Geography Reference
In-Depth Information
3-45 Direct Transformation of Oblique Quasi-Spherical
Longitude/Latitude
2
4
1
,A
2
The standard representation of a point
X
being an element of
E
is given in terms of surface
2
normal ellipsoidal longitude/latitude
{
Λ, Φ
}∈{
R
|
0
≤
Λ<
2
π,
−
π/
2
<Φ<
+
π/
2
}
,which
2
A
1
,A
2
excludes North Pole and South Pole of
E
. Accordingly,
{
Λ, Φ
}
constitutes only a first chart
2
A
1
,A
2
of
E
, i.e.
A
1
cos
Φ
cos
Λ
X
1
=
X
=
1
− E
2
sin
2
Φ
,
A
1
cos
Φ
cos
Λ
X
2
=
Y
=
1
,
(3.129)
E
2
sin
2
Φ
−
E
2
)cos
Φ
X
3
=
Z
=
A
1
(1
−
1
.
E
2
sin
2
Φ
−
A
1
,A
2
The ellipsoidal coordinates
Λ
and
Φ
are called
surface normal
since the surface normal of
E
enjoys the
spherical image
N
=
E
1
cos
Φ
cos
Λ
+
E
2
cos
Φ
sin
Λ
+
E
3
sin
Φ.
(3.130)
2
A
1
,A
2
2
A
1
,A
2
A minimal atlas of
, has to be based on two charts given by
Grafarend and Syffus
(
1995
). The direct mapping of type (
3.129
), namely
E
,whichcoversallpointsof
E
{
Λ, Φ
}→{
X,Y,Z
}
,
has the inverse
tan
Λ
=
Y
X
,
1
Z
√
X
2
+
Y
2
.
tan
Φ
=
(3.131)
1
−
E
2
By means of (
3.104
), (
3.108
), and (
3.109
), one alternatively derives the direct mapping equations
and inverse mapping equations
X
=
R
(
A, B
)[cos
A
cos
B
cos
Ω −
sin
A
cos
B
sin
Ω
cos
I
+sin
B
sin
Ω
sin
I
]
,
Y
=
R
(
A, B
[cos
A
cos
B
sin
Ω
+sin
A
cos
B
cos
Ω
cos
I
−
sin
B
cos
Ω
sin
I
]
,
(3.132)
Z
=
R
(
A, B
)[sin
A
cos
B
sin
I
+sin
B
cos
I
]
,
tan
Λ
=
cos
A
cos
B
sin
Ω
+sin
A
cos
B
cos
Ω
cos
I
sin
B
cos
Ω
sin
I
cos
A
cos
B
cos
Ω −
sin
A
cos
B
sin
Ω
cos
I
+sin
B
sin
Ω
sin
I
,
−
(3.133)
1
sin
A
cos
B
sin
I
+sin
B
cos
I
cos
2
A
cos
2
B
+(sin
A
cos
B
cos
I
tan
Φ
=
sin
B
sin
I
)
2
.
1
−
E
2
−
Let us here additionally collect the result of the transformation
{
A, B
}→{
Λ, Φ
}
by the following
Corollary
3.9
.
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