Geography Reference
In-Depth Information
X
=
X
cos
Ω
+
Y
sin
Ω, Y
=
Z, Z
=
−
−
X
sin
Ω
+
Y
cos
Ω.
(H.6)
Example H.1 (An example:
Ω
= 270
◦
(cos
Ω
=0
,
sin
Ω
=
−
1)).
As an example, let us choose
Ω
= 270
◦
(cos
Ω
=0
,
sin
Ω
=
−
1) so that we obtain (
H.7
)or(
H.8
),
identified as western, southern, and Greenwich.
E
1
=
−
E
2
,
E
2
=
−
E
3
,
E
3
=
E
1
,
(H.7)
X
=
Y, Y
=
Z, Z
=
X.
−
−
(H.8)
2
(
X,Y
) directed towards Greenwich.
Indeed, the meta-equator is elliptic in the plane
P
End of Example.
The example may motivate the exotic choice of
Ω
= 270
◦
,I
= 270
◦
. Figure
H.2
illustrates the
special transverse frame of reference
{
E
1
,
E
2
,
E
3
;0
}
.
Fig. H.2.
The frame of reference
{
E
1
,
E
2
,
E
3
;0
}
and the transverse frame of reference
{
E
1
,
E
2
,
E
3
;0
}
for the
special choice
Ω
= 270
◦
,
I
= 270
◦
A
1
,A
2
While (
H.1
) is a representation of the biaxial ellipsoid
E
of semi-major axis
A
1
and semi-
minor axis
A
2
in terms of
{
X,Y,Z
}
coordinates along the orthonormal basis
{
E
1
,
E
2
,
E
3
}
,(
H.9
)
is the analogous representation of
E
A
1
,A
2
in terms of
{X
,Y
,Z
}
along the transverse orthonormal
basis
{
E
1
,
E
2
,
E
3
}
.
2
3
|
(
X
2
+
Z
2
)
/A
1
+
Y
2
/A
2
=1
,
R
+
+
E
A
1
,A
2
=
{
X
∈
R
A
1
>A
2
∈
R
}.
(H.9)
The meta-equator
X
2
/A
1
+
Y
2
/A
2
= 1 is elliptic, while the meta-meridian
X
2
+
Z
2
=
A
1
is
circular. These properties of the meta-equator and the meta-meridian motivate the introduction
of surface normal ellipsoidal meta-longitude/meta-latitude
{
A
∗
,B
∗
}
, namely in order to parame-
A
1
,A
2
terize
accordingto(
H.10
) in contrast to (
H.11
) with respect to surface normal ellipsoidal
longitude/latitude
E
Λ
∗
,Φ
∗
}
and with respect to relative eccentricity
E
2
=(
A
1
−
A
2
)
/A
1
.
{
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