Geography Reference
In-Depth Information
24
C
10
(3): The Ten Parameter Conformal Group as a Datum
Transformation in Three-Dimensional Euclidean Space
In Chap.
21
, we already transformed from a global three- dimensional geodetic network into a
regional or local geodetic network. We aimed at the
analysis of datum parameters,
namely
seven
parameters
of type
translation, rotation and scale,
as elements of the
global conformal group C
7
(3).
Here we concentrate on a transformation which leaves
locally
angles and distance ratios equiv-
ariant (covariant, form invariant): the
ten parameter
conformal group
C
10
(3) in three-dimensional
Euclidean
space. It forms the basis of the
geodetic datum transformation
whose
ten parameters
are determined by
robust adjustment
of two data sets of Cartesian coordinates in
3
.Numerical
E
results are presented.
Three-dimensional datum transformations play a central role in contemporary Euclidean point
positioning. Recently
Kampmann
(
1993
) applied for the overdetermined three-dimensional datum
adjustment problem (7 datum parameter transformation/similarity transformation)
robust tech-
niques
(
l
1
-,
l
2
-,
l
∞
-norm optimization).
Strauss and Walter
(
1993
) made an attempt to eliminate
the
correlations
of the pseudo-observational equations generated by the 7 datum parameter trans-
formation (GPS-WGS84 Cartesian coordinates versus local LPS- Bessel Cartesian coordinates).
In contrast,
Fotiou and Rossikopoulos
(
1993
) adjusted the overdetermined datum transformation
by
variance-component techniques
, in particular allowing also
a
ne
transformation by the param-
eters. Similarly
Brunner
(
1993
)advocated
a
ne transformations
for the analysis of deforming
networks. For the synthesis of datum transformations.
Wolf
(
1990
) proved that centralized Carte-
sian coordinates constitute
correlated
pseudo-observations with a
singular
variance-covariance
matrix.
Three-dimensional geodetic datum transformations in three-dimensional Euclidean space are
generated by the
transformation group
which leaves
angles
and
distance ratios equivariant
.This
transformation group is the
ten parameter conformal group
C
10
(3). Surprisingly geodesists have
only used the
seven parameter
conformal subgroup. Accordingly we aim here at a solution of
the
ten parameter
overdetermined datum problem.
C
10
(3) has an interesting history. It has been
originally applied by
Bateman
(
1910
)and
Cunningham
(
1910
) who proved that the
Maxwell
equations of electromagnetism in vacuo
are
equivariant
with respect to the
15 parameter conformal
group
C
(1,3) in four-dimensional pseudo-Euclidean space (
Minkowski space
). This result led
Haantjes
(
1937
,
1940
)and
Schouten and Haantjes
(
1936
) to the representation of
C
(
n
), e.g.
C
(3) in Sect.
24-1
.Formore
detail
of
C
(
n
) we refer to
Barut
(
1972
),
Bayen
(
1976
),
Beckers
et al.
(
1976
),
Boulware et al.
(
1970
),
Carruthers
(
1971
),
Freund
(
1974
),
Fulton et al.
(
1962
),
Kastrup
(
1962
,
1966
), Mariwalla (1971),
Mayer
(
1975
),
Soper
(
1976
)and
Wess
(
1960
).
Section
24-2
contains the
numerical
overdetermined
ten
parameter determination of the con-
formal group
C
10
(3) in three-dimensional space, namely with respect to two data sets of ter-
restrial and GPS three-dimensional Cartesian coordinates and robust adjustment techniques.
Search WWH ::
Custom Search