3. DEGENERATE CONFIGURATIONS AND 
SIMULATED EXAMPLES 
3.1 Some remarks on degenerate configurations 
Some degenerate cases, where a solution is not 
possible, are given in this chapter; however, a thor- 
ough investigation on the existence and uniqueness of 
the solutions is not presented. The examples refer to 
the use of straight lines in space resection and relative 
orientation. 
In space resection using straight lines as control 
features the position of the projection centre can not 
be determined, if the lines are parallel or if they 
intersect in an identical point. In these cases the 
projection centre can be moved infinitely in the 
direction of the parallel lines or in the direction of 
the intersection point respectively, while the bundle of 
projecting rays or planes remains unchanged. These 
degeneracy applies to point to line correspondences as 
well as line to line correspondences. A differential 
uncertainty is given, if the orientation of the lines and 
the direction vectors of differential changes of the 
rotation parameters in object space are identical. 
In relative orientation of three images using straight 
lines as tie features a degeneracy occurs, if the 
projection centres are collinear and at least one of the 
straight lines and the line connecting the projection 
centres are coplanar. In this case the three cor- 
responding projecting planes are identical and the 
position and orientation of these straight lines cannot 
be determined. This case can be avoided by not using 
this kind of lines for the orientation. Another 
degenerate configuration occurs, if the projection 
centres again are collinear and the object lines are 
coplanar. This degeneracy, which can be explained by 
means of projective geometry, essentially restricts the 
application of the method in aerial triangulation. 
3.2 Simulated examples 
In order to study the effect of the location and 
orientation of linear features in basic orientation tasks 
simulated configurations were investigated by deriving 
the theoretical precision of the orientation 
parameters. However, only a few examples and some 
of the main results concerning space resection and 
relative orientation are given in this paper. 
In space resection using straight lines as control 
features the resulting precision mainly depends on the 
spatial location and orientation of the lines in object 
space. A control point version was chosen, where 3 
points are arranged regularly on a circle in a 
horizontal plane and the projection centre is located 
above the centre of the circle, which is the origin of 
the object coordinate system. The image coordinates 
117 
were assumed to be uncorrelated and equally precise, 
and the control points or lines to be error-free. 
For point to line corresponden- 
ces with horizontal control lines 
and a regular distribution of the 
points on the circle, the same 
precision of the orientation para- 
meters as in the control point | 
version can be achieved by a con- 
figuration, where the opposite 
straight - lines are in pairs 
orthogonal and the directions are either radial or tan- 
gential. (Fig. 1). The simulations showed that for 
configurations with irregular orientation of the lines 
in space the resulting precision of the orientation 
parameters is slightly worse. If the configurations are 
approaching one of the degenerate cases as mentioned 
above (some are shown in Fig. 2), the precision 
deteriorates significantly. 
  
  
  
  
Fig. 1 
  
  
  
  
  
  
Fig. 2: Some degenerate configurations 
In relative orientation the investigations on the use of 
straight lines confirmed the limited suitability of the 
method for aerial applications. This is due to the 
requirement that the straight lines have to be pro- 
jected in at least three images, which is a severe 
restriction in case of standard overlap, and due to the 
singularity in case of collinear projection centres and 
coplanar object lines. The simulations showed that the 
precision deteriorates, when approaching these 
degenerate configurations, e.g. in case of 
approximately flat terrain the resulting precision is 
very low. For curved lines this situation is much more 
encouraging. It was shown that the relative orientation 
with two circular features is possible, however, only 
with moderate accuracy. In order to obtain com- 
parable accuracy results as with a tie points version at 
least 4 or 5 lines are needed. 
4. CONCLUSIONS 
In this paper an overview on the use of general 
geometric features for orientation tasks was given and 
minimum configurations were derived. Furthermore, 
some degenerate configurations were mentioned and 
the results of some simulated examples were 
summarized. 
An extension of the concept to more complex lines or 
surfaces is feasible and similar investigations can be 
performed. The use of general features in conjunction 
with line scanners is another possible field of