Geography Reference
In-Depth Information
Indeed, for an ellipsoid-of-revolution, there exist also three aspects which are called
normal,
oblique
,and
transverse
. Again, these aspects generate special ellipsoidal coordinates of the
ellipsoid-of-revolution, taking into account
one Killing vector of symmetry
. The oblique frame
of reference of
A
1
,A
2
2
O
E
is based upon the centric oblique plane
P
, which intersects the ellipsoid-
2
O
A
1
,A
2
of-revolution and passes the origin
O
. Such an oblique plane
P
intersects
E
in an elliptic
oblique equator, also called
meta-equator
, which is oriented by two Kepler elements
, called
the longitude
Ω
of the ascending node and the inclination
I
. The transverse frame of reference is
obtained by choosing an inclination
I
=
π/
2.
{
Ω, I
}
3-41 The Direct and Inverse Transformations of the Normal Frame
to the Oblique Frame
along the principal axes of the
ellipsoid-of-revolution of semi-major axis
A
1
and semi-minor axis
A
2
:
Let us orientate a set of orthonormal base vectors
{
E
1
,
E
2
,
E
3
}
2
3
|
(
X
2
+
Y
2
)
/A
1
+
Z
2
/A
2
=1
,A
1
>A
2
,A
1
∈
R
+
,A
2
∈
R
+
E
A
1
,A
2
:=
{
X
∈
R
}.
(3.100)
Against this frame of reference
{
E
1
,
E
2
,
E
3
|O}
at the origin
O
, we introduce the oblique frame
of reference
{
E
1
,
E
2
,
E
3
|O}
at the origin
O
built on an alternative set of orthonormal base
vectors which are related by means of a rotation:
⎡
⎤
⎡
⎤
E
1
E
2
E
3
E
1
E
2
E
3
⎣
⎦
=R
1
(
I
)R
3
(
Ω
)
⎣
⎦
.
(3.101)
This rotation is illustrated by Fig.
3.8
. The rotation around the 3 axis is denoted by
Ω
,theright
ascension of the ascending node, while the rotation around the intermediate 1 axis is denoted by
I
, the inclination. R
1
and R
3
are orthonormal matrices such that the following relation holds:
⎡
⎤
⎡
⎤
cos
Ω
sin
Ω
0
−
sin
Ω
cos
Ω
0
0
10 0
0cos
I
sin
I
0
−
sin
I
cos
I
⎣
⎦
⎣
⎦
R
1
(
I
)R
3
(
Ω
)=
0 1
⎡
⎤
cos
Ω
sin
Ω
0
−
sin
Ω
cos
I
cos
Ω
cos
I
sin
I
sin
Ω
sin
I
⎣
⎦
,
=
(3.102)
−
cos
Ω
sin
I
cos
I
3
×
3
.
R
1
(
I
)R
3
(
Ω
)
∈
R
Accordingly, the following vector equation defines a representation of the placement vector
X
in
the orthonormal bases
{
E
1
,
E
2
,
E
3
}
and
{
E
1
,
E
2
,
E
3
}
, respectively:
3
3
E
i
X
i
.
E
i
X
i
=
E
1
X
+
E
2
Y
+
E
3
Z
=
E
1
X
+
E
2
Y
+
E
3
Z
=
X
=
(3.103)
i
=1
i
=1
Note that the corresponding Cartesian coordinate transformations are dual to the following sys-
tems of coordinate transformations:
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