
Q
¼
1. There are 9
components of
Q
that satisfy 6 conditions given by (A.168)
1
. There are therefore
only three independent components of
Q
. However, there are 36 components of
Q
that satisfy the 21 conditions given by (A.168)
2
and hence 15 independent
1
correspond to
Q
¼
1
and
Q
¼
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