Geography Reference
In-Depth Information
Let us present the summary of the Gauss mapping “ellipsoid-of-revolution to sphere” based
upon the stretch equation (
9.50
) in form of Lemma
9.1
.
ar
cos
φ
A
1
cos
Φ
(1
−E
2
sin
2
Φ
)
1
/
2
ar
cos
φ
N
(
Φ
)cos
Φ
.
=
(9.50)
Lemma 9.1 (Gauss mapping “ellipsoid-of-revolution to sphere”).
The
mapping equations
from the ellipsoid-of-revolution adequately parameterized by
{
Λ
e
,Φ
e
}
to
the sphere adequately parameterized by
{
λ
s
,φ
s
}
of type
conformal
read
λ
s
=
λ
0
+
a
(
Λ
e
Λ
0
)
,
−
(9.51)
tan
π
=
c
a
tan
π
a
1
aE/
2
4
+
φ
s
4
+
Φ
e
E
sin
Φ
e
1+
E
sin
Φ
e
−
,
(9.52)
2
2
and
a
=cos
Φ
0
N
0
M
0
+tan
2
Φ
0
=
1+
E
2
cos
4
B
0
,E
2
=
E
2
E
2
,
1
−
tan
4
+
φ
2
1
/a
tan(
4
+
2
)
1
−
E
sin
Φ
0
c
=
E/
2
=
1+
E
sin
Φ
0
=
exp(
q
0
/a
)
exp
Q
0
,
(9.53)
ln
c
=
1
a
q
0
−
Q
0
,
tan
φ
0
=
M
0
N
0
tan
Φ
0
.
Relative to the equidistant mapping of the “fundamental point”,
P
0
(
Λ
0
,Φ
0
)
→
p
0
=
p
(
λ
0
,φ
0
),
there hold the conditions
Λ
0
=1
,Λ
0
=0
,Λ
0
=0
.
(9.54)
The mean spherical radius reads
r
=
M
0
N
0
.
(9.55)
End of Lemma.
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