Geography Reference
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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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