Geography Reference
In-Depth Information
M
r
:
{
R
3
:
δ
ij
}
{
R
3
:
δ
ij
}
M
r
)”
Table 22.2
Metric topology of the right manifold
“
topology of
(
right manifold
∈
R
2
X
=
e
1
x
+
e
2
y
(22.22)
{
R
3
,δ
ij
}
dS
2
=
δ
11
dx
2
+
δ
22
dy
2
=
dx
2
+
dy
2
“
rightmetricof
(22.23)
δ
11
:=
∂
X
= 1
∂
X
∂x
∂x
|
(22.24)
δ
22
:=
∂
X
= 1
∂
X
∂y
∂y
|
(22.25)
δ
12
:=
∂
X
= 0
∂
X
∂y
∂x
|
(22.26)
“
left Cauchy-Green deformation tensor
”
“
Jacobi map
” (pullback operation)
dx
dy
=
x
L
x
B
y
L
y
B
dL
dB
=
J
l
dL
dB
(22.27)
ds
2
=
dx
2
+
dy
2
=(
x
L
dL
+
x
B
dB
)
2
+(
y
L
dL
+
y
B
dB
)
2
=
x
L
+
y
L
dL
2
+
x
2
B
+
y
B
dB
2
+2(
x
L
y
B
+
x
B
y
L
)
dLdB
(22.28)
dS
2
=
dL dB
J
l
J
l
dL
(22.29)
dB
C
l
:=
J
l
J
l
(22.30)
dS
2
=
dL dB
x
L
+
y
L
dL
dB
x
L
x
B
+
y
L
y
B
(22.31)
x
2
B
+
y
B
x
L
x
B
+
y
L
y
B
C
l
=
x
L
+
y
L
x
L
x
B
+
y
L
y
B
∈ R
2
×
2
(22.32)
x
2
B
+
y
B
x
L
x
B
+
y
L
y
B
dS
2
=
dL dB
C
l
dL
dB
(22.33)
2
r
, namely the plane, have been identified by
small
letters
. Equations (
22.7
), (
22.9
)and(
22.10
) express the coordinates of the metric tensor in
Gauss
ellipsoidal coordinates
(
L, B
), while Eq.(
22.8
)asthe
left Cauchy-Green deformation tensor
refers
to the metric of the plane as already outlined, originally described by
Cartesian coordinates
(
x, y
).
Equation (
22.11
) is a representation of the surface element of
all quantities relating to the right manifold
M
A
1
,A
2
in terms of
Gauss ellipsoidal
coordinates
(
L, B
). Equation (
22.5
) defines harmonicity of the mapping functions
x
(
L, B
)
,y
(
L, B
)
,
namely in terms of the
Laplace Beltrami operator
in
Gauss ellipsoidal coordinates
(
L, B
).
The
Laplace Beltrami operator
as a generator of harmonic maps takes a much simpler from if
we use
isometric coordinates. {L, Q}
also called
conformal, iso-thermal
or
Mercator coordinates
in Table
22.3
, namely as transformations
{L, B}→{L, Q}
by Eqs. (
22.34
)-(
22.36
).
The “ln
tan
” function of Eq.(
22.36
)isreferredtoasthe
Lambert function
or
isometric latitude.
The
placement vector
X
E
3
is given in terms of isometric longitude
L
—identical to
Gauss surface normal longitude
—and isometric latitude
Q
by Eq. (
22.37
). Finally, Eqs.(
22.38
)
and (
22.39
) summarize the conformal representation of the metric tensor in terms of
isometric
coordinates
(
L, Q
)of
Mercator type, Λ
(
Q
) is called the
left factor of conformality
. Finally let us
specialize the first theorem by means of relative isometric coordinates
l
:=
L
∈
E
A
1
,A
2
⊂
R
−
L
0
,q
:=
Q
−
Q
0
relative to a
Taylor point
(
L
0
,Q
0
)) inside the
Reference Meridian Strip.
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