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tan
4
+
φ
2
tan
a
4
+
Φ
2
1
−
E
sin
Φ
0
c
a
=
aE/
2
,
1+
E
sin
Φ
0
tan
4
+
φ
2
1
/a
tan(
4
+
Φ
2
)
1
−
E
sin
Φ
0
c
=
E/
2
.
(9.65)
1+
E
sin
Φ
0
End of Proof.
The principal stretches of the conformal mapping “ellipsoid-of-revolution to sphere” are explicitly
given by (
9.66
).
Λ
1
=
Λ
2
=
Λ
=
a
√
M
0
N
0
cos
φ
, a
as given above
,
(9.66)
N
(
Φ
)cos
Φ
φ
= 2 arctan
c
a
tan
π
aE/
2
a
1
−
E
sin
Φ
1+
E
sin
Φ
4
+
Φ
π
2
.
−
2
(9.67)
9-3 The Equal Area Mappings “Ellipsoid-of-Revolution to Plane”
The equal area mappings from the ellipsoid-of-revolution to the plane: the condition of equal
area, the standard integrals, authalic latitude.
First, we postulate the condition of equal area
Λ
1
Λ
2
= 1 and subsequently we take advantage of
the standard integrals that are collected in Box
9.7
.
Λ
1
Λ
2
=1
⇔
ar
cos
φ
A
1
cos
Φ
(1
r
A
1
(1
− E
2
)
d
φ
d
Φ
(1
E
2
sin
2
Φ
)
1
/
2
E
2
sin
2
Φ
)
3
/
2
= 1
−
−
(9.68)
⇔
r
2
cos
φ
d
φ
=
A
1
(1
−
E
2
)
a
d
Φ
cos
Φ
E
2
sin
2
Φ
)
2
.
(9.69)
(1
−
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