Legenstein, Dietmar 
  
The differentials with respect to the rotation tensors and the centre of projection are not quoted explicitly but can be 
found in [Legenstein 1997]. 
2.3 Rate of Convergence for the Different Conditions of Contour 
The first testruns with ellipsoids have shown that the different conditions of contour lead to completely different rates of 
convergence. For too inaccurate approximations — greater than the radius of convergence — one of the equations leads to 
divergence while the others converge. The clue to the understanding of this behaviour can be found in the examination 
of the differential geometric loci (cf. 2.2-5). The mathematical argument can be found in the tensors I” and Y. These 
tensors represent normal projections, orthogonal projecting a vector onto the normal planes in the direction of n in case 
of I, in the direction of s in case of V. Vector fields of the normal vector of the geometric loci in the different centres of 
initialisation (for selected values) were plotted. If points with zero normal vectors or curls in the vector field are 
detected, this explains the different rate of convergence. The following images show normal vectors for differential loci. 
The ellipse on the left-hand side ® represents the curve, the right one © the points satisfying the condition of contour. 
The left main vertex of the ellipse O is the centre of the surface, the right one the centre of projection X. 
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Figure 3 shows a curl-free (conservative) vector field. There is only one point with zero normal vector. This point is the 
middle of the centre of the surface and the centre of projection. In figure 4 a curl around the centre of the ellipsoid can 
be seen. In addition the normal vector is zero in this centre. This curl causes a big change in the direction of the 
geometric locus in these areas. This is especially disturbing because the point of contour is in that area. In figure 5 there 
are two curls. One of them is in the centre of the ellipsoid, the other one in the centre of projection. The normal vector is 
zero on the line from the centre of projection to the centre of the surface. 
To determine the areas of convergence the differential loci of the contour condition are intersected with the surface 
condition. Both conditions have to be fulfilled for the observation of a point on contour. In addition no measurements 
are needed because both observations are fictitious. The intersection point is used as the next approximation value for 
the linearisation. In the favourable case the intersection point converges to the point of contour. 
The equations for the geometric loci for the conditions of contour (2.1-1, 2.1-2, 2.1-3) can be derived from (2.2-5) in a 
straightforward manner using the differentials calculated above (2.2-14, 2.2-16, 2.2-17): 
E: (s'F; + n° )dx =—n“s, (2.3-1) 
E, (sr F n )dx, --n's, (2.3-2) 
E.: (rr; en V8. )dx, - -n's, (2.3-3) 
  
478 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000.