Geography Reference
In-Depth Information
Let us now take care of the
polar coordinate
and base our analysis on the transformation of
reference frames, in particular, on the orthonormal Euclidean triad, corotating with the Earth,
called
, and on the moving frame, called
South, East, Vertical
, an orthonormal triad
in an astronomical orientation, namely
{
E
1
,
E
2
,
E
3
}
{
E
1
∗
,
E
2
∗
,
E
3
∗
}
:seeBox
8.9
.Wehereusethesymbol
of a star to identify the antipolar star orientation.
Γ
Gr
refers to the
gravity vector
at Greenwich,
while
denotes the
global rotation vector
of the Earth. By contrast,
E
1
∗
refers to the South unit
vector,
E
2
∗
refers to the East unit vector, and
E
3
∗
completes the orthonormal triad as the local
vertical vector. The Euler rotation matrix
R
E
(
Λ
0
,Φ
0
,
0) and the rotation matrices
R
3
(
Λ
0
)and
R
2
(
π/
2
Ω
−
Φ
0
) are provided by (
8.97
). In Fig.
8.12
,
E
1
∗
and
E
2
∗
are compactly illustrated.
Box 8.9 (
{
E
1
,
E
2
,
E
3
}
and
{
E
1
∗
,
E
2
∗
,
E
3
∗
}
, Euler rotation matrix, Euler parameters).
Transformation of fixed and moving frame
(
{
E
1
,
E
2
,
E
3
}
versus
{
E
1
∗
,
E
2
∗
,
E
3
∗
}
):
∂
X
/∂Φ
∂
X
/∂Φ
∂
X
/∂Λ
∂
X
/∂Λ
E
3
∗
;=+
∂
X
/∂H
E
1
∗
:=
−
E
2
∗
:= +
(8.93)
∂
X
/∂H
(South)
,
(East)
,
(Vertical)
,
⎡
⎤
⎡
⎤
E
1
∗
E
2
∗
E
3
∗
E
1
E
2
E
3
⎣
⎦
=
R
E
(
Λ
0
,Φ
0
,
0)
⎣
⎦
,
(8.94)
E
3
,
E
2
:=
−
Γ
Gr
×
Ω
−
Γ
E
1
:=
E
2
×
,
(8.95)
×
Ω
Gr
Ω
Ω
E
3
:=
.
Euler rotation matrix:
R
E
(
Λ
0
,Φ
0
,
0) :=
R
3
(0)
R
2
(
π/
2
−
Φ
0
)
R
3
(
Λ
0
)
,
(8.96)
⎡
⎤
⎡
⎤
cos
Λ
0
sin
Λ
0
0
−
sin
Φ
0
0
cos
Φ
0
01 0
cos
Φ
0
0 in
Φ
0
−
⎣
⎦
,
R
2
(
π/
2
⎣
⎦
.
R
3
(
Λ
0
)=
sin
Λ
0
cos
Λ
0
0
0
−
Φ
0
)=
(8.97)
0
1
Euler parameters:
⎡
⎤
cos
Λ
0
sin
Φ
0
sin
Λ
0
sin
Φ
0
−
cos
Φ
0
⎣
⎦
.
R
E
(
Λ
0
,Φ
0
,
0) =
sin
Λ
0
cos
Λ
0
0
cos
Λ
0
cos
Φ
0
sin
Λ
0
cos
Φ
0
sin
Φ
0
−
(8.98)
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