592 G Generalized Mollweide projection
Thus, the left Cauchy–Green deformation tensor c AB is generated by
c 11 =
∂x
∂Λ
2
+
∂y
∂Λ
2
= A
2
1 P
2 g
2 ,
c 12 = c 21 =
∂x
∂Λ
∂x
∂Φ
+
∂y
∂Λ
∂y
∂Φ
= A
2
1 ΛP g(P
g + P g
) ,
c 22 =
∂x
∂Φ
2
+
∂y
∂Φ
2
= A
2
1 Λ
2 (P
g + P g
)
2 + A
2
1 f
2 ,
(G.13)
{c AB } = A
2
1
P
2 g
2
ΛP P
g
2 + ΛP
2 gg
ΛP P
g
2 + ΛP
2 gg
Λ
2 P
2 g
2 + Λ
2 P
2 g
2 + 2Λ
2 P P
gg
+ f
2
.
(G.14)
The infinitesimal distance dS between two points X and X +dX, both elements of the biaxial ellipsoid
E
2
A 1 ,A 2
, is in terms of the first chart represented by (G.15). Simultaneous diagonalization of the two
matrices {c AB } and {G AB } being positive–definite leads to the general eigenvalue problem (G.16)
with the eigenvalues (principal distortions) {Λ 1 , Λ 2 } given by (G.17).
dS
2 = G AB dU
A dU
B
∀ {G AB } =
N
2 (Φ) cos
2 Φ
0
0
M
2 (Φ)
= A
2
1
P
2
0
0 M
2
,
(G.15)
c AB − Λ
2
S G AB
= 0 ,
(G.16)
Λ
4
S [Λ
2 (P
g + P g
)
2 + f
2 + g
2 M
2 ] + g
2 f
2 = 0
⇔
Λ
2
S =
1
2
(Λ
2 P
2 g
2 + Λ
2 P
2 g
2 + 2ΛP P
gg
+ f
2 + g
2 M
2 )±
±
1
4
(Λ 2 P 2 g 2 + ΛP P gg + f 2 + g 2 M 2 ) 2 + 1
⇔
Λ
2
S = a ± b ⇔ Λ
2
1 = a + b , Λ
2
2 = a − b .
(G.17)
The postulate of an equiareal mapping Λ
2
1 Λ
2
2 = 1 leads to (a + b)(a − b) = a
2
− b
2 = 1 or gf
2 = 1 or
g = f
or g
−1 = f
. Obviously, the postulate of a conformal mapping Λ
2
1 = Λ
2
2 cannot be fulfilled
since a + b = a − b, b = 0 holds, in general.
End of Proof.
In summarizing, we are led to the equiareal pseudo-cylindrical mapping equations (G.18) with the
principal distortions (G.19).
x = A 1 Λ
cos Φ
1 − E 2 sin
2 Φ
1
f (Φ)
= N (Φ)Λ cos Φ
1
f (Φ)
= A
2
1 Λ
cos Φ
1 − E 2 sin
2 Φ
1
dy
dΦ
,
y = A 1 f (Φ) ,
(G.18)
Λ
2
S =
1
2
(Λ
2 P
2 + Q
2 + 1) ±
1
4
(Λ 2 P 2 + Q 2 + 1) + 1
∀ P
(Φ) = −
sin Φ
1 − E 2 sin
2 Φ
+
E
2 sin Φ cos
2 Φ
(1 − E 2 sin
2 Φ) 3/2 .
(G.19)