86 RADIAL TRIANGULATION, DISCUSSION 
we had to deal not with simple rhomboids but 
with rhomboids which had variable numbers of 
wing points; I think it went up to six or eight. 
It would have been hardly possible to match this 
case with the classical method of adjusting 
rhomboids. 
Finally, I want to mention that some of the 
ideas developed here we have found also in a 
paper presented by Mr Dmochowski of Poland. 
However, there are some points of disagreement 
because in computing the rhomboid directly 
with the observations he gets also the closing 
error at the end of the rhomboid, but instead of 
taking the average of the two points which he 
gets there, he considers the centre of gravity 
of this aerial triangulation as the best possible 
adjustment. This is not in accordance with the 
least squares solutions. A second objection we 
have against his method of adjusting the strip 
of rhomboids, is that his method comes down 
to what you can call an interpolation method of 
the closing error at the strip, but instead of 
distributing this closing error more or less 
regularly he makes it a function of the individual 
closing error of the rhomboid. That means his 
corrections are not a smooth function of the 
length of the strip, but jumped. 
Mr J. Visser: I will base my comments on 
the investigation of Professor Roelofs and the ac- 
curacy of numerical radial triangulation. Again, 
a radial triangulator — and this time one of a 
much improved design — is readily available. 
Every person charged with the planning of 
procedures in photogrammetric activities should 
think about possible applications of this numer- 
ical radial triangulation, in particular if this 
method may profitably replace spatial triangula- 
tion on the one hand and template triangulation 
on the other. Of course, this only applies to 
projects where the planimetric co-ordinates ex- 
clusively are required. In practice, I think this 
concerns mainly relatively flat areas. It is ob- 
vious that the operation of a radial triangulator 
such as the Wild RTI is much simpler than that 
of a universal first order instrument, and that 
consequently inexperienced operators can be 
used. Further, from a simple cross-calculation it 
will be clear that the cost of the numerical radial 
triangulation will be about $42 lower per trian- 
gulated model than spatial triangulation. This 
is due to the higher speed of the observations in 
radial triangulation, and to the much lower cost 
of amortisation of the instrument. 
Now about the attainable accuracies. When 
block adjustment methods, such as the ITC- 
Jerie are applied, it is only the deformation with- 
in the units formed by two successive models 
which are of importance, and not the propaga- 
tion of errors through the triangulated strips. Ip 
his 1956 publication, Professor Roelofs derived 
the formula for a systematic error in transfer of 
scale and azimuth from one model to the next in 
numerical radial triangulation. This, indeed, 
gives a good insight into the deformations per 
unit which are to be expected and which deter. 
mine the so-called relative accuracy of the block 
adjustment. 
When considering the possibility of applica- 
tion of numerical radial triangulation in the 
National Cartographic Centre in Teheran, we 
used a set of photographs which were taken 
from actual flights in Iran. The tilts were larger 
than those used by Professor Roelofs in his 
numerical example. We used a series with 
maximum tilt component of 1.7 grades and 
averages of 0.48 grades, and 0.57 grades. 
Assuming the ground to be relatively flat, dis- 
regarding the influence of ground elevation, we 
found that in principal point triangulation sys- 
tematic errors of the scale transfer of the order 
of 0.15 promille were no exception; and also 
that errors of 0.30 promille could be expected. 
In nadir point triangulation these errors, how- 
ever, practically never were larger than 0.12 
promille. The systematic errors in the azimuth 
transfer are practically negligible in both cases. 
As Professor Roelofs did, we chose as com- 
parison the errors and scale and azimuth 
transfer in spatial air triangulation, caused by 
observational errors only. We found that for 
rightangle photography the mean square error 
in scale and azimuth transfer was about 0.12 
promille, so if we may assume that the errors 
in scale and azimuth transfer were caused by 
observational errors in radial triangulation to be 
equal to those in spatial triangulation — and it 
is Mr Van der Weele who can give an answer to 
this, and who will do so in his I T C-publication 
which will appear in the very near future — 
then indeed nadir point triangulation may 
replace the more expensive spatial triangulation. 
This, however, is not entirely true for principal 
point triangulation. 
Prof B. HALLERT: We have practised a 
method for radial triangulation founded on 
image co-ordinate methods only, and this has 
worked very well. We have quite a large ex- 
perience from it, and it seems particularly from 
the point of view of theory of errors that in this 
case we get very clear information about the 
propagation of from the image co- 
ordinates to the final results. Perhaps this could 
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