Geography Reference
In-Depth Information
15-1 The Equations Governing Conformal Mapping
The equations governing conformal mapping and their fundamental solution. The Korn-
Lichtenstein equations, the Laplace-Beltrami equations.
A
1
,A
1
,A
2
(ellipsoid-of-
revolution, spheroid, semi-major axis
A
1
, semi-minor axis
A
2
) embedded in a three-dimensional
Euclidean manifold
E
Here, we are concerned with a conformal mapping of the biaxial ellipsoid
E
3
,δ
ij
}
with a standard canonical metric
δ
ij
, the Kronecker delta of
ones in the diagonal, of zeros in the off-diagonal, namely by means of (
15.4
), introducing surface
normal ellipsoidal longitude
L
and surface normal ellipsoidal latitude
B
.
3
=
{
R
A
1
cos
B
cos
L
A
1
cos
B
sin
L
, X
3
=
A
1
(1
−
E
2
)sin
B
X
1
=
, X
2
=
1
1
1
,
(15.4)
E
2
sin
2
B
E
2
sin
2
B
E
2
sin
2
B
−
−
−
E
2
:= (
A
1
−A
2
)
/
(
A
1
)=1
−A
2
/A
1
denotes the first
numerical eccentricity
squared. According to
[
L, B
]
π/
2
,
+
π/
2], we exclude from the domain [
L, B
] North Pole and South Pole.
Thus, [
L, B
] constitute only a first chart of
∈
[
−
π,
+
π
]
×
[
−
A
1
,A
1
,A
2
A
1
,A
1
,A
2
basedupontwo
charts, which covers all points of the ellipsoid-of-revolution, is given in all detail by
Grafarend
and Syffus
(
1995
).
Conformal coordinates
E
: a minimal atlas of
E
(isometric coordinates, isothermal coordinates) are constructed
from the surface normal ellipsoidal coordinates
{
x, y
}
as solutions of the Korn-Lichtenstein
equations (conformal change from one chart to another chart: c: Cha-Cha-Cha)
{
L, B
}
x
L
x
B
=
−
y
L
y
B
,
1
G
11
G
22
G
12
G
11
(15.5)
−
G
22
G
12
G
12
−
subject to the integrability conditions
x
LB
=
x
BL
,y
LB
=
y
BL
(15.6)
or
Δ
LB
x
:=
G
11
x
B
−
+
G
22
x
L
−
G
12
x
L
G
12
x
B
G
11
G
22
−
G
11
G
22
−
=0
,
G
12
G
12
B
L
(15.7)
Δ
LB
y
:=
G
11
y
B
− G
12
y
L
+
G
22
y
L
− G
12
y
B
G
11
G
22
− G
12
G
11
G
22
− G
12
=0
,
B
L
x
L
x
B
y
L
y
B
=(
x
L
y
B
−
x
B
y
L
)
>
0
(15.8)
Search WWH ::
Custom Search