191 
transition 
n=2 point —> straight line 
n=3 point —> plane 
n>3 point —> hyperplane. 
In an orthonormal vector space V 3 C can be derived 
easily from the well-known condition of coplanarity 
(f1g.4) of the position vectors yn, ys and the 
vector yo (base vector or epipolar axis) between 
the centers of projection Op and Os. Using (1.2.3) 
with Br*=R, Bs*=S the position vectors read yR=Rx R , 
ys=Sx s and the condition turns to 
(yn ,yo ,ys)=yR T 
0 
y0 3 
— yo 2 
-yos 
0 
yoi 
yo 2 
-yo 1 
0 
ys=XR T R T DoSxs 
0. 
Hence C results from the product 
C = R T DoS, (2.1.2) 
has the important attribute det(C)=0 because of 
det(Do)=0 and contains the components of the base 
vector in skew-symmetric form. Thus h R must equal 
the null vector if Os is projected and vice versa, 
because according to (1.2.3) XK 3 = -S _1 yo and there 
fore 
hx R =-CS~ 1 yo=-R T DoSS _1 yo=-R T yoxyo=0. 
Consequently the epipoles follow from the homo 
geneous systems 
C T R _1 yo= C t xk r =0 in Pr (2.1.3) 
CS _1 yo= Cxk 3 =0 in Ps, 
which can be solved because of the already men 
tioned attribute det(C)=0. By means of (2.1.3) 
moreover 
(h R ) t Xk r = (Cx 3 ) T x« R = (x 3 ) t C t xk r = (x s ) T 0=0 
follows and # therefore the important fact, that the 
coordinates of the epipole satisfy identically 
every straight line h R =Cx 3 or, in other words, 
every linear entity (h R ) T x R =0 contains the epipole 
xk r and represents therefore an epipolar line. In 
this way it is possible to calculate the matching 
epipolar line in Pr/Ps directly for an arbitrary 
point of Ps/Pr, i.e. the geometric locus of its 
homologous point. Along this line the point must be 
situated by non-geometric informations as the 
profile of gray levels and its derivatives. 
2.2 Dete.rnitnat.i on._and aggllQa.tlQn_.Qf-C 
C consists of nine unknown components Cjk. with 
regard to Its homogeneity, C can be normalized 
dividing 1t by one component, so that eight 
unknowns are left to be determined. The norming 
component is not allowed to vanish and, for prac 
tical computations, it should be the numerically 
largest one. 
Introducing now mensurable affine coordinates, the 
condition (2.1.1) changes, because of (1.2.5), that 
is x r -(1/o)Uur and x 3 =(1/o)Uus, to 
(Ui R ) T Cui 3 = 0 (2.2.1) 
with 
C = U T R T DoSU . (2.2.2) 
The procedure based on (2.2.1) requires coordinates 
of eight homologous points of Pr and Ps 1n oppo 
sition to conventional photogrammetry, where only 
five orientation points are needed. One row of the 
resulting 8x9-system reads (without i) 
C00+Ufl 1 Cl0+UR 2 C20+US 1 C01+UR 1 U3 1 Cl1+UR 2 US 1 C21+ 
+US 2 C02+UR 1 US 2 C12+UR 2 US 2 C22=0. 
In general it is not possible to predict the most 
stable component, but in the majority of practical 
photogrammetric applications with orthonormali zed 
coordinates the values of C12 or C21 will be larger 
then those of the other components (Rirmer, 1963). 
Setting Zjk=Cjk/c2i a linear system Az=a is ob 
tained, from which the normalized components follow 
by z=A"'a. C consists therefor of the calculated 
Zjk and Z2i=1. 
coordini 
elementi 
2.3 Reli 
The rei. 
model s 
structec 
row 
zoo 
20 1 
Zo 2 
Z10 
22 0 
Z 1 1 
Z1 2 
Z22 
-a 
0 
1 
0 
0 
0 
0 
0 
0 
0 
0 
1 
1 
1 
0 
1 
0 
1 
0 
0 
0 
2 
3 
1 
0 
1 
0 
1 
0 
0 
1 
0 
7 
1 
us 1 
us 2 
UR 1 
UR 2 
UR 1 US 1 
UR 1 US 2 
ur 2 us 2 
UR 2 US 1 
Tab. 1: Coefficients of the 8x8-system A for the 
computation of the normalized components 
Zjk=cjk/c2i. The absolute vector a con 
sists of the coefficients of -C2l. 
y = yOR 
which a 
orthonoi 
and wh< 
centers 
homogen« 
equivali 
y = irTi 
or else 
TRXR = 
Considering three of the homologous points as basic 
points G, the affine coordinates may be referred to 
the axis across them (fig.3).In this way three rows 
of A contain very simple coefficients (tab. 1) and 
yield the relations 
ZOO=0, ZOI=-Zl0-Z11, Z02=-Z20-Z22, 
arise. 
Tr yo R “ 
because 
yield tl 
over, tl 
whereby three unknowns* which indicate only 
relations between the affine systems, may be 
eliminated. The remaining five equations show the 
well-known fact, that the intersection of five 
corresponding rays of the two bundles of projection 
satisfy the relative orientation of the images Pr 
and Ps. 
In order to avoid singularities of A, the coor 
dinates of the correlation points do not dare to be 
linear!ly dependent. Thus they should not be points 
of planes passing three other points including the 
centers of projection. 
If C has the structure of (2.2.2),the first columns 
of the 3x3-matrices (2.1.3) ace because of u T = 
= (1,u\u 2 ) the absolute vectors and the solutions 
of these linear systems give directly the affi Re 
Tr yx = 
of any 
produce 
already 
Al 1 th< 
orienta 
determ1 
Their s 
the horn 
buted i 
hedron