Geography Reference
In-Depth Information
x
=0
f
(
x
)=
1+
x
1
− x
E/
2
1+
x
1
− x
E/
2
−
1
(1
=1+
1
1!
E
2
−
x
)
−
(1 +
x
)(
−
1)
(1
− x
)
2
x
+O(2)=
=1+
Ex
+O(2)
.
Alternative :
artanh
x
=
x
+
x
3
3
+
x
5
5
+
x
7
7
+
x
9
9
+
x
11
11
+O(
x
13
)
,
E
cos
Δ
=
x
1
,
(8.62)
artanh(
E
cos
Δ
)=
=
E
cos
Δ
+
E
3
3
cos
3
Δ
+
E
5
5
cos
5
Δ
+
E
7
7
cos
7
Δ
+
E
9
9
cos
9
Δ
+
E
11
11
cos
11
Δ
+O(
x
13
)
.
Question: “Is the conformal mapping of the ellipsoid-of-
revolution to a tangential plane at the North Pole UPS?”
Answer: “Let us work out this subject in the following pas-
sage in more detail.”
Let us introduce the stereographic projection of the point
P ∈
E
A
1
,A
2
of the ellipsoid-of-revolution
E
A
1
,A
2
at the North Pole N. The
South Pole S has been chosen as the perspective center, also called
O
∗
, the center of the projection.
Q
=
π
(
P
) is the point on the
z
axis generated by an orthogonal projection. Consult Fig.
8.5
for
further geometrical details. Naturally,
NS
p
=
QS
P
denotes the characteristic parallactic angle
of the central projection
p
=
π
(
P
):
A
1
,
4
2
to the point
p
=
π
(
P
), an element of the tangent space
T
N
E
2
A
2
=
√
X
2
+
Y
2
r
tan
NS
p
=tan
QS
P
⇔
A
2
+
Z
⇒
(8.63)
A
2
+
Z
√
X
2
+
Y
2
=2
A
1
cos
Φ
2
A
2
A
2
r
=
A
2
1
,
E
2
sin
2
Φ
+
A
1
(1
−
−
E
2
)sin
Φ
√
1
−E
2
sin
2
Φ
+
√
(1
−E
2
)sin
Φ
,
2
A
1
cos
Φ
r
=
f
(
Φ
)
f
(
Φ
)
→ f
(
Δ
)
,
(8.64)
√
1
−E
2
cos
2
Δ
+
√
(1
−E
2
)cos
Δ
.
2
A
1
sin
Δ
r
=
f
(
Δ
)
The projective equations document a radial function
r
=
f
(
Δ
) which differs remarkably from the
equations of an azimuthal conformal mapping. Definitely, the azimuthal conformal mapping of
the ellipsoid-of-revolution is not UPS.
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