Geography Reference
In-Depth Information
D(
L
i
,L
j
)=(
δ
im
+
b
0
ip,m
t
p
)D(
l
m
,l
n
)(
δ
jn
+
b
0
jq,n
t
q
)+
b
0
D(
t
p
,t
q
)
b
0
.
(21.65)
ip
jq
(Summation convention over repeated indices,
{
i, j, m, n
}∈{
1
,
2
}
and
{
p, q
}∈{
1
,...,
9
}
.)
21-4 Gauss-Krueger/UTM Coordinates: From a Local to a Global
Datum
Transformation of conformal coordinates of type Gauss-Krueger or UTM from a local datum
(regional, National, European) to a global datum (WGS 84).
A key problem of contemporary geodetic positioning is the transformation of
mega data sets
of
conformal coordinates of type Gauss-Krueger or UTM from a
local datum
, also called regional,
National, European etc., to a
global datum
, for instance, the
World Geodetic System 1984
(
WGS
84
) with reference to
Boyle
(
1987a
,
b
). As an example, let us refer to the mega data sets of
more than 150 Mio. Gauss-Krueger conformal coordinates of Germany, where the West German
conformal coordinates relate to the
Bessel reference ellipsoid
, while the East German conformal
coordinates relate to the
Krassowsky reference ellipsoid
. Thanks to the satellite Global Positioning
System (“global problem solver”: GPS) and advanced computer software, geodetic positions are
given as conformal coordinates of type Gauss-Krueger or UTM relating to the reference ellipsoid
of the World Geodetic System (WGS 84). In connection with a chart, GPS-derived conformal
coordinates of type Gauss-Krueger or UTM can only be used when they are transformed from
the global datum to the local datum, which the chart is based upon. Or vice versa: the conformal
coordinates of type Gauss-Krueger or UTM which are presented in a chart of local datum have
to be transformed to the global datum. The transformation of conformal coordinates (Gauss-
Krueger or UTM) from a local datum to a global one and vice versa is the objective of our
contribution.
As outlined by means of the commutative diagram of Fig.
21.4
, Cartesian coordinates
{
X
1
,X
2
,X
3
}
(capital letters: global datum) are first transformed into Cartesian coordinates
x
1
,x
2
,x
3
(small letters: local datum) by means of the conformal group C
7
(3). Notably the con-
formal group C
7
(3) (seven parameters in a three-dimensional Weitzenbock space: three parameters
for translation, three parameters for rotation, one scale parameter) leaves angles and distance
ratios invariant (equivariant, form invariant). Such a datum transformation is called a
rectangular
datum transformation
.
Second, in contrast, surface normal ellipsoidal coordinates of type ellipsoidal longitude, ellip-
soidal latitude, ellipsoidal height
{
}
{
L, B, H
}
and
{
l,b,h
}
replace the Cartesian coordinates as user
coordinates.
{
L, B, H
}
refer to a global datum, in particular, to a global reference ellipsoid-
2
A
1
,A
2
of-revolution
E
(semi-major axis
A
1
, semi-minor axis
A
2
, relative eccentricity squared
E
2
:= (
A
1
−
A
2
)
/A
1
), while
2
a
1
,a
2
{
l,b,h
}
refer to a local reference ellipsoid-of-revolution
E
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