Geography Reference
In-Depth Information
of symmetric matrices,
G
l
positive-definite, we succeed to compute and illustrate the
principal distortions
of the Riemann
mapping. We conclude with a global distortion analysis generated by charting the ellipsoid-of-
revolution
Solving the general eigenvalue/eigenvector problem for the pair
{
C
l
,
G
l
}
2
A
1
,A
2
by means of conformal Gauss-Krueger coordinates, parallel Soldner coordinates
and normal Riemann coordinates summarized by the
Airy measure
of total deformation or total
distortion for a symmetric strip [
E
. A special
highlight is Table
20.5
, comparing those three coordinate systems in favor of
normal Riemann
coordinates
. Assume a celestial body like the Earth can be
globally
modeled by an ellipsoid-
of-revolution. Then in terms of the Airy measure of total deformation on a symmetric strip
[
−l
E
,
+
l
E
]
×
[
−b
N
,
+
b
N
] given by Table
20.5
, normal coordinates produce the minimal Airy
global distortion when compared to parallel Soldner coordinates and conformal Gauss-Krueger
coordinates. Let us therefore push forward the geodetic application of normal Riemann coordi-
nates.
−
l
E
,
+
l
E
]
×
[
−
b
N
,
+
b
N
] relative to a point
{
L
0
,B
0
}
20-1 Geodesic, Geodesic Circle, Darboux Frame, Riemann
Coordinates
Riemann polar/normal coordinates. Frobenius matrix, Gauss matrix, Hesse matrix, Christof-
fel symbols. Surface fundamental forms.
2
A
1
,A
2
(biaxial ellipsoid, spheroid, with semi-major
A
1
, with semi-minor axis
A
2
, and with relative eccentricity
E
2
:= (
A
1
−
Let there be given the
ellipsoid-of-revolution
E
A
2
)
/A
1
). It is embedded
3
:=
3
,δ
IJ
in
E
{
R
}
, the three-dimensional Euclidean space of canonical metric
I
=
{
δ
IJ
}
of
Kronecker type
. The Latin indices
I
and
J
are elements of
{
1
,
2
,
3
}
.
2
A
1
,A
2
3
(
X
2
+
Y
2
)
/A
1
+
Z
2
/A
2
=1
E
:=
{
X
∈
R
|
}
.
(20.1)
A
1
,A
2
The ellipsoid-of-revolution
and
{U, V }
constituted by
{
ellipsoidal longitude, ellipsoidal latitude
}
and
{
meta-longitude, meta-
latitude
}
, in particular (
20.2
), for
open sets
(
20.3
) illustrated in Fig.
20.2
.
E
is
globally covered
by the union of the
two charts
{
L, B
}
A
1
cos
V
cos
U
X
I
=+
E
1
A
1
cos
B
cos
L
√
1
−E
2
sin
2
B
1
X
II
=+
E
1
E
2
sin
2
U
cos
2
V
−
E
2
)cos
V
sin
U
A
1
(1
−
√
1
−E
2
sin
2
B
+
E
2
A
1
cos
B
cos
L
1
versus
+
E
2
+
(20.2)
E
2
sin
2
U
cos
2
V
−
A
1
sin
V
+
E
3
A
1
(1
−
E
2
)sin
B
√
1
−E
2
sin
2
B
+
E
3
1
,
E
2
sin
2
U
cos
2
V
−
π
2
<B<
+
π
0
<L<
2
π,
−
2
,
π
2
<V <
+
π
0
<U<
2
π,
−
2
.
(20.3)
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