on 
nd 
ANALYTICAL AERIAL TRIANGULATION, THOMPSON 7 
plications if the matrix R,,, has nine non-zero elements. Then for each of the points 
used we have to pre-multiply by (%;41 Yırı 1) and post-multiply by (x, y, 1)” which 
involves eight multiplications only. Hence, for a complete iteration on six points, we 
require sixty multiplications which is a saving of about 30% on any previous method. 
The advantage becomes greater as more points are used in the solution but, from what 
has been said above, these redundant points are best introduced as a final step in the 
process. 
5. The intermediate stereographic projection. 
In a recent paper (Thompson 1959c) the representation of space points by stereo- 
graphic projection in the complex plane was used to arrive at an interesting solution of 
the problem of absolute orientation. This raises the possibility of using such a represen- 
tation for relative orientation. This is not the place in which the problem can be set out 
in detail, but the main outline is worth considering, although, so far, no simplification is 
apparent over the more conventional attacks that Schut and I have been discussing. Let 
the points in the complex plane that represent two corresponding rays be z, and z, res- 
pectively; and let the base vector be represented by b, (its diametrically opposite direc- 
tion being represented by b,). The condition that the two rays should be coplanar with 
the base is expressed by saying that the cross-ratio of the four points z,, #9, b, and b, 
should be real. 
That is to say the expression 
  
1 (04 — 24) (bg — 23) 
hie (bae) 
must be real, the condition for which is that, 
C—C=0 
If we adopt the scheme discussed in paragraph 4 then the unknowns will be b, and the 
Cayley-Klein parameters that describe the rotation of the right hand photograph. If the 
representation of the observed right hand ray in the complex plane is w, then we will 
have, 
aW» d 6 
—jw, +a 
“a 
“ 
where a and f are the Cayley-Klein parameters. 
A small amount of detailed algebra will show that these equations do not lead to 
any direct solution of the problem as was possible in absolute orientation. An iterative 
proeess might be practieable, though insufficient work has been done on this to allow 
any conclusions to be drawn. An obvious saving is that an orthogonal matrix does not 
have to be constructed at each iteration. 
6. The iteration process. 
I am not entirely in agreement with Schut about the necessity of setting up linear 
equations with modified coefficients at each iteration. I do not quarrel with his argu- 
ments which are sound and lead to a second order process that resembles Newton's pro- 
cess in one variable: I only doubt its value in this context. 
If the linear equations can be set up once and for all and inverted, the resulting 
programme is simplified and this is, I think, not offset by slower convergence necessi- 
tating more iterations. Is not this exactly what a digital computer likes to do? Does it 
not like the repetition of simple routines? 
I do not think the evaluation of trial values should be over-simplified by supposing 
that the points fall in standard positions on the model when they do not; and this has