16 ANALYTICAL AERIAL TRIANGULATION, SCHUT 
of the two photographs and the angle between the two bases are not much more than 
one degree two iterations are sufficient to reach convergence, that is, to find the orien- 
tation as it follows from the given measurements with an accuracy of a few seconds of 
arc or better. Of course this requires that the computations be performed with sufficient 
accuracy. 
10. In the method developed by Schmid for the third triangulation procedure the con- 
dition equation B X X,; X,,,— 0 is differentiated with respect to the orientation ele- 
ments. The parameters in the matrix for initial approximate orientation of a photograph 
are rotations a4, a, and ag. However, the parameters for correction of the attitude are 
corrections to those rotations. This simulates the procedure on first order plotters. 
As a result the derivatives of the vectors X, (k—i, itl) are vectors i' X X,a,, 
j X Xa, and k' X X,as, in which i’, j’ and k' are unit vectors in the direction of the 
axes of rotation. Only the primary axis remains fixed parallel to a coordinate axis. 
Choosing e.g. a primary a,-axis, a secondary a,-axis and a tertiary a4-axis, we have 
il'-i 
j = +cos a} + sin a,k 
k’ = +sin agi — sin a; cos a,} + Cosa, COS ask 
and so: 
0X, 
BLUT —Z,j + Y,k 
01 
0X, LZ PE d 
So (+cos a, 2, — Sin a, Y,)i + sin a, X,} — cos a, Xk 
0 X, 
— (—sina, eos aZ; — cos a, cos asY;)i 
0 ag 2 K 2* Kk 
+ (cos a; cos a54X; — sin 052 ,,)] 
+ (sin agY, t sina, cos aS X,) k (kid irl 
Thus the derivatives to secondary and tertiary rotation are more complicated than the 
corresponding ones in paragraph 6. 
Computed corrections to the rotations can now be added to the approximate values. 
The orthogonal matrix is then computed from the new values. Matrix multiplication is 
thus avoided. This economy compensates only partly for ihe use of the more complicated 
linear equation. 
It follows that the method is perfectly sound but that it requires somewhat more 
computation than is required when any of the sets in paragraph 5 is chosen as para- 
meters in each iteration. 
An additional feature of the method is differentiation of the condition equation with 
respect to the elements of interior orientation which are treated as adjustable quantities. 
11. In the Ordnance Survey Method which uses the first triangulation procedure, the 
pre-determined solution is derived by simplification of the parallax equation. Of course, 
this solution can also be obtained by simplification of the equation B X X;: X;,, — 0. 
In each iteration of the orientation three matrix elements are chosen as parameters. 
These parameters are added to the corresponding elements of the matrix of approximate 
orientation and the sums are used as parameters for the computation of an improved 
orientation matrix. This is possible because the special 6-point method specifies approx- 
imately vertical photographs, and consequently all matrices concerned are fairly close to 
the unit matrix. 
This procedure eliminates the matrix multiplication after each iteration. That may 
be justified here because the method with its use of a pre-determined solution of the 
linear equations requires very little computation for each iteration. However, if not all