Geography Reference
In-Depth Information
subject to
1
3
5
175
441
4
E
2
64
E
4
256
E
6
16384
E
8
65536
E
10
E
0
=1
−
−
−
−
−
(8.42)
and
F
2
=
8
E
2
+
16
E
4
+
213
2048
E
6
+
255
4096
E
8
+
166479
655360
E
10
,
256
E
4
+
21
21
+
533
120563
F
4
=
256
E
6
8192
E
8
−
327680
E
10
,
(8.43)
151
+
155
4096
E
8
+
2767911
F
6
=
6144
E
6
9175040
E
10
,
1097
273697
F
8
=
131072
E
8
−
4587520
E
10
.
End of Lemma.
The
Equidistant Polar Mapping
(EPM) of the ellipsoid-of-revolution is summarized in
Lemma
8.10
, which is based upon the direct mapping equations, its left principal stretches,
the left eigenvectors, the left maximal angular distortion, and the inverse mapping equations that
are collected in Box
8.5
.
Lemma 8.10 (Equidistant Polar Mapping (EPM), equidistant mapping of the ellipsoid-of-
revolution to the tangential plane at the North Pole).
The
equidistant mapping
of the spheroid to the tangential plane at the North Pole of an oblate
ellipsoid-of-revolution, in short,
Equidistant Polar Mapping (EPM)
, is parameterized by
x
=
f
(
Δ
)cos
Λ, y
=
f
(
Δ
)sin
Λ,
(8.44)
subject to the
left Cauchy-Green eigenspace
. The radial function
r
=
f
(
Δ
)that
represents the meridian arc length from the North Pole to a point on the meridian
Λ
= constant
is given either in the form of an elliptic integral of the second kind or in the series expansion of
Box
8.5
.
{
E
Λ
Λ
1
(
Δ
)
,
E
Φ
}
End of Lemma.
Box 8.5 (Equidistant mapping of the ellipsoid-of-revolution to the tangential plane at the
North Pole).
Parameterized mapping:
α
=
Λ, r
=
f
(
Δ
)
, Δ
:=
π/
2
−
Φ, f
(
Δ
)
→
f
(
Φ
)
,
(8.45)
x
=
r
cos
α
=
f
(
Φ
)cos
Λ, y
=
r
sin
α
=
f
(
Φ
)sin
Λ.
Series expansion, equidistant mapping of the family of meridians:
f
(
Φ
)=
A
1
E
0
π
2
− Φ
− E
2
sin 2
Φ − E
4
sin 4
Φ − E
6
sin 6
Φ − E
8
sin 8
Φ
E
1
0sin10
Φ
+O(E
12
)
.
−
(8.46)
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