151 
cosßcosy cos a sin y + sin a sinß cos y sinasiny-cosasinßcosy 
-cosßsiny cosacosy-sinasinßsiny sin a cos y + cos a sin ß sin y 
sin ß -sinacosß cosacosß 
days, where 68 points were observed. The purpose of this 
campaign was the enlargement the EUREF reference frame into 
the continental part of Croatia, and to establish dense 
geodynamic network covering the area around the Adriatic sea. 
At this campaign 5 EUREF points in Slovenia and 10 points in 
Croatia were observed once more. 
TRANSFORMATION OF COORDINATE SYSTEMS 
With the availability of high precision coordinate systems the 
transformation of three dimensional coordinate systems is 
becoming frequent computation. There are several ways of 
transforming one coordinate system to another. At this place we 
will only mention two of them. The most common 
transformation is similarity transformation in which the scale 
factor is the same in all directions. This transformation 
preserves shape, but the distances and point positions may 
change. The second is so called orhogonal transformation in 
which the scale factor is unity. This transformation preserves 
the distances and the shape, but the positions of points do 
change on transformation. 
Similarity transformation 
If we define x=(jc| ,,jc 2 ,,^3,/), i=l,...,n the coordinates of n>3 
points with respect to {O.bi.b^b^} and x'=(jc 1>/ '^c 2 ,/’^3,/'), 1=1,.-»« 
the coordinates of the same points with respect to {0,b\',b 2 ',b 3 '}. 
The 7 similarity transformation parameters (t',oc,(3,y,dm) can be 
determined with the 7 coordinates given in both coordinate 
systems. In the case when the number of common coordinates 
given in both systems is larger than 7 we get the overdetermined 
linear (linearized) system of unknown parameters which are 
estimated by least squares solution. 
Transformation written in the form of equation (1) where m= 1 
is an orthogonal transformation. It is used when the presumption 
of equal scale factor for both coordinate systems can be 
accepted. In this case only six transformation parameters relate 
both coordinate systems (t',a,(3,y). 
ASTROGEODETIC NETWORK OF SLOVENIA 
AND ETRS 89 
Similarity transformation is usual transformation procedure in 
geodesy, where coordinate systems established by different 
measuring techniques are orthogonal but of different scale. 
Let S={0,b l ,b 2 ,b ? ,} and S’={0,b\,b 2 ',bT,'} be Cartesian coordinate 
systems, and let x=(x!,x2,x 3 ) and x'=(xi',x2',x 3 ') be a Cartesian 
coordinates of a point with respect to the coordinate systems S 
and S' respectively. 
In the general form a similarity transformation between 
coordinate systems is written: 
x'=/»Rx + t' (1) 
where R is an orthogonal transformation matrix RR'=I, 
m=(l+dm) is the scale factor, t'=(ti,t 2 ,t 3 ') is the vector of the 
coordinates of the coordinate system's S origin in the coordinate 
system 5'. Every orhogonal matrix R can be described by three 
parameters, where there are several possibilities to choose these 
three representation parameters. Usual representation of these 
parameters in geodesy are Euler- or Cartan- angles. Here we 
mention Cartan- angles, where every orthogonal matrix can be 
represented as: 
R = R 3 (y)R 2 (p)R,(a) 
As we already mentioned, the astrogeodetic network of 
Slovenia is classical triangulation network. In classical 
triangulation networks angles are determined with much higher 
accuray than baseline lenghts and in general all such networks 
exhibit quite large scale errors. Some previous investigations 
(Jenko, 1986) confirmed the hypothesis that the situation is the 
same also in the astrogeodetic network of Slovenia. 
For the 27 of first order triangulation points, besides the official 
coordinates in the national coordinate system, also the positions 
in ETRS 89 coordinate system are available. Both sets of 
coordinates of these points enable some comparisons of 
coordinate systems. Points with positions given in both system 
are in Figure 2 marked with the filled circles and squares. 
For the transformation procedure Cartesian coordinates of the 
points in the astrogeodetic net have to be computed. These 
coordinates were computed using the following relations: 
X 
(N +h) cos 9 cos À 
y 
- 
(N + /i)cos9sin^ 
z 
[(1 - e 2 )N + /r]sin 9 
e'=2 f-f 1 
where N = . ^ — , 
yj 1 - e 2 sin 2 9 
(2) 
with: 
R,(y) 
R,(a) 
cosy 
siny 
0' 
cosß 
0 
-sinß 
-sin y 
cosy 
0 
, R,(ß) = 
0 
1 
0 
0 
0 
1 
sinß 
0 
cosß 
'1 
0 
0 
' 
0 cosa sina 
0 -sina cosa 
The orthogonal rotation matrix R is decomposed into three 
consecutive rotations around the b\, b 2 and ¿3 axes. In the 
general form of the Cartan- representation of an orthogonal 
matrix is: 
R = R(a,ß,y) = 
Parameters of official reference elliposid i.e. the Bessel 
reference elliposid are a=6377397,155m and 
l//=299,1528128533. Ellipsoidal height /? of each point in the 
net is computed from h = H + N, where the geoidal height N is 
computed with the existing geoid model (Colic et al., 1992a) 
Transformation between national coordinate system and 
ETRS 89 
We have performed the 7-parameter similarity and 6-parameter 
orthogonal transformations of ETRS 89 coordinate system to 
the national coordinate system. This way the estimation of some 
attributes of the official national coordinate system became 
possible. 
The 7-parameter transformation is the most common procedure 
when combining different coordinate systems. But in general it