150 RELATIVE ORIENTATION IN MOUNTAINOUS TERRAIN, VAN DER WEELE 
and: (b) for flat terrain: 
Z2 + X2 XY 
= 4 
AZ = 7 A+ a AB TCuX.4AC0 YS ADU RAE". . s (NA) 
in which: 4AA' — Ao" — Ag” 
AB’ = Ao! — Ao" 
1 
AC' — —2. 4g" + b (4b, — Ab,") -- Ao 
Z 
AD’ = Aw” + 40 + Ax! — Ax" 
| 
| 
Z | 
, | 
and: AE’ = ^ (4b,” — Ab,") + b. Ag" + Ab," + 20 | 
It may be seen from formula V.5, that, in the case of flat terrain, the only effects 
related to the relative orientation which can be separated from those of the elements of 
absolute orientation, are those of Ap and Aw. The well-known ¢-curvature, and w-twist, 
in the model, are the results of errors in these elements, and to correct for them we need 
to know the elevations of at least five terrain points, well distributed over the model. 
In the case of mountaincus terrain, the elements of absolute orientation are con- 
tained only in the pseudo-elements AC, AD and AG. If at least seven well-distributed 
points are known (and this knowledge should include all three dimensions of the model) 
we may expect, theoretically to be able to determine corrections to the relative orien- 
tation, for the elements 4g, /o,/x, and Ab,. 
It will be obvious that the determination of /b,, that is of the scale of the model, 
will, generally, be more accurate if we use the X and Y coordinates of two points having 
approximately the same altitude and, as this method of determination is usually em- 
ployed, the pseudo-element, AF, may be considered as known. In that case, six known 
elevations will be sufficient to correct for Ag, Aw and Ax. 
A numerical solution of this problem involves solving six equations. Unlike the case 
of numerical orientation based on y-parallaxes, it will not be possible, in this case, to 
use a general set of formulae, for a pre-requisite would be an assumption of the terrain 
form, with known elevations at predetermined positions. For this reason, a numerical 
solution cannot be considered as a practical possibility. It is not possible, either, to give 
an emperical solution in a general way, as this, also, depends on the position of the 
known points in the model. 
Even in flat terrain, one can hardly expect to improve the precision of a relative 
orientation, by applying corrections based on given elevations (see, for instance, W. 
Loscher, in Schweizerische Zeitschrift fiir Vermessungswesen, Kulturtechnik, und Photo- 
grammetrie, 1959, No. 8, p. 273). Since the above formulae show that, for mountainous 
terrain, the problem is even more complex, a real improvement of the relative orien- 
tation cannot be expected in this way. If a closer correspondence of heights, in the given 
points, is required, a practical solution seems to be that the model should be oriented in 
turn on different combinations of three points. A better solution does not seem, up to 
now, to exist. 
It should be noted that the remaining errors in elevation will not, generally, be 
caused be real errors in the relative orientation. They will, more probably, be due to er- 
rors in the photographs, caused by such factors as unflatness of glass plates, film defor- 
mation, lens distortion, etc., to errors inherent in the instrument, or, possibly to errors 
in the terrestrial determination of the heights. Errors in the reconstruction of the interior 
orientation have an influence which is proportional to the height-differences in the 
model and thus, they may be of primary importance in this respect.