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d
t
d
Φ
a
2
cos
2
t
(
a
2
Λ
2
sin
2
t
+
b
2
cos
2
t
)
G
11
G
22
a
4
Λ
2
sin
2
t
cos
2
t
G
11
G
22
−
−
=
μ ±
μ
2
− γ,
Λ
1
Λ
2
=1
μ
2
(
μ
2
⇔
−
−
γ
)=1
⇔
γ
=1
⇔
a
4
Λ
2
sin
2
t
cos
2
t
]
d
t
d
Φ
2
[
a
2
cos
2
t
(
a
2
Λ
2
sin
2
t
+
b
2
cos
2
t
)
−
=
G
11
G
22
(G.25)
⇔
d
Φ
=
G
11
G
22
=
M
(
Φ
)
N
(
Φ
)cos
Φ
=
A
1
(1
ab
cos
2
t
d
t
cos
Φ
(1
− E
2
sin
2
Φ
)
2
.
E
2
)
−
Let us collect the previous results in Corollary
G.2
.
Corollary G.2 (Generalized Kepler equation, generalized Mollweide projection for the biaxial
ellipsoid).
Under the equiareal pseudo-cylindrical mapping equations (
G.26
), where parallel circles are
mapped onto straight lines and meridians are mapped onto ellipses
x
2
/a
2
(
Λ
)+
y
2
/b
2
=1with
a
(
Λ
)=
aΛ
,the
generalized Kepler equation
(
G.27
) has to be solved.
x
(
Λ, Φ
)=
aΛ
cos
t
(
Φ
)
,y
(
Φ
)=
b
sin
t
(
Φ
)
,
(G.26)
ab
cos
2
t
d
t
cos
Φ
d
Φ
=
A
1
(1
E
2
)
−
E
2
sin
2
Φ
)
2
.
(G.27)
(1
−
End of Corollary.
Next, let us integrate the
generalized Kepler equation
.
d
Φ
=
A
1
(1
E
2
)
(1
− E
2
sin
2
Φ
)
2
,
cos
2
t
(
Φ
)d
t
=
A
1
(1
E
2
)
d
t
−
1
cos
2
t
(
Φ
)
cos
Φ
−
cos
Φ
(1
− E
2
sin
2
Φ
)
2
d
Φ.
(G.28)
ab
ab
The
forward substitution x
=
E
sin
Φ
leads to (
G.29
).
t
x
cos
2
t
d
t
=
A
1
(1
E
2
)
Eab
−
1
(1
− x
2
)
2
d
x
(G.29)
0
0
x
2
)
+artanh
x
.
1
2
t
+
1
4
sin 2
t
=
A
1
(1
−
E
2
)
x
abE
2(1
−
The
backward substitution
finally leads to the integrated generalized Kepler equations (
G.30
).
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