Geography Reference
In-Depth Information
x
=
AΛ,
ln
1+
E
sin
Φ
1
=
y
=
A
(1
−
E
2
)
4
E
2
E
sin 1
Φ
E
sin
Φ
+
(F.21)
E
2
sin
2
Φ
−
1
−
artanh(
E
sin
Φ
)+
.
=
A
(1
−
E
2
)
2
E
E
sin
Φ
E
2
sin
2
Φ
1
−
End of Corollary.
Proof.
For the proof, let us depart from the setup
x
=
AΛ
and
y
=
f
(
Φ
)as special case of (
F.11
) with
L
(
Φ
)=1and
g
(
Φ
) = 1 such that the equator is mapped equidistantly. The left Cauchy-Green
deformation tensor
c
AB
is generated by
c
11
=
A
2
,c
12
=0,and
c
22
=
f
2
(
Φ
) such that the left
principal stretches, following (
F.12
)-(
F.14
), amount to (
F.22
), which leads by partial integration
(decomposition into fractions) subject to the condition
f
(
Φ
= 0) = 0 directly to (
F.21
).
N
(
Φ
)cos
Φ
, Λ
2
=
f
(
Φ
)
A
Λ
1
=
M
(
Φ
)
,
Λ
1
Λ
2
=1
⇔
(F.22)
d
f
=
A
−
1
N
(
Φ
)
M
(
Φ
)cos
Φ
d
Φ,
cos
Φ
E
2
)
d
f
=
A
(1
−
E
2
sin
2
Φ
)
2
d
Φ.
(1
−
End of Proof.
Corollary F.3 (Ellipsoidal equiareal projection of pseudo-sinusoidal type).
The ellipsoidal equiareal projection of pseudo-sinusoidal type (generalized Sanson-Flamsteed pro-
jection) in the class of pseudo-cylindrical projections is represented in terms of surface normal
longitude
Λ
and latitude
Φ
of
2
E
A,B
by (
F.23
) mapping a parallel circle equidistantly.
A
cos
Φ
x
=
1
Λ,
E
2
sin
2
Φ
−
y ≈ A
(1
− E
2
)
Φ
+
3
8
E
2
(2
Φ −
sin 2
Φ
)+O(
E
4
)
≈
(F.23)
A
Φ
8
E
2
(2
Φ
+3sin2
Φ
)+O(
E
4
)
.
1
≈
−
End of Corollary.
Proof.
For the proof, let us depart from the setup (
F.24
), which leads under the postulate of an equiareal
mapping via (
F.8
) of Corollary
F.1
to (
F.25
), expressing the arc length of the meridian, namely
in terms of the standard elliptic integral of second kind, here instead by series expansion.
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