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β
Φ
+
3
256
E
4
(12
Φ −
8 sin 2
Φ
+ sin 4
Φ
)+O(
E
6
)
+
8
E
2
(2
Φ −
sin 2
Φ
)+
15
×
(F.28)
ln
1+
E
sin
Φ
1
,
+
α
4
E
2
E
sin
Φ
E
sin
Φ
+
E
2
sin
2
Φ
−
1
−
1
1
f
(
Φ
)
=
α
+
β
αy
gL
+
βy
gSF
,
f
1
(
Φ
)=
α
+
β
(
αy
gL
+
βy
gSF
)
,
(F.29)
cos
Φ
y
gL
(
Φ
)=
A
(1
E
2
)
(1
− E
2
sin
2
Φ
)
2
=
A
−
1
N
(
Φ
)
M
(
Φ
)cos
Φ,
−
E
2
)
A
(1
−
y
gSF
(
Φ
)=
E
2
sin
2
Φ
)
3
/
2
=
M
(
Φ
)
,
(F.30)
(1
−
1
f
(
Φ
)
=
x
=
x
(
Λ, Φ
)=
N
(
Φ
)
M
(
Φ
)cos
ΦΛ
N
(
Φ
)
M
(
Φ
)cos
ΦΛ
(
α
+
β
)
αA
−
1
N
(
Φ
)
M
(
Φ
)cos
Φ
+
βM
(
Φ
)
=
(
α
+
β
)
A
cos
Φ
α
cos
Φ
+
β
1
=
Λ.
(F.31)
E
2
sin
2
Φ
−
Thus,wehaveprovenLemma
F.5
.
Lemma F.5 (Vertical mean of the generalized Lambert projection and of the generalized Sanson-
Flamsteed projection of the biaxial ellipsoid
E
2
A,B
).
The vertical 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 rep-
resented by (
F.32
).
(
α
+
β
)
A
cos
Φ
α
cos
Φ
+
β
1
x
=
x
(
Λ, Φ
)=
Λ,
E
2
sin
2
Φ
−
E
2
)
α
+
β
A
(1
−
y
=
y
(
Φ
)
≈
×
(F.32)
β
Φ
+
3
8 sin 2
Φ
+ sin 4
Φ
)+O(
E
6
)
+
sin 2
Φ
)+
15
8
E
2
(2
Φ
256
E
4
(12
Φ
×
−
−
ln
1+
E
sin
Φ
1
.
+
α
4
E
2
E
sin
Φ
E
sin
Φ
+
E
2
sin
2
Φ
−
1
−
End of Lemma.
The
second variant
of mixed equiareal mapping of the biaxial ellipsoid onto the plane is gen-
erated as weighted mean of the
horizontal coordinates
{
x
gL
,x
gSF
}
with respect to the gen-
eralized 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 coor-
dinate mean
if (
F.33
)holdswhere
α
and
β
are weight coecients.
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