70 Commission III Invited paper 
Radial Triangulation 
by R. ROELOFS 
Delft (Netherlands). 
1. Introduction. 
All methods of Radial Triangulation (R.T.) have in common that, in contrast with 
Spatial Triangulation, they produce no heights but planimetry only. This means that the 
application of R.T. is restrieted to special cases, which nevertheless occur frequently: 
a. when only the planimetry and no heights have to be determined, as for instance, for 
topographic maps of flat or rather flat country or for cadastral and statistical maps, 
inasfar as the accuracy requirements can be fulfilled; 
b. when, for showing the shape of che terrain on the map, it is sufficient to use form 
lines — instead of countour lines — to be sketched from observation in a table stereo- 
Scope, as, for example, when mapping the results of photo-interpretation ; 
c. when the requirements for height-accuracy are so high that it is impossible or un- 
economical to determine heights photogrammetrically, in which case they have to be 
measured by terrestrial methods. Example: maps for irrigation or drainage projects. 
Obvious advantages of R.T. over Spatial Triangulation are: 
1. relatively inexpensive equipment; very evident in the case of mechanical R.T., but an 
important feature also in the case of numerical R.T.: 
no need to compensate for radial distortion; 
9. greater speed of operation, no relative orientations having to be carried out; 
less high requirements for experience of the observers; 
9. relatively simple computations in numerical R.T. as compared with numerical spatial 
triangulation. 
In this report a brief review will be given of recent investigations and applications, 
new developments and future possibilities in the field of numerical and mechanical R.T.; 
graphieal R.T. will not be taken into account, it being practically obsolete. 
2. Numerical Radial Triangulation. 
In its original form this type of R.T. consists in the measurement of directions in 
the plane of each photograph, from a certain point, the “radial point", and computing 
the chain of rhomboids thus produced, to obtain rectangular ground coordinates of al! 
chain-points. 
The Principal Point is mostly used as radial point, its location being very easy, as 
distinet from that of the Nadir Point, for whose location the tilt of the camera must be 
known. In the first case the ease of location is paid for by the occurrence of systematic 
errors in the directions measured and consequently in the rectangular coordinates com- 
puted. In the second case, after the measured directions have been corrected for tilt, they 
still are affected by the uncertainty of the tilt-determination, which results in a devia. 
tion of the rectangular coordinates of the chain-points. 
This question of errors in R.T. is as old as R.T. itself; numerous investigators have 
occupied themselves with this problem, not excepted the author of the present report. 
Most investigations are limited however to the study of directional errors, the ques- 
tion of the effect of these errors on the final result being left unanswered. In a paper 
presented at the Stockholm-congress 1956, [1], the author tried to give more eloquent 
evidence by studying the errors in scale-transfer and azimuth-transfer, these being the 
quantities which propagate through the whole strip of photographs. Quite recently the 
author decided to go a step further by investigating for the same examples the influence