Geography Reference
In-Depth Information
We gain the parametric representation by
Λ
=
Λ
∗
and
√
X
2
+
Y
2
=
√
X
∗
2
+
Y
∗
2
. The integral
dependent of the reduced latitude then is obtained as follows.
:=
(
X,Y,Z
):
X
2
+
Y
2
A
2
=1
+
Z
2
2
A
1
,A
2
3
,
E
⊂
R
(14.10)
A
1
A
1
=
√
1
Z
√
X
2
+
Y
2
=(1
E
2
)tan
Φ
=
A
2
A
2
A
1
tan
Φ
∗
,
−
−
E
2
⇒
tan
Φ
∗
=
√
1
− E
2
tan
Φ
=
A
2
A
1
tan
Φ
(14.11)
⇒
E
2
)
Φ
0
d
Φ
f
(
Φ
)=
A
1
(1
−
E
2
sin
2
Φ
)
3
/
2
(1
−
=
A
1
Φ
∗
0
d
Φ
∗
(1
−
E
2
cos
2
Φ
∗
)
.
(14.12)
Let us here also define the elliptic integral of the second kind (for example, consult Appendix
C
or
Gradsteyn and Ryzhik 1983
, namely p. 905, formula 8.1113). The definition (
14.13
)leadsfor
f
(
Φ
) to the representation (
14.14
). The principal stretches are easily computed as (
14.15
).
E
2
)
Φ
0
E
2
sin
2
Φ
)
3
/
2
, f
(
Φ
∗
)=
A
1
Φ
∗
d
Φ
∗
√
1
d
Φ
f
(
Φ
)=
A
1
(1
−
−
E
2
cos
2
Φ
∗
,
(1
−
0
f
(
Δ
∗
)=
A
1
π/
2
Δ
∗
E
2
sin
2
Δ
∗
=
A
1
π/
2
0
d
Δ
∗
1
d
Δ
∗
1
E
2
sin
2
Δ
∗
−
−
−
(14.13)
A
1
Δ
∗
0
d
Δ
∗
1
E
2
sin
2
Δ
∗
=
A
1
[
E
(
π/
2
,E
)
−
E
(
Δ
∗
,E
)]
,
−
−
f
(
Φ
)=
A
1
E
(
π/
2
,E
)
−
E
π
2
−
arctan
A
2
A
1
tan
Φ
,E
,
(14.14)
Λ
1
=
1
−
E
2
sin
2
Φ
cos
Φ
, Λ
2
=1
.
(14.15)
14-22 Special Normal Cylindric Mapping (Normal Conformal,
Equidistant: Equator)
First, let us here apply the postulate of conformal mapping. Similar as before, we obtain the
following set of formulae.
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