Geography Reference
In-Depth Information
X
1
=
X
1
cos
Ω
X
2
sin
Ω
cos
I
+
X
3
sin
Ω
sin
I
=:
X,
X
2
=
X
1
sin
Ω
+
X
2
cos
Ω
cos
I
−
X
3
cos
Ω
sin
I
=:
Y,
−
(3.104)
X
3
=
X
2
sin
I
+
X
3
cos
I
=:
Z,
or
X
1
=+
X
1
cos
Ω
+
X
2
sin
Ω
=:
X
,
X
2
=
X
1
sin
Ω
cos
I
+
X
2
cos
Ω
cos
I
+
X
3
sin
I
=:
Y
,
−
(3.105)
X
3
=+
X
1
sin
Ω
sin
I
X
2
cos
Ω
sin
I
+
X
3
cos
I
=:
Z
.
−
3-42 The Intersection of the Ellipsoid-of-Revolution and the Central
Oblique Plane
In order to obtain an oblique equatorial plane, namely an oblique equator, we intersect the
ellipsoid-of-revolution
E
A
1
,A
2
of semi-major axis
A
1
and semi-minor axis
A
2
and the central oblique
plane
L
(two-dimensional linear manifold through the origin
O
). Subsequently, the oblique
equatorial plane as well as its normal enables us to establish an oblique
quasi-spherical coordinate
system
. Our first result is summarized in Corollary
3.7
.
2
O
2
A
1
,A
2
2
O
Corollary 3.7 (The intersection of
E
and
L
).
The intersection of the ellipsoid-of-revolution
E
A
1
,A
2
and the c
entral o
bli
que plane
L
2
O
is the ellipse
of semi-major axis
A
1
=
A
1
and semi-minor axis
A
2
=
A
1
√
1
E
2
/
√
1
−
−
E
2
cos
2
I
with respect
to the relative eccentricity
E
2
:= (
A
1
− A
2
)
/A
1
:
1
A
E
:=
,A
1
2
:=
x
∈
R
2
x
A
1
y
A
2
=1
,A
1
=
A
1
,A
2
=
A
1
√
1
− E
2
/
√
1
− E
2
cos
2
I,A
1
>A
2
.
+
(3.106)
End of Corollary.
For short, the proof of Corollary
3.7
has been given in
Engels and Grafarend
(
1995
, pp. 42-43).
Compare with Fig.
3.8
, which illustrates the oblique elliptic equator as well as the orthogonal
projection of a point
X
E
A
1
,A
2
onto the oblique equatorial plane, respectively. Note that in a
following section, the oblique equator is used to establish the following elliptic cylinder:
∈
2
A
C
:=
,A
:=
X
∈
R
3
1
2
.
X
2
A
1
+
Y
2
=1
,Z
∈
R
(3.107)
A
2
A
1
,A
2
We here note that the points of
E
are conformally mapped just to lay down the foundation
of a cylindric map projection.
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