576 F Mixed cylindric map projections
The second variant of mixed equiareal mapping of the biaxial ellipsoid onto the plane is generated
as weighted mean of the horizontal coordinates {x gL , x gSF } with respect to the generalized Lambert
coordinate x gL (→ (F.21)) and the generalized Sanson–Flamsteed coordinate x gSF (→ (F.23)). In
contrast, the vertical coordinate y is constructed from the postulate of equiareal pseudo-cylindrical
mapping equations of the type (F.20).
Definition F.6 (Equiareal mapping of pseudo-cylindrical type: horizontal coordinate mean).
An equiareal mapping of pseudo-cylindrical type of the biaxial ellipsoid is called horizontal coordinate
mean if (F.33) holds where α and β are weight coefficients.
x = x(Λ, Φ) =
=
αx gL + βx gSF
α + β
=
A
2 (1 − E
2 ) cos Φ
(1 − E 2 sin
2 Φ) 2 Λ
1
f (Φ)
,
y = y(Φ) =: f (Φ) .
(F.33)
End of Definition.
This time, in constructing the unknown function f (Φ), the horizontal coordinate mean x(Λ, Φ), i. e.
(F.34), is the basis generating the vertical coordinate y(Φ), i. e. (F.35)–(F.38).
αx gL + βx gSF
α + β
=
A
2 (1 − E
2 ) cos Φ
(1 − E 2 sin
2 Φ) 2
Λ
f (Φ)
,
(F.34)
x gL = A Λ , x gSF =
A cos ΦΛ
1 − E 2 sin
2 Φ
,
(F.35)
f
(Φ) =
A
2 (1 − E
2 ) cos ΦΛ
(1 − E 2 sin
2 Φ) 2
α + β
αx gL + βx gSF
,
f
(Φ) =
A
2 (1 − E
2 ) cos Φ Λ
(1 − E 2 sin
2 Φ) 2
α + β
αA Λ + β(A cos Φ)Λ/
1 − E 2 sin
2 Φ
=
=
(α + β)A(1 − E
2 ) cos Φ(1 − E
2 sin
2 Φ)
−3/2
α
1 − E 2 sin
2 Φ + β cos Φ
,
(F.36)
f (Φ = 0) = 0
⇔
f (Φ) =
Φ
0
f
(Φ
∗ )dΦ
∗ ,
(F.37)
y(Φ) = A(1 − E
2 )(α + β)
Φ
0
cos Φ
∗ (1 − E
2 sin
2 Φ
∗ )
−3/2
α
1 − E 2 sin
2 Φ ∗ + β cos Φ ∗
dΦ
∗ .
(F.38)