Geography Reference
In-Depth Information
Theorem 22.2 (Harmonic maps, isometric coordinates (conformal, isothermal) of an ellipsoid of
revolution, variational formulation).
Let
x
=
ξ
(
q,l
)
,y
=
η
(
q,l
) denote the mapping equations in terms of isometric coordinates (con-
formal, isothermal) of Mercator type of
2
2
subject to the
Reference Meridian Strip L
0
,
E
A
1
,A
2
→
P
namely
2
(
q,l
)
∈{
(
q,l
)
∈
|L
W
− L
0
≤ l ≤ L
E
− L
0
,Q
s
− Q
0
≤ q ≤ Q
N
− Q
0
}
(22.41)
namely to Cartesian coordinates (
x, y
)
∈
P
2
of the chart
{
2
,δ
ij
}
of the ellipsoid of revolution
E
A
1
,A
2
, coordinated by (
q,l
). The functions
ξ
(
q,l
)
,η
(
q,l
) are assumed to be analytical. If the
distortion energy
L
E
dL
Q
N
Q
S
1
2
dQA
2
(
Q
)
tr
C
l
G
−
l
=min
ξ,η
(22.42)
L
W
is minimal over a
Reference Meridian Strip {L
w
≤ L ≤ L
E
,Q
S
≤ Q ≤ Q
N
}
with respect to the
West-East isometric longitude
L
w
,L
E
and South-North isometric latitude
Q
S
,Q
N
of type fixed
boundaries.
L
E
dL
Q
N
Q
S
dQ
ξ
L
+
ξ
Q
+
η
L
+
η
Q
= in
ξ
(
Q,L
)
,η
(
Q,L
)
1
2
(22.43)
L
W
l
E
dl
q
N
q
S
dq
ξ
l
+
ξ
q
+
η
l
+
η
q
=min
ξ
(
q,l
)
,η
(
q,l
)
1
2
(22.44)
l
W
l
E
dl
q
N
q
S
dq
=min
ξ
(
q,l
)
,η
(
q,l
)
1
2
gradξ
2
gradη
2
+
(22.45)
l
W
is assumed to be minimal the mapping equations, namely the functions
ξ
(
q,l
)
,η
(
q,l
)are
har-
monic,
Δξ
(
q,l
)=0
,Δη
(
q,l
)=0
,
(22.46)
Λ
2
(
Q
)
[
∂
2
∂l
2
+
∂
2
1
Δ
:=
∂q
2
]
(22.47)
is the
Laplace-Beltrami operator
in isometric coordinates of
Mercator type
subject to
G
l
=
Λ
2
(
Q
)
10
(22.48)
01
C
l
=
ξ
l
+
η
l
η
l
η
q
(22.49)
η
q
η
l
η
q
The coordinates of the left metric tensor
G
l
of conformal type and the coordinates of the
left
Cauchy-Green deformation tensor
C
l
where the
left factor of conformality
is given by
A
1
cos
2
(
lam
−
1
Q
)
A
1
cos
2
Q
1
− E
2
sin
2
Q
A
2
(
Q
)=
1
− E
2
sin
2
(
lam
−
1
Q
)
=
(22.50)
expQ
=tan
π
1
−
E
sin
B
1+
E
sin
B
E/
2
4
+
B
(22.51)
2
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