Mixed Cylindric Map Projections - Map Projections: Cartographic Information Systems - page 761

E 2 sin 2 Φ ) − 3 / 2 cos Φ (1 −

α 1 E 2 sin 2 Φ + β cos Φ ≈ −

cos Φ
α + β cos Φ ×

≈

(F.41)

1+ E 2 (4 α +3 β cos Φ )sin 2 Φ
2( α + β cos Φ )

× +

+O( E 6 ) ,

+ E 4 (24 α 2 +37 αβ cos Φ +15 β 2 cos 2 Φ )sin 4 Φ
8( α + β cos Φ ) 2

Φ

cos Φ ∗ (1 E 2 sin 2 Φ ∗ ) − 3 / 2

−

d Φ ∗ ≈

α 1 E 2 sin 2 Φ ∗ + β cos Φ ∗

−

0

1+ E 2 1

2 β 2 +

α 2

2 α

β 2 arctan ( β

− α )tan Φ / 2

β α 2

α 2

≈ −

− − β 2

(F.42)

1+ E 2 3

2 β 2 +

α 2

+ Φ
β

4 −

+ E 2 ( α

− β cos Φ )(2 α +3 β cos Φ )sin Φ

4 β 2 ( α + β cos Φ )

+O( E 4 ) .

For the term-wise integration in ( F.42 ) by MATHEMATICA, uniform convergence of the kernel
has been necessary. In summarizing, we have proven Lemma F.7 .

Lemma F.7 (Horizontal mean of the generalized Lambert projection and of the generalized
Sanson-Flamsteed projection of the biaxial ellipsoid E

2 A,B ).

The horizontal mean of the generalized Lambert projection and of the generalized Sanson-
Flamsteed projection of the biaxial ellipsoid leads to the equiareal mapping of pseudo-cylindrical
type represented by ( F.43 ).

α + .

AΛ
α + β

β cos Φ

x = x ( Λ, Φ )= 1 (F.43)

E 2 sin 2 Φ

−

α>β :

E 2 )( α + β ) y = y ( Φ ) ≈ A (1 − ×

1+ E 2 (1 −

2 β 2 ) +

2 α
β α 2 − β 2 arctan ( β

− α )tan Φ / 2

α 2

α 2

× (F.44)

− β 2

1+ E 2 3

2 β 2 +

α 2

+ Φ
β

4 −