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be identified without inconsistencies. Sometimes, the angular parameter
ω
is called
ambiguity
.
As it is outlined in Boxes
3.18
and
3.19
,the
identity postulate
(
3.90
) leads to trigonometric
equations for
(given
ω, I
,and
Ω
). Accordingly, we
are able to transform forward and backward between the oblique frames of reference subject to
[
E
1
,
E
2
,
E
3
]
∗
=[
e
1
0
,
e
2
0
,
e
3
0
]
∗
,
[
E
1
,
E
2
,
E
3
]
∗
=[
e
1
,
e
2
,
e
3
]
∗
. Compare with Fig.
3.10
,
which illustrates the commutative diagram for oblique frames of reference. The essential formulae
for transforming
E
→
{
ω, I, Ω
}
(given
λ
0
and
φ
0
)and
{
λ
0
,φ
0
}
E
are collected in Lemmas
3.4
,
3.5
, and Corollary
3.6
.
These transformation formulae are summarized and numerically tested in the following section.
e
0
as well as
e
0
→
Fig. 3.10.
Commutative diagram for oblique frames of reference
⎡
⎤
E
1
E
2
E
3
⎣
⎦
=
⎡
⎤
⎦
=R
2
π
⎡
⎤
E
1
E
2
E
3
φ
0
R
3
(
λ
0
)
e
1
e
2
e
3
⎣
⎣
⎦
=
=R
3
(
ω
)R
1
(
I
)R
3
(
Ω
)
2
−
(3.90)
⎡
⎤
e
1
0
e
2
0
e
3
0
⎣
⎦
=
Box 3.18 (Transformation between oblique frames of reference: first design, second design).
(i)
E → E
⎡
⎤
⎡
⎤
E
1
E
2
E
3
E
1
E
2
E
3
⎣
⎦
=R
3
(
ω
)
⎣
⎦
,
(3.91)
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