Geography Reference
In-Depth Information
We here note that
A
1
is the semi-major axis,
A
2
is the intermediate semi-major axis
A
2
<A
1
,
and finally
A
3
is the semi-minor axis
A
3
<A
2
<A
1
. The eccentricity of the intersection ellipses
is given by (
J.77
)inthe
{
1
,
2
}
=
{
X,Y
}
plane and by (
J.78
)inthe
{
1
,
3
}
=
{
X,Z
}
plane.
E
12
=
1+
A
2
/A
1
,
(J.77)
E
13
=
1+
A
3
/A
1
.
(J.78)
Furthermore, we here point out that
elliptic heights
on top of a triaxial ellipsoid can be expressed
by (
J.79
) subject to (
J.80
).
X
=
A
1
W
+
H
(
L, B
)
cos
B
cos
L,
Y
=
A
1
(1
−
E
12
)
W
+
H
(
L, B
)
cos
B
sin
L,
(J.79)
Z
=
A
1
(1
−
E
13
)
W
+
H
(
L, B
)
sin
B,
W
=
1
E
13
sin
2
B
E
12
cos
2
B
sin
2
L.
−
−
(J.80)
J-32 Position, Orientation, Form Parameters: Case Study Earth
Let us here assume that we refer to the
geoid
as the equipotential surface of gravity close to
the
Mean Sea Level
fitted to the triaxial ellipsoid. Actually, with respect to the biaxial ellipsoid,
fitting the triaxial ellipsoid is 65% better. The difference of axes in the equatorial plane
A
1
−−A
2
rounds up to 69m. With respect to the center of the best fitting triaxial ellipsoid, the mass center
of the Earth is displaced approximately by 11-15m. The orientation of the triaxial axes with
respect to the principal axes is given by
A
1
=6
,
378
,
173
.
435 m (14
◦
53
42
westerly of the Greenwich meridian)
,
A
2
=6
,
378
,
103
.
9m
, A
3
=6
,
356
,
754
.
4m
,
(J.81)
A
3
=21
,
419
.
0m
.
The reciprocal polar flattening is provided by
1
α
13
=
A
1
−
A
2
=69
.
5m
, A
1
−
A
1
A
3
= 297
.
781194
.
(J.82)
A
1
−
The polar flattening is provided by
α
13
=3
.
35817
×
10
−
3
,
1
− α
13
=0
.
99664383
.
(J.83)
The reciprocal equatorial flattening is provided by
1
α
12
:=
A
1
A
1
−
A
2
=91
,
650
.
826
.
(J.84)
The equatorial flattening is provided by
α
12
=1
.
091097
10
−
3
,
1
×
−
α
12
=0
.
999989089
.
(J.85)
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