Geography Reference
In-Depth Information
Universal Mercator Projection (UMP):
x
=
A
1
L
=:
p
UMP
,
(16.10)
y
=
A
1
ln
tan
π
E/
2
=:
q
UMP
.
1
4
+
B
E
sin
B
1+
E
sin
B
−
2
Universal Polar Stereographic Projection (UPS):
1
−
E
1+
E
E/
2
tan
π
1+
E
sin
B
1
− E
sin
B
E/
2
2
A
1
B
2
x
=
√
1
4
−
cos
L
=:
p
UPS
,
(16.11)
−
E
2
1
E/
2
tan
π
1+
E
sin
B
1
E/
2
2
A
1
E
1+
E
−
B
2
√
1
− E
2
x
=
4
−
sin
L
=:
q
UPS
.
−
E
sin
B
Once one system of conformal coordinates is established, we can use it as the input for another sys-
tem of conformal coordinates (conformal change from one conformal chart to another conformal
chart, c:c:cha-cha-cha). Accordingly, the Korn-Lichtenstein equations reduce to the d'Alembert-
Euler equations (
16.12
) (more known as the Cauchy-Riemann equations) subject to the inte-
grability conditions (
16.13
)or(
16.14
), which is automatically
orientation preserving
according
to (
16.15
). Here, we have denoted
{
p, q
}
as being generated by (
16.10
)(UMP)orby(
16.11
)
(UPS).
x
p
=
y
q
,
q
=
−
y
p
,
(16.12)
x
pq
=
y
qp
,
pq
=
y
qp
,
(16.13)
Δ
LB
x
:=
x
pp
+
x
qq
=0
, Δ
LB
y
:=
y
pp
+
y
qq
=0
,
(16.14)
x
q
y
p
=
x
p
+
y
p
=
y
q
+
x
q
>
0
.
x
p
y
q
−
(16.15)
A fundamental solution of the d'Alembert-Euler equations (
16.12
) (Cauchy-Riemann equations)
subject to the integrability conditions (
16.14
) in the class of polynomials is provided by (
16.16
),
or in matrix notation, based on the Kronecker-Zehfuss product
⊗
and transposition T, provided
by (
16.17
).
x
=
α
0
+
α
1
q
+
β
1
p
+
α
2
(
q
2
p
2
)+
β
2
2
pq
+
−
1)
s
r
q
r−
2
s
p
2
s
+
1)
s+1
r
r/
2
(
r
+1)
/
2
N
N
+
α
r
(
−
β
r
(
−
2
s
2
s
−
1
r
=3
s
=0
r
=3
s
=1
q
r−
2s+1
p
2
s−
1
,
y
=
β
0
+
β
1
q
α
1
q
+
β
2
(
q
2
p
2
)+
α
2
2
pq
+
−
−
(16.16)
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