G-2 The generalized Mollweide projections for the biaxial ellipsoid 593
G-2 The generalized Mollweide projections for the biaxial ellipsoid
The characteristics of the spherical Mollweide projection within the class of pseudo-cylindrical mappings are as follows. The graticule parallel circles are mapped on parallel straight lines, while meridians
on ellipses. We shall keep these properties for the ellipsoidal Mollweide projection by the “Ansatz”
x(Λ, Φ) = aΛ cos t, y(Φ) = b sin t .
(G.20)
The polar coordinate t can be interpreted as the reduced latitude of the ellipse x
2 /a
2 (Λ) + y
2 /b
2 = 1,
where a(Λ) = aΛ holds. Note that for longitude Λ = 0, a(Λ) = 0 follows. Therefore, the central
Greenwich meridian Λ = 0 is mapped onto a straight line. Of course, any other central meridian could
have been chosen alternatively. Again, we compute the left Cauchy–Green deformation tensor, this
time represented in terms of t(Φ) as follows.
∂x
∂Λ
= a cos t(Φ) ,
∂x
∂Φ
= −aΛ sin t(Φ)
dt
dΦ
,
∂y
∂Λ
= 0 ,
∂y
∂Φ
= b cos t(Φ)
dt
dΦ
,
(G.21)
(G.13), (G.21)
⇒
{c AB } =
⎡
⎣
a
2 cos
2 t(Φ)
−a
2 Λ sin t(Φ) cos t(Φ)
dt
dΦ
−a
2 Λ sin t(Φ) cos t(Φ)
dt
dΦ a
2 Λ
2 sin
2 t(Φ)
dt
dΦ
2 + b
2 cos
2 t(Φ)
dt
dΦ
2
⎤
⎦ ,
(G.22)
(G.14), (G.15), (G.22) ⇒
c AB − Λ
2
S G AB
= 0
⇔
a
2 cos
2 t − Λ
2
S G 11
−a
2 Λ sin t cos t
dt
dΦ
−a
2 sin t cos t
dt
dΦ
(a
2 Λ
2 sin
2 t + b
2 cos
2 t)
dt
dΦ
2 − Λ
2
S G 22
= 0
⇔
Λ
4
S G 11 G 22 − Λ
2
S [G 11 (a
2 Λ
2 sin
2 t + b
2 cos
2 t)
dt
dΦ
2
+ G 22 a
2 cos
2 t]+
+a
2 cos
2 t(a
2 Λ
2 sin
2 t + b
2 cos
2 t)
dt
dΦ
2 − a
4 Λ
2 sin
2 t cos
2 t
dt
dΦ
2 = 0
⇔
Λ
4
S − Λ
2
S
G 11 (a
2 Λ
2 sin
2 t + b
2 cos
2 t)
dt
dΦ
2
G 11 G 22
+
G 22 a
2 cos
2 t
G 11 G 22
+
+
a
2 cos
2 t(a
2 Λ
2 sin
2 t + b
2 cos
2 t)
dt
dΦ
2
G 11 G 22
−
a
4 Λ
2 sin
2 t cos
2 t
dt
dΦ
2
G 11 G 22
= 0 ,
(G.23)
Λ
2
S =
1
2
G 11 (a
2 Λ
2 sin
2 t + b
2 cos
2 t)
dt
dΦ
2
G 11 G 22
+
G 22 a
2 cos
2 t
G 11 G 22
±
±
1
4
G 11 (a
2 Λ
2 sin
2 t + b
2 cos
2 t)
dt
dΦ
2
G 11 G 22
+
G 22 a
2 cos
2 t
G 11 G 22
2
−
−
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
dt
dΦ
2
1/2
= µ ±
µ 2 − γ ,
(G.24)