(p'xb).p" 0 the condition of coplanarity 
(p’ x b).p" = 0. (1.2.2) 
The vector product is equivalent to p’xb=p’TB, if 
0 —bs b2 
B = b3 0 -b1 , 
-b2 bi 0 
and by means of p = R x (1.2.2) converts to 
p''Bp' - x'TR'TBR'x" z x'TCx" 0. (1.2.3) 
It contains the matrix C of correlation as it is 
used in (Rinner 1963) and put into projective re- 
lationships by (Thompson 1968). A more detailed 
structure may be obtained from 
> 
1t " 
Cz |j iB | 1jk | = 
KT 
(ixi').b (3x1).b (k'x1').b 
z[|'(d'xj».b (jxjo.b- (kxj).5b |, (1.2.4) 
(i"xk).b  (j'xk.b (k'xk’).b 
which shows the connexion with the unit vectors of 
the two camera systems. 
C has two important properties (Thompson 1968): 
1. From 
CTxo' = 0 and C xo" = 0 1.2.5) 
(rank(C) = 2) result the coordinates xo of the 
epipoles. 
2. The (dualistic) transformations 
and n° = CC x’ (1.2.6) 
deliver the coefficients of the epipolar lines 
h'°.x'=0 and h'-x"=0, i.e. 
the geometric loci of homologous points. 
Due to the homogeneity of (1.2.3) only a matrix 
Z z (1/032)C  (z32=1) (1.2.7) 
can be calculated (Rinner 1963), where c3z is the 
probably biggest component, but it can be used in- 
stead of C without any limitation,since (1.2.5) is 
homogeneous too and the h of (1.2.6) contains coef- 
ficients of homogeneous equations,where common fac- 
tors do not have any influence. As for further con- 
siderations of this paper, the calculation of the 
  
1.3 co ruction of the model 
Regarding R - E , from the two formulas (1.1.1) of 
reconstruction results their difference 
ÀN'XN"- Au'xN! z— b |-YN |-»v? 
and the successive scalar multiplications by a 
vector ynT=(c,0,xn) yield therefrom because of 
xn(1).yn(i) = 0 the expressions 
b.yn" b.yn’ 
I 
AN and Aw" (1.3.1) 
XN! .yu" XN" .yu' 
for the stretching factors, depending only on the 
base and the image coordinates of the normal case. 
If rotational relative orientation is to be used, 
the base takes the form b" (1,0,0) and the 
formulas of (1.3.1) change by means of b.yw-c, 
XN' .yu"zCc(xu'-xu"), xu".yu'zc(xu"-xu?) to 
1 
AN = An’= An" ; (1.3.2) 
XN! -XN " 
that is the reciprocal of the x-parallax. Aw ap- 
proaches infinity, if xn’=xn", indicating parallel 
projectors, or in other words, images of points in 
infinity. 
Knowing An, from (1.1.3) arises the simple formula 
of reconstruction 
X = Xo + ANXN, (1.3.3) 
which delivers the coordinates of the model. The 
well-known effect of double determination from P’ 
and P" enables the check of calculation and from 
(1.3.3) results analogously to (1.1.6) the ex- 
pression 
/(X-Xo).(X-Xo) 
ÀN (1.3.4) 
A XN . XN 
as a final test of the reconstruction from the 
normal case. 
2. DETERMINATION OF THE PARAMETERS OF 
TRANSFORMAT ION 
2.1 Ihe rotational relative orientation 
This procedure is well-known from analog photo- 
grammetry and is executed in such a way that the 
base remains unchanged, that is b'z(1,0,0), the 
left image P’ is moved only by tip ¢’ and swing K', 
the right image P" by tilt Q', tip 9" and swing K'. 
Thus the movement of P" is to be described by the 
orientation matrix (Wolf 1974, p. 533) 
coordinates of the epipoles is 
of main interest. 
One restriction must be obeyed, COS COS K -cosósinK sinó 
which results from possible 1i- R'=| sin@sinécosÆcos@sink -sin@sindsink+cosQcosk -sinQcos® (2.1.1) 
nearities among the rows of the -cos$sinécosK*tsingsinK ^ cosQsinésinKtsingcosK  ^cosQcosó 
(8x8)-matrix for the determi- 
nation of the eight components Pr 
of Z. In order to avoid such singularities, in and the movement of P'(Q'z0) by 
space the points of correlation should not coincide , 
with planes passing three other points. Thus the cOSécOSK  -cosóésinK sine 
model should be clearly spatial and the points R’ = sink cosK 0 (2.1.2) 
well-distributed. -sinécosK ^ sinésinK  cosó 
702 
The correlation matrix (1.2.4) results now because 
of ba2-bs-0 in