594 G Generalized Mollweide projection
Λ
2
1 Λ
2
2 = 1 ⇔ µ
2
− (µ
2
− γ) = 1 ⇔ γ = 1
⇔
[a
2 cos
2 t(a
2 Λ
2 sin
2 t + b
2 cos
2 t) − a
4 Λ
2 sin
2 t cos
2 t]
dt
dΦ
2
= G 11 G 22
⇔
ab cos
2 t
dt
dΦ
=
G 11 G 22 = M (Φ)N (Φ) cos Φ = A
2
1 (1 − E
2 )
cos Φ
(1 − E 2 sin
2 Φ) 2 .
(G.25)
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 = 1 with 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
dt
dΦ
= A
2
1 (1 − E
2 )
cos Φ
(1 − E 2 sin
2 Φ) 2 .
(G.27)
End of Corollary.
Next, let us integrate the generalized Kepler equation.
dt
dΦ
=
A
2
1 (1 − E
2 )
ab
1
cos 2 t(Φ)
cos Φ
(1 − E 2 sin
2 Φ) 2 , cos
2 t(Φ) dt =
A
2
1 (1 − E
2 )
ab
cos Φ
(1 − E 2 sin
2 Φ) 2 dΦ . (G.28)
The forward substitution x = E sin Φ leads to (G.29).
t
0
cos
2 t dt =
A
2
1 (1 − E
2 )
Eab
x
0
1
(1 − x 2 ) 2 dx
⇔
1
2
t +
1
4
sin 2t =
A
2
1 (1 − E
2 )
abE
x
2(1 − x 2 )
+ ar tanh x
.
(G.29)
The backward substitution finally leads to the integrated generalized Kepler equations (G.30).
1
2
t +
1
4
sin 2t =
=
A
2
1
ab
1 − E
2
E
E sin Φ
2(1 − E 2 sin
2 Φ)
+
1
2
ar tanh (E sin Φ)
.
(G.30)
Due to ar tanh (E sin Φ) =
1
2 ln
1+E sin Φ
1−E sin Φ , we alternatively obtain (G.31).
2t + sin 2t =
=
A
2
1
ab
1 − E
2
E
ln
1 + E sin Φ
1 − E sin Φ
+
2E sin Φ
1 − E 2 sin
2 Φ
.
(G.31)