578 F Mixed cylindric map projections
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).
x = x(Λ, Φ) =
A Λ
α + β
α +
β cos Φ
1 − E 2 sin
2 Φ
.
(F.43)
α > β:
y = y(Φ) ≈ A(1 − E
2 )(α + β)×
×
2α
β
α 2 − β 2
arctan
(β − α) tan Φ/2
α 2 − β 2
1 + E
2 (1 −
α
2
2β 2 )
+
+
Φ
β
1 + E
2
3
4
−
α
2
2β 2
+
+E
2 (α − β cos Φ)(2α + 3β cos Φ) sin Φ
4β 2 (α + β cos Φ)
+ O(E
4 )
.
(F.44)
α < β:
y = y(Φ) ≈ A(1 − E
2 )(α + β)×
×
−
α
β
β 2 − α 2
ln
β + α cos Φ +
β 2 − α 2 sin Φ
α + β cos Φ
1 + E
2
1 −
α
2
2β 2
+
+
Φ
β
1 + E
2
3
4
−
α
2
2β 2
+
+E
2 (α − β cos Φ)(2α + 3β cos Φ) sin Φ
4β 2 (α + β cos Φ)
+ O(E
4 )
.
(F.45)
α = β = 1:
x = x(Λ, Φ) ≈
1
2
A Λ
1 − E 2 sin
2 Φ + cos Φ
1 − E 2 sin
2 Φ
,
y = y(Φ) ≈ 2A(1 − E
2 )×
×
Φ
1 +
E
2
4
− tan
Φ
2
1 +
E
2
2
+ E
2 1 − cos Φ
4(1 + cos Φ)
(2 + 3 cos Φ) sin Φ + O(E
2 )
.
(F.46)
End of Lemma.