572 F Mixed cylindric map projections
The infinitesimal distance dS between two points X and X +dX, both elements of the biaxial ellipsoid
E
2
A,B , represented in terms of the first chart is
dS
2 = G AB dU
A dU
B
∀
{G AB } =
N
2 (Φ) cos
2 Φ
0
0
M
2 (Φ)
=
L
2
0
0 M
2
.
(F.14)
Simultaneous diagonalization of the two matrices {c AB } and {G AB } being positive–definite leads to
the general eigenvalue problem (F.15), leading to the left principal stretches (F.16), solved by (F.17),
subject to (F.18).
|c AB − Λ
2
S G AB | = 0 =
L
2 g
2
− Λ
2
S L
2
ΛLg(L
g + Lg
)
ΛLg(L
g + Lg
) Λ
2 (L
g + Lg
)
2 + f
2
− Λ
2
S M
2
,
(F.15)
Λ
4
S − Λ
2
S
Λ
2 (L
g + Lg
)
2
M 2
+
f
2
M 2 + g
2
+
g
2 f
2
M 2 = 0 ,
(F.16)
Λ 1 =
1
2
(a +
a 2 − 4b) , Λ 2 =
1
2
(a −
a 2 − 4b) ,
(F.17)
a := Λ
2 (L
g + Lg
)
2 /M
2 + f
2 /M
2 + g
2 ,
b := g
2 f
2 /M
2 .
(F.18)
The postulate of equiareal mapping Λ 1 Λ 2 = 1 leads to (F.19) being equivalent to b = 1 or g = Mf
or g
2 f
2 /M
2 = 1. Obviously, the postulate of a conformal mapping Λ 1 = Λ 2 cannot be fulfilled since
a
2 = 4b leads to a nonlinear functional F (g
4 , g
2 , g
4 , g
2 , f
4 , f
2 ).
1
2
(a +
a 2 − 4b)
1
2
(a −
a 2 − 4b) = 1
(F.19)
End of Proof.
In summarizing, we are led to the equiareal pseudo-cylindrical mapping equations of Box F.2.
Box F.2 (Equiareal pseudo-cylindrical mapping).
x =
A
2 (1 − E
2 ) cos Φ
(1 − E 2 sin
2 Φ) 2 Λ
1
f (Φ)
= M (Φ)N (Φ) cos Φ Λ
1
f (Φ)
,
y = f (Φ) .
(F.20)
F-2 Mixed equiareal cylindric mapping: biaxial ellipsoid onto plane
The variants of mixed equiareal mappings of the biaxial ellipsoid onto the plane are generated by
the weighted mean of the normal equiareal cylindric mapping (for the sphere S
2
R this is the Lambert
equiareal projection) and the equiareal pseudo-cylindrical mapping of sinusoidal type (for the sphere
S
2
R this is the Sanson–Flamsteed equiareal projection). The separate mappings are beforehand collected
in Corollary F.2.
The infinitesimal distance dS between two points X and X +dX, both elements of the biaxial ellipsoid
E
2
A,B , represented in terms of the first chart is
dS
2 = G AB dU
A dU
B
∀
{G AB } =
N
2 (Φ) cos
2 Φ
0
0
M
2 (Φ)
=
L
2
0
0 M
2
.
(F.14)
Simultaneous diagonalization of the two matrices {c AB } and {G AB } being positive–definite leads to
the general eigenvalue problem (F.15), leading to the left principal stretches (F.16), solved by (F.17),
subject to (F.18).
|c AB − Λ
2
S G AB | = 0 =
L
2 g
2
− Λ
2
S L
2
ΛLg(L
g + Lg
)
ΛLg(L
g + Lg
) Λ
2 (L
g + Lg
)
2 + f
2
− Λ
2
S M
2
,
(F.15)
Λ
4
S − Λ
2
S
Λ
2 (L
g + Lg
)
2
M 2
+
f
2
M 2 + g
2
+
g
2 f
2
M 2 = 0 ,
(F.16)
Λ 1 =
1
2
(a +
a 2 − 4b) , Λ 2 =
1
2
(a −
a 2 − 4b) ,
(F.17)
a := Λ
2 (L
g + Lg
)
2 /M
2 + f
2 /M
2 + g
2 ,
b := g
2 f
2 /M
2 .
(F.18)
The postulate of equiareal mapping Λ 1 Λ 2 = 1 leads to (F.19) being equivalent to b = 1 or g = Mf
or g
2 f
2 /M
2 = 1. Obviously, the postulate of a conformal mapping Λ 1 = Λ 2 cannot be fulfilled since
a
2 = 4b leads to a nonlinear functional F (g
4 , g
2 , g
4 , g
2 , f
4 , f
2 ).
1
2
(a +
a 2 − 4b)
1
2
(a −
a 2 − 4b) = 1
(F.19)
End of Proof.
In summarizing, we are led to the equiareal pseudo-cylindrical mapping equations of Box F.2.
Box F.2 (Equiareal pseudo-cylindrical mapping).
x =
A
2 (1 − E
2 ) cos Φ
(1 − E 2 sin
2 Φ) 2 Λ
1
f (Φ)
= M (Φ)N (Φ) cos Φ Λ
1
f (Φ)
,
y = f (Φ) .
(F.20)
F-2 Mixed equiareal cylindric mapping: biaxial ellipsoid onto plane
The variants of mixed equiareal mappings of the biaxial ellipsoid onto the plane are generated by
the weighted mean of the normal equiareal cylindric mapping (for the sphere S
2
R this is the Lambert
equiareal projection) and the equiareal pseudo-cylindrical mapping of sinusoidal type (for the sphere
S
2
R this is the Sanson–Flamsteed equiareal projection). The separate mappings are beforehand collected
in Corollary F.2.