C10(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
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C10 (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 C7 (3). Here we concentrate on a transformation which leaves locally angles and distance ratios equivariant (covariant, form invariant): the ten parameter conformal group C10 (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 E3 . Numerical 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 techniques (l1 -, l2 -, 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 transformation (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 affine transformation by the parameters. Similarly Brunner (1993) advocated affine transformations for the analysis of deforming networks. For the synthesis of datum transformations. Wolf (1990) proved that centralized Cartesian 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. For more 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 conformal group C10 (3) in t
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