Geography Reference
In-Depth Information
Table 22.28
Simultaneous diagonalization of the matrix pair (
G
l
,
G
r
):
left eigenvectors
dL
dB
=
L
x
L
y
B
x
B
y
dx
dy
=
J
r
dL
(22.170)
dB
=
x
L
x
B
y
L
y
B
−
1
=
y
B
−
1
x
L
y
B
−
x
B
J
r
=
J
−
1
l
(22.171)
−
y
L
x
L
x
B
y
L
G
11
(
y
B
dx
x
B
dy
)
2
dS
2
=
G
11
dL
2
+
G
22
dB
2
=
1
(
x
L
y
B
−x
B
y
L
)
2
−
y
L
dx
+
x
L
dy
)
2
+
G
22
(
−
(22.172)
(
x
L
y
B
−x
B
y
L
)
2
[
G
11
y
B
+
G
22
y
L
dx
2
+
G
11
x
2
B
+
G
22
x
L
dy
2
1
=
−
2(
G
11
x
B
y
B
+
G
22
x
L
y
L
)
dxdy
]
dS
2
=
dx dy
J
r
J
r
dx
(22.173)
dy
C
r
:=
J
r
J
r
⎡
⎤
dx
dy
G
22
y
L
(
x
L
y
B
−x
B
y
L
)
2
G
11
y
B
+
G
11
x
B
y
B
+
G
22
x
L
y
L
(
x
L
y
B
−x
B
y
L
)
2
−
dS
2
=
dx dy
⎣
⎦
(22.174)
G
11
x
2
B
+
G
22
x
L
G
11
x
B
y
B
+
G
22
x
L
y
L
(
−
x
B
y
L
)
2
x
B
y
L
)
2
x
L
y
B
−
(
x
L
y
B
−
1
(
x
L
y
B
−
G
11
y
B
+
G
22
y
L
−
(
G
11
x
B
y
B
−
G
22
x
L
y
L
)
∈
2
×
2
C
r
=
(22.175)
G
11
x
2
B
+
G
22
x
L
x
B
y
L
)
2
−
(
G
11
x
B
y
B
−
G
22
x
L
y
L
)
dS
2
=
dx dy
C
r
dx
dy
(22.176)
In the final step integration by parts or
Green's First Identity
will be applied to Eqs.(
22.206
)
and (
22.207
) subject to Eq.(
22.211
). By means of Tables
22.32
and
22.33
, respectively, we identify
{
in Eqs.(
22.212
), (
22.213
)andEqs.(
22.217
), (
22.218
)forthe
x
-variation as well
as the
y
-variation.
Green's First Identities
Eqs.(
22.214
)and(
22.218
)aregiveninaspecialform
since the boundary curve is considered
fixed
with respect to
x
-and
y
-variation, quite in contrast
to the surface variation. The
quantor
grad f, grad g
}
S
l
guarantees the local representation Eqs.(
22.215
)
and (
22.224
) for regular functional kernels. Finally, Eqs.(
22.216
)and(
22.225
) is a representation
of the
Laplace-Beltrami equation
in terms of “surface normal”
∀
δx
∈
{
ell
ipso
idal longitude
L
, elli
psoid
al
and the coordinates of the metric tensor
G
l
,namely
√
G
11
=
N
(
B
)coscos
B
,
√
G
22
=
latitude
B
}
M
(
B
).
Appendix 3
The
Euler-Lagrange equation of minimal distortion energy in isometric coordinates of Mercator
type
The
Laplace-Beltrami equation
Eqs.(
22.216
)and(
22.225
) take a much simpler form when they
are derived in
isometric coordinates
also called
conformal or isothermal.
Let us prove Theo-
rem
22.2
based upon isometric coordinates
{
L
,
Q
}
of Mercator type summarized in Table
22.3
,
Eqs.(
22.34
)-(
22.40
).
Search WWH ::
Custom Search