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conventional equatorial frame of reference and in the oblique equatorial frame of reference. (iii)
We finally derive the forward as well as backward equations of transformation between them.
Solution (the first problem).
The first problem can be solved (i) by representing the placement vector
x
(
λ
0
,φ
0
,r
)intermsof
the chosen spherical coordinates
{
λ
0
,φ
0
,r
}
, (ii) by computing the triplet of partial derivatives
{
D
λ
0
x
,D
φ
0
x
,D
r
x
}
, which are normalized by the Euclidean norms
D
λ
0
x
,
D
φ
0
x
,and
D
r
x
,
leading to the triplet
,
and these three triplet terms are called
South, East
,and
Vertical
. Finally, we relate the base
vectors
−
e
φ
0
:=
D
φ
0
x
/
D
φ
0
x
,
+
e
λ
0
:=
D
λ
0
x
/
D
λ
0
x
,and+
e
r
:=
D
r
x
/
D
r
x
{
e
1
0
,
e
2
0
,
e
3
0
|
x
0
}
:=
{
e
λ
0
,
e
φ
0
,
e
r
|
x
0
}
to
{
e
1
,
e
2
,
e
3
|O}
. This is outlined in Box
3.3
.
For geometrical details, consult Figs.
3.6
and
3.7
.
End of Solution (the first problem).
Solution (the second problem).
The second problem, the representation of the placement vector
x
in the orthonormal equa-
torial frame of reference
{
e
1
,
e
2
,
e
3
|O}
at the origin
O
as well as in the oblique frame of ref-
erence
{
e
1
0
,
e
2
0
,
e
3
0
|O}
(which is called
meta-equatorial
)attheorigin
O
in terms of spherical
coordinates
, is solved by forward and
backward transformations. This is outlined in Boxes
3.3
and
3.4
. For geometrical details, consult
again Figs.
3.6
and
3.7
. Here, we meet the particular problem to
parallel transport
the oblique
frame of reference
{
e
1
0
,
e
2
0
,
e
3
0
|
x
(
λ
0
,φ
0
,r
)
}
(which is defined at the point
x
(
λ
0
,φ
0
,r
)) in the
Euclidean sense from (
λ
0
,φ
0
,r
) to the origin
O
in order to generate the centric frame of reference
{
e
1
0
,
e
2
0
,
e
3
0
|O}
.
{
λ, φ, r
}
as well as in meta-spherical coordinates
{
α,β,r
}
End of Solution (the second problem).
Box 3.3 (Establishing an oblique frame of reference (meta-equatorial) of the sphere).
(i) Placement vector towards the meta-North Pole:
x
(
λ
0
,φ
0
,r
)=
e
1
r
cos
φ
0
cos
λ
0
+
e
2
r
cos
φ
0
sin
λ
0
+
e
3
r
sin
φ
0
.
(3.37)
(ii) Jacobi map:
D
λ
0
x
=
−
e
1
r
cos
φ
0
sin
λ
0
+
e
2
r
cos
φ
0
cos
λ
0
,
D
φ
0
x
=
−
e
1
r
sin
φ
0
cos
λ
0
−
e
2
r
sin
φ
0
sin
λ
0
+
e
3
r
cos
φ
0
,
(3.38)
D
r
x
=+
e
1
cos
φ
0
cos
λ
0
+
e
2
cos
φ
0
sin
λ
0
+
e
3
sin
φ
0
.
(iii) Meta-equatorial (oblique) frame of reference:
{
South, East, Vertical
}
:=
:=
−
D
φ
0
x
||
=:
D
λ
0
x
D
r
x
,
,
(3.39)
D
φ
0
x
D
λ
0
x
D
r
x
=:
{
e
1
0
,
e
2
0
,
e
3
0
|
[
λ
0
,φ
0
,r
]
}
,
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