Geography Reference
In-Depth Information
x
=
x
(
Λ, Φ
)=
A
cos
Φ
1
− E
2
sin
2
Φ
=
Λ
=[
N
(
Φ
)cos
Φ
]
Λ
=
L
(
Φ
)
Λ,
(F.24)
y
=
f
(
Φ
)
,
g
(
Φ
)=1
, f
(
Φ
)=
M
(
Φ
)
,
M
(
Φ
∗
)d
Φ
∗
=
Φ
0
Φ
E
2
)d
Φ
∗
(1
− E
2
sin
2
Φ
∗
)
3
/
2
.
A
(1
−
f
(
Φ
)=
(F.25)
0
Note the integral kernel expansion as powers of
E
2
is
uniformly convergent.
f
(
Φ
)
E
2
)
≈
A
(1
−
×
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.26)
End of Proof.
Note that the coordinate lines
Λ
= const., the
meridians
, are mapped close to a sinusoidal arc as
can been seen by
Φ
in (
F.23
). Here, we find the reason for our term “pseudo-sinusoidal”. This is
in contrast to the coordintae lines
Φ
= const.: the
parallel circles
are mapped onto straight lines
parallel to the
x
axis represented by
x
=c
1
Λ
and
y
=
c
2
,where
c
1
,c
2
are constants.
The
first variant
of mixed equiareal mapping of the biaxial ellipsoid onto the plane is generated
as
weighted mean
of the
vertical coordinates {y
gL
,y
gSF
}
with respect to the
generalized Lam-
bert coordinate y
gL
(
→
(
F.21
)) and the
generalized Sanson-Flamsteed coordinate y
gSF
(
→
(
F.23
)).
In contrast, the
horizontal coordinate x
is constructed from the postulate of equiareal pseudo-
cylindrical mapping equations of the type (
F.20
).
Definition F.4 (Equiareal mapping of pseudo-cylindrical type: vertical coordinate mean).
An equiareal mapping of pseudo-cylindrical type of the biaxial ellipsoid is called
vertical coordinate
mean
if (
F.27
)holdswhere
α
and
β
are weight coecients.
x
=
x
(
Λ, Φ
)=
=
A
2
(1
E
2
)cos
Φ
(1
− E
2
sin
2
Φ
)
2
Λ
−
1
f
(
Φ
)
,
(F.27)
y
=
y
(
Φ
)=
=
αy
gL
+
βy
gSF
α
+
β
=:
f
(
Φ
)
.
End of Definition.
The unknown function
f
(
Φ
) is constructed as follows. Obviously, the
vertical coordinate mean
y
(
Φ
), i.e. (
F.28
), is the basis which generates the
horizontal coordinate x
(
Λ, Φ
), namely (
F.29
),
subject to (
F.30
)and(
F.31
).
A
(1
−
E
2
)
α
+
β
f
(
Φ
)
≈
×
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