in order for the 
23/2 a33] 
natrix Q and the 
ised, the derived 
used after some 
me values (This 
nulation data of 
Is are considered 
'entre are used as 
> of the radius is 
  
  
  
  
  
  
  
Figure 1. The significance of the reference system selection 
in development creation. Up: Developed image created 
without implementing any rotations. Down: Developed 
image created with implementation of the calculated 
rotation angles. 
  
Error in the x position (cm) 
ce 
  
  
  
  
  
This new system should have its origin placed at the centre of 
the model. In addition, the orientation of the system is very 
important, because this factor is critical for the way that the 
object of interest is represented on the projection plane. It is 
easily understood that remaining rotations can result in unreal 
and unwanted distortions in the final product. 
The rotations to be implemented were calculated with respect to 
the position of the object of interest and a reference plane 
defined by the edge of the dome. 
The rotations ® and ¢ are calculated in order for the plane 
defined by the X-Y axes of the new system to become parallel 
to the reference plane, whereas the x rotation is calculated in 
order for the object of interest to be centred. In this way, the 
representation of the object on the projection plane will have no 
distortions caused by remaining rotations. However, this last 
stage of calculations is completely optional and it may be 
omitted. 
The basic algorithm developed for the definition of the rotations 
is a least squares adjustment that is based on the equation of a 
plane (3). 
(3) 
P=ax+by-z+c=0 
a, b , c = constant coefficients 
X, y, Z — object coordinates in ground coordinate 
system 
where 
After these coefficients are determined by the least squares 
adjustment, they are geometrically interpreted. The following 
code illustrates how the geometrical interpretation of the 
coefficients can be achieved 
----—- Geometric interpretation of the a, b, c coefficients----- 
®, = atan(b) 
Q, — atan((-a)/(b*sin(o,)*cos(o,))) 
After the angles o, and q, are calculated, the x, may also be 
calculated using only one point that is considered to lie 
somewhere close to the centre of the object. Knowing the 
-90 -60 -30 0 30 60 90 ; m 
de point's coordinates on the geodetic reference system, the origin 
Longitude (degrees) : : 
and the o, and q, angles, the coordinates of the selected point 
the unknowis o] | ^ — — | dR -3cm .... dR-6cm — dR-9cm can be calculated in the new system. In this case, only the x and 
Considering the 
nd the radius (R) 
tes of the control 
  
  
  
  
Figure 2. The way that the error in the radius of the model 
effects the x position on the projection plane. 
  
  
  
  
  
  
y coordinates are of interest, as these two values are used for the 
calculation of the «, angle that shall be implemented. 
s adjustment has pe (4) 
om the values of 
ng variations and 30 K, = atan(dx/dy) 
it, which may be B 
ustment. ê 10 where dx, dy = the x and y coordinates of the selected point 
E 0 after implementing translation and rotation 
5 E In Figure (1), an attempt is made to show the importance of this 
TION =” v process. In the first case, the picture is developed without taking 
130 into account the rotations between the object and the initial 
ic system, which 90 -60 -30 0 30 60 90 reference system. In the second case, all the calculations have 
3D space. Such à Latitude (degrees) been made and the result is a rather improved developed image. 
poses and this is = om dR=é6cm--dk-0em In this occasion, the derived values of the angles were about 
dinate system. BE 0.5-3 degrees. However, these rather small values have brought 
  
Figure 3. The way that the error in the radius of the model 
a significant change to the result and thus should not be left out 
of the calculations. 
  
effects the y position on the projection plane. 
—465—