190 
an arb 
from 
(1.4. 
ived fi) 
may 
comput 
also t 
r oject1 
3d by t 
he foe 
> defin 
Fig. 3: Optical rectification P—>Pn; the optical system is represented by 
its principal plane H 
the vanishing line, represented in P M by the 
equation 
1/Mo + (1/iii-1/no)uM 1 +(1/p2-1/Mo)uM 2 = om t um = 0, 
> syst 
: setti 
are x 
2=X3/x 
X’=Pf 
(1.5.| 
which results from the inverse form u=H* 1 um of 
(1.3.2) analogously. The parallelism denotes, that 
two coefficients of S already are known, that is 
Bi=ai , 82 = 02 in P , 
0mi=omi, 8m2=Qm2 in Pm. 
(1.5.4) 
In analogy to (1.5.2) the coefficients a can be 
transformed by 
result from 
cotv = h3/h2 , 
cotv’ = h 3 ’/h2’ = (-ho/f + hs)/h2 = -ho/h2f + cotv 
because of (1.5.3). From the equation of G ho=-h2Y- 
-h3Z follows and tnerefore the condition 
f cotv’ = Y + (2 + f)cotv 
between the inclinations and the focal length of 
the rectification lens. This form refers to an 
arbitrary point of G. At the optical axis where Y=0 
and Z=Zo, the condition reads 
QM T =a T M' 1 or a T =OM T M 
(1.5.5) f cotv’ = (Zo+f)cotv 
x -p,-i| As the two planes intersect in S, this relation 
at ion yi el ds using (1.5.4) the three linear equations 
(1.5.Î 
8m 0 , ÛM 1 , ÛM 2 
m~P0 P2-po 
P1 0 
0 U2 
£ Bo ,Q1 ,02 
]• 
and gives the inclination v’ of Pm when P is 
specified, and the required focal length, when P 
and Pm are predetermined. The eighth degree of 
freedom is already consumed by the condition of 
Scheimpflug. 
(1.5.Î from which the two unknowns result in 
2. Correlation between projective spaces 
:1on ar Bmo = 1 + 1/uo and Go = 1+po. 
By means of these parameters, S is fixed in both 
plane coordinate systems of image and map, which 
can be rotated and translated in their planes - 
oving s consuming six degrees of freedom - until the inter 
act section lines are identical and become parallel to 
e recti the X-axis. Now the determination of the remaining 
The setting element (inclination of one plane) may be 
performed in the normal section [Y,Z] as it is done 
usually in photogrammetric text books (e.g .Wolf, 
1974). There, the planes are represented by the 
lines 
entedIG: ho+h 2 Y+h 3 Z=0 , Gm: ho ’+h 2 ’Y’+ha ’Z’=0 
and the inclinations against the principal plane 
2.1 Definition of the concept 
Correlation in general is subsumed in the sto 
chastic concepts. In connection with projective 
geometry its meaning is a certain kind of trans 
formation between projective spaces Pr" and Ps n of 
equal dimension n, the so-called dual transfor 
mation, which converts points of one space into 
linear entities of the other one. Its general form 
is [Fuchs, 1988) 
(x R ) T C x 3 = 0 (2.1.1) 
wherein the x are homogeneous coordinates of homo 
logous points of Ps and Pr and C is the "matrix of 
correlation". If x s is the vector of a known point 
of Ps, the transformation h R = Cx s yields the coef 
ficients of the linear function (h R ) T x R =0, which 
represents in Pr - depending on the dimension - the