Geography Reference
In-Depth Information
To Cartesian coordinates (
x, y
)
∈
R
2
of the chart
{
R
2
,δ}
of the ellipsoid of revolution
E
A
1
,A
2
,
A
1
∈
R
is equipped the metric topology of a two-dimensional
Euclidean space (“flat space”). The functions
{x
(
L − B
)
,y
(
L − B
)
}
areassumedtobeanalytical.
If the distortion energy
+
,
R
+
A
2
<A
1
.Thechartof
E
A
1
,A
2
dS
l
tr
C
l
G
−
l
=
min
1
2
(22.2)
with respect to
West-East longitude
L
W
,
L
E
and South-North Latitude
B
S
,
B
N
of type fixed boundaries
In particular over a
Reference Meridian Strip
{
L
W
≤
L
≤
L
E
,B
S
≤
B
≤
B
N
}
E
2
(sin
B
)
2
2
cos
B
x
L
= in
x
(
L,B
)
,y
(
L,B
)
L
E
dL
B
N
B
S
E
2
)
G
11
+
x
2
B
G
22
+
y
L
G
33
+
y
B
1
2
A
1
(1
−
dB
(22.3)
1
G
44
−
L
W
L
E
dL
B
N
B
S
E
2
(sin
B
)
2
2
cos
B
Grad x
= in
x
(
L,B
)
,y
(
L,B
)
1
2
A
1
(1
− E
2
)
2
+
Grad y
2
dB
(22.4)
1
−
L
W
Is assumed to be minimal, the mapping equations, namely the functions
x
(
L, B
)
,y
(
L, B
)are
harmonic,
Δ
(
L, B
)=0
,Δy
(
L, B
) = 0
(22.5)
1
G
11
+
1
G
22
∂
∂L
∂
∂L
∂
∂B
∂
∂B
Δ
:=
(22.6)
Is the
Laplace-Beltrami operator
in
Gauss
ellipsoidal coordinates subject to
G
l
=
N
2
cos
2
B
0
M
2
(22.7)
0
C
l
=
x
L
+
y
L
x
L
x
B
+
y
L
y
B
(22.8)
x
2
B
+
y
B
x
L
x
B
+
y
L
y
B
The coordinates of the left metric tensor
G
l
and the coordinates of the
left Cauchy-Green defor-
mation tensor
C
l
,where
E
2
)(1
E
2
sin
2
B
)
−
3
/
2
M
:=
A
1
(1
−
−
(22.9)
N
:=
A
1
(1
− E
2
sin
2
B
)
−
1
/
2
(22.10)
are the principal radii, also called “
meridianal
”and“
normal
”.
E
2
:= (
A
1
− A
2
)
/A
1
denotes the
relative eccentricity of
E
A
1
,A
2
.Its
left surface element
dS
l
:=
dLdB
det
G
l
=
dL dB
A
1
(1
− E
2
)
1
− E
2
sin
2
B
2
(22.11)
has been represented un terms of
Gauss ellipsoidal coordinates
(
L
,
B
).
End of Theorem.
Though details for the proof of Theorem
22.1
are given in Appendix 1, we like to give here
a short review of the
metric topology
of the left manifold
2
M
r
,namelythe
ellipsoid-of-revolution
2
A
1
,A
2
2
2
,inTable
22.2
.Bymeans
E
,inTable
22.1
and of the right manifold
M
r
,namelythe
plane
P
2
A
1
,A
2
of Eq.(
22.12
) we define the algebraic geometry of the ellipsoid
E
of rotational symmetry. The
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